stochastic signals and processes lec. 8 samuel schmidt 25 ... · lec. 8 samuel schmidt 25-10-2011...
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Stochastic signals and processes Lec 8
Samuel Schmidt 25-10-2011
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
How to filter a signal Y(t) so it became as similar to a desired signal S(t) as possible
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
bull Noise contaminated signal
bull Estimate the linear filter L[middot] so Ŝ(t) become as close to S(t) as possible
Y(t)=S(t)+N(t)
Ŝ(t)=L[Y(t)]
Wiener filter
0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
Y(t
)
Y(t)
0 1 2 3 4 5-5
0
5
Time (s)
Se(t
)
Se(t)
0 1 2 3 4 5-5
0
5
Time (s)
e(t
)
e(t)
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Y(t)S(t) and N(t) are zero mean stationary processes
bull L[middot] is a linear filter
bull FIR filters (M-order)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Or like
119884 119899 = 120596119896119883(119899 minus 119896)
119872minus1
119896=0
bull In linear system the order of multiplication and addition is irrelevant
bull Defined by super position
][][][][ 2121 nxTbnxTanxbnxaT
bull Linear algebra Two vectors is orthogonal if the dot product is zero
bull That corresponds zero correlation between two signals
0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
N(t
)N(t)
-2 -15 -1 -05 0 05 1 15 2-4
-3
-2
-1
0
1
2
3
4
S
N
Scatter between S(t) and N(t)
-5 -4 -3 -2 -1 0 1 2 3 4 5-4
-3
-2
-1
0
1
2
3
4Scatter between Y(t) and N(t)
N(t
)
X(t)
Orthogonal (r=00) Non-orthogonal (r=069)
S(t) vs N(t) Y(t) vs N(t)
bull If two process are orthogonal for all delays in a give range the process are also orthogonal after linear transformation
IF
Then
bull Error ε(t)= Ŝ(t) - S(t)
bull Mean square error
E[|ε(t)|2]= E[| Ŝ(t) - S(t) |2]
E[|ε(t)|2]=E[| L[Y(ξ)] - S(t) |2]
E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]
0 1 2 3 4 5-5
0
5
Time (s)
Se(t
)
Se(t)
- 0 1 2 3 4 5
-5
0
5
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
e(t
)
e(t)=
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull The Minimum mean-square error is obtained when the error ε(t) is Orthogonal to the unfiltered signal Y(ξ) where ξ is in the range [ti tf]
E[Y(ξ) e(t)]=E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
bull That means thats the
filter is optimal when the
Correlation between Y(t+k)
and e(t) is low for all k
-5 -4 -3 -2 -1 0 1 2 3 4 5-08
-06
-04
-02
0
02
04
06
08
1
Y(t)
e(t
)
bull The mean-square error is mean when the correlation between the e(t) and the signal is zero that is the optimal filter
0 05 1 15 2 25 3 35 4 45 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Time (s)
Y(t) and e(t)
Y(t)
e(t)
The Minimum mean-square error can be estimated
Since E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
Is E[(S(t)- L[Ŝ(t)] ) L[Y(ξ)] ]=0
Thereby is the Minimum mean-square
em=E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]= E[(S(t)-L[Y(ξ)]) S(t)]
em=E[(S(t)-L[Y(ξ)]) S(t)]
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
How to filter a signal Y(t) so it became as similar to a desired signal S(t) as possible
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
bull Noise contaminated signal
bull Estimate the linear filter L[middot] so Ŝ(t) become as close to S(t) as possible
Y(t)=S(t)+N(t)
Ŝ(t)=L[Y(t)]
Wiener filter
0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
Y(t
)
Y(t)
0 1 2 3 4 5-5
0
5
Time (s)
Se(t
)
Se(t)
0 1 2 3 4 5-5
0
5
Time (s)
e(t
)
e(t)
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Y(t)S(t) and N(t) are zero mean stationary processes
bull L[middot] is a linear filter
bull FIR filters (M-order)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Or like
119884 119899 = 120596119896119883(119899 minus 119896)
119872minus1
119896=0
bull In linear system the order of multiplication and addition is irrelevant
bull Defined by super position
][][][][ 2121 nxTbnxTanxbnxaT
bull Linear algebra Two vectors is orthogonal if the dot product is zero
bull That corresponds zero correlation between two signals
0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
N(t
)N(t)
-2 -15 -1 -05 0 05 1 15 2-4
-3
-2
-1
0
1
2
3
4
S
N
Scatter between S(t) and N(t)
-5 -4 -3 -2 -1 0 1 2 3 4 5-4
-3
-2
-1
0
1
2
3
4Scatter between Y(t) and N(t)
N(t
)
X(t)
Orthogonal (r=00) Non-orthogonal (r=069)
S(t) vs N(t) Y(t) vs N(t)
bull If two process are orthogonal for all delays in a give range the process are also orthogonal after linear transformation
IF
Then
bull Error ε(t)= Ŝ(t) - S(t)
bull Mean square error
E[|ε(t)|2]= E[| Ŝ(t) - S(t) |2]
E[|ε(t)|2]=E[| L[Y(ξ)] - S(t) |2]
E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]
0 1 2 3 4 5-5
0
5
Time (s)
Se(t
)
Se(t)
- 0 1 2 3 4 5
-5
0
5
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
e(t
)
e(t)=
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull The Minimum mean-square error is obtained when the error ε(t) is Orthogonal to the unfiltered signal Y(ξ) where ξ is in the range [ti tf]
E[Y(ξ) e(t)]=E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
bull That means thats the
filter is optimal when the
Correlation between Y(t+k)
and e(t) is low for all k
-5 -4 -3 -2 -1 0 1 2 3 4 5-08
-06
-04
-02
0
02
04
06
08
1
Y(t)
e(t
)
bull The mean-square error is mean when the correlation between the e(t) and the signal is zero that is the optimal filter
0 05 1 15 2 25 3 35 4 45 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Time (s)
Y(t) and e(t)
Y(t)
e(t)
The Minimum mean-square error can be estimated
Since E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
Is E[(S(t)- L[Ŝ(t)] ) L[Y(ξ)] ]=0
Thereby is the Minimum mean-square
em=E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]= E[(S(t)-L[Y(ξ)]) S(t)]
em=E[(S(t)-L[Y(ξ)]) S(t)]
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
How to filter a signal Y(t) so it became as similar to a desired signal S(t) as possible
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
bull Noise contaminated signal
bull Estimate the linear filter L[middot] so Ŝ(t) become as close to S(t) as possible
Y(t)=S(t)+N(t)
Ŝ(t)=L[Y(t)]
Wiener filter
0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
Y(t
)
Y(t)
0 1 2 3 4 5-5
0
5
Time (s)
Se(t
)
Se(t)
0 1 2 3 4 5-5
0
5
Time (s)
e(t
)
e(t)
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Y(t)S(t) and N(t) are zero mean stationary processes
bull L[middot] is a linear filter
bull FIR filters (M-order)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Or like
119884 119899 = 120596119896119883(119899 minus 119896)
119872minus1
119896=0
bull In linear system the order of multiplication and addition is irrelevant
bull Defined by super position
][][][][ 2121 nxTbnxTanxbnxaT
bull Linear algebra Two vectors is orthogonal if the dot product is zero
bull That corresponds zero correlation between two signals
0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
N(t
)N(t)
-2 -15 -1 -05 0 05 1 15 2-4
-3
-2
-1
0
1
2
3
4
S
N
Scatter between S(t) and N(t)
-5 -4 -3 -2 -1 0 1 2 3 4 5-4
-3
-2
-1
0
1
2
3
4Scatter between Y(t) and N(t)
N(t
)
X(t)
Orthogonal (r=00) Non-orthogonal (r=069)
S(t) vs N(t) Y(t) vs N(t)
bull If two process are orthogonal for all delays in a give range the process are also orthogonal after linear transformation
IF
Then
bull Error ε(t)= Ŝ(t) - S(t)
bull Mean square error
E[|ε(t)|2]= E[| Ŝ(t) - S(t) |2]
E[|ε(t)|2]=E[| L[Y(ξ)] - S(t) |2]
E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]
0 1 2 3 4 5-5
0
5
Time (s)
Se(t
)
Se(t)
- 0 1 2 3 4 5
-5
0
5
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
e(t
)
e(t)=
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull The Minimum mean-square error is obtained when the error ε(t) is Orthogonal to the unfiltered signal Y(ξ) where ξ is in the range [ti tf]
E[Y(ξ) e(t)]=E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
bull That means thats the
filter is optimal when the
Correlation between Y(t+k)
and e(t) is low for all k
-5 -4 -3 -2 -1 0 1 2 3 4 5-08
-06
-04
-02
0
02
04
06
08
1
Y(t)
e(t
)
bull The mean-square error is mean when the correlation between the e(t) and the signal is zero that is the optimal filter
0 05 1 15 2 25 3 35 4 45 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Time (s)
Y(t) and e(t)
Y(t)
e(t)
The Minimum mean-square error can be estimated
Since E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
Is E[(S(t)- L[Ŝ(t)] ) L[Y(ξ)] ]=0
Thereby is the Minimum mean-square
em=E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]= E[(S(t)-L[Y(ξ)]) S(t)]
em=E[(S(t)-L[Y(ξ)]) S(t)]
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull Noise contaminated signal
bull Estimate the linear filter L[middot] so Ŝ(t) become as close to S(t) as possible
Y(t)=S(t)+N(t)
Ŝ(t)=L[Y(t)]
Wiener filter
0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
Y(t
)
Y(t)
0 1 2 3 4 5-5
0
5
Time (s)
Se(t
)
Se(t)
0 1 2 3 4 5-5
0
5
Time (s)
e(t
)
e(t)
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Y(t)S(t) and N(t) are zero mean stationary processes
bull L[middot] is a linear filter
bull FIR filters (M-order)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Or like
119884 119899 = 120596119896119883(119899 minus 119896)
119872minus1
119896=0
bull In linear system the order of multiplication and addition is irrelevant
bull Defined by super position
][][][][ 2121 nxTbnxTanxbnxaT
bull Linear algebra Two vectors is orthogonal if the dot product is zero
bull That corresponds zero correlation between two signals
0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
N(t
)N(t)
-2 -15 -1 -05 0 05 1 15 2-4
-3
-2
-1
0
1
2
3
4
S
N
Scatter between S(t) and N(t)
-5 -4 -3 -2 -1 0 1 2 3 4 5-4
-3
-2
-1
0
1
2
3
4Scatter between Y(t) and N(t)
N(t
)
X(t)
Orthogonal (r=00) Non-orthogonal (r=069)
S(t) vs N(t) Y(t) vs N(t)
bull If two process are orthogonal for all delays in a give range the process are also orthogonal after linear transformation
IF
Then
bull Error ε(t)= Ŝ(t) - S(t)
bull Mean square error
E[|ε(t)|2]= E[| Ŝ(t) - S(t) |2]
E[|ε(t)|2]=E[| L[Y(ξ)] - S(t) |2]
E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]
0 1 2 3 4 5-5
0
5
Time (s)
Se(t
)
Se(t)
- 0 1 2 3 4 5
-5
0
5
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
e(t
)
e(t)=
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull The Minimum mean-square error is obtained when the error ε(t) is Orthogonal to the unfiltered signal Y(ξ) where ξ is in the range [ti tf]
E[Y(ξ) e(t)]=E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
bull That means thats the
filter is optimal when the
Correlation between Y(t+k)
and e(t) is low for all k
-5 -4 -3 -2 -1 0 1 2 3 4 5-08
-06
-04
-02
0
02
04
06
08
1
Y(t)
e(t
)
bull The mean-square error is mean when the correlation between the e(t) and the signal is zero that is the optimal filter
0 05 1 15 2 25 3 35 4 45 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Time (s)
Y(t) and e(t)
Y(t)
e(t)
The Minimum mean-square error can be estimated
Since E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
Is E[(S(t)- L[Ŝ(t)] ) L[Y(ξ)] ]=0
Thereby is the Minimum mean-square
em=E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]= E[(S(t)-L[Y(ξ)]) S(t)]
em=E[(S(t)-L[Y(ξ)]) S(t)]
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
Y(t
)
Y(t)
0 1 2 3 4 5-5
0
5
Time (s)
Se(t
)
Se(t)
0 1 2 3 4 5-5
0
5
Time (s)
e(t
)
e(t)
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Y(t)S(t) and N(t) are zero mean stationary processes
bull L[middot] is a linear filter
bull FIR filters (M-order)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Or like
119884 119899 = 120596119896119883(119899 minus 119896)
119872minus1
119896=0
bull In linear system the order of multiplication and addition is irrelevant
bull Defined by super position
][][][][ 2121 nxTbnxTanxbnxaT
bull Linear algebra Two vectors is orthogonal if the dot product is zero
bull That corresponds zero correlation between two signals
0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
N(t
)N(t)
-2 -15 -1 -05 0 05 1 15 2-4
-3
-2
-1
0
1
2
3
4
S
N
Scatter between S(t) and N(t)
-5 -4 -3 -2 -1 0 1 2 3 4 5-4
-3
-2
-1
0
1
2
3
4Scatter between Y(t) and N(t)
N(t
)
X(t)
Orthogonal (r=00) Non-orthogonal (r=069)
S(t) vs N(t) Y(t) vs N(t)
bull If two process are orthogonal for all delays in a give range the process are also orthogonal after linear transformation
IF
Then
bull Error ε(t)= Ŝ(t) - S(t)
bull Mean square error
E[|ε(t)|2]= E[| Ŝ(t) - S(t) |2]
E[|ε(t)|2]=E[| L[Y(ξ)] - S(t) |2]
E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]
0 1 2 3 4 5-5
0
5
Time (s)
Se(t
)
Se(t)
- 0 1 2 3 4 5
-5
0
5
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
e(t
)
e(t)=
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull The Minimum mean-square error is obtained when the error ε(t) is Orthogonal to the unfiltered signal Y(ξ) where ξ is in the range [ti tf]
E[Y(ξ) e(t)]=E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
bull That means thats the
filter is optimal when the
Correlation between Y(t+k)
and e(t) is low for all k
-5 -4 -3 -2 -1 0 1 2 3 4 5-08
-06
-04
-02
0
02
04
06
08
1
Y(t)
e(t
)
bull The mean-square error is mean when the correlation between the e(t) and the signal is zero that is the optimal filter
0 05 1 15 2 25 3 35 4 45 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Time (s)
Y(t) and e(t)
Y(t)
e(t)
The Minimum mean-square error can be estimated
Since E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
Is E[(S(t)- L[Ŝ(t)] ) L[Y(ξ)] ]=0
Thereby is the Minimum mean-square
em=E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]= E[(S(t)-L[Y(ξ)]) S(t)]
em=E[(S(t)-L[Y(ξ)]) S(t)]
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Y(t)S(t) and N(t) are zero mean stationary processes
bull L[middot] is a linear filter
bull FIR filters (M-order)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Or like
119884 119899 = 120596119896119883(119899 minus 119896)
119872minus1
119896=0
bull In linear system the order of multiplication and addition is irrelevant
bull Defined by super position
][][][][ 2121 nxTbnxTanxbnxaT
bull Linear algebra Two vectors is orthogonal if the dot product is zero
bull That corresponds zero correlation between two signals
0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
N(t
)N(t)
-2 -15 -1 -05 0 05 1 15 2-4
-3
-2
-1
0
1
2
3
4
S
N
Scatter between S(t) and N(t)
-5 -4 -3 -2 -1 0 1 2 3 4 5-4
-3
-2
-1
0
1
2
3
4Scatter between Y(t) and N(t)
N(t
)
X(t)
Orthogonal (r=00) Non-orthogonal (r=069)
S(t) vs N(t) Y(t) vs N(t)
bull If two process are orthogonal for all delays in a give range the process are also orthogonal after linear transformation
IF
Then
bull Error ε(t)= Ŝ(t) - S(t)
bull Mean square error
E[|ε(t)|2]= E[| Ŝ(t) - S(t) |2]
E[|ε(t)|2]=E[| L[Y(ξ)] - S(t) |2]
E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]
0 1 2 3 4 5-5
0
5
Time (s)
Se(t
)
Se(t)
- 0 1 2 3 4 5
-5
0
5
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
e(t
)
e(t)=
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull The Minimum mean-square error is obtained when the error ε(t) is Orthogonal to the unfiltered signal Y(ξ) where ξ is in the range [ti tf]
E[Y(ξ) e(t)]=E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
bull That means thats the
filter is optimal when the
Correlation between Y(t+k)
and e(t) is low for all k
-5 -4 -3 -2 -1 0 1 2 3 4 5-08
-06
-04
-02
0
02
04
06
08
1
Y(t)
e(t
)
bull The mean-square error is mean when the correlation between the e(t) and the signal is zero that is the optimal filter
0 05 1 15 2 25 3 35 4 45 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Time (s)
Y(t) and e(t)
Y(t)
e(t)
The Minimum mean-square error can be estimated
Since E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
Is E[(S(t)- L[Ŝ(t)] ) L[Y(ξ)] ]=0
Thereby is the Minimum mean-square
em=E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]= E[(S(t)-L[Y(ξ)]) S(t)]
em=E[(S(t)-L[Y(ξ)]) S(t)]
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull Y(t)S(t) and N(t) are zero mean stationary processes
bull L[middot] is a linear filter
bull FIR filters (M-order)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Or like
119884 119899 = 120596119896119883(119899 minus 119896)
119872minus1
119896=0
bull In linear system the order of multiplication and addition is irrelevant
bull Defined by super position
][][][][ 2121 nxTbnxTanxbnxaT
bull Linear algebra Two vectors is orthogonal if the dot product is zero
bull That corresponds zero correlation between two signals
0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
N(t
)N(t)
-2 -15 -1 -05 0 05 1 15 2-4
-3
-2
-1
0
1
2
3
4
S
N
Scatter between S(t) and N(t)
-5 -4 -3 -2 -1 0 1 2 3 4 5-4
-3
-2
-1
0
1
2
3
4Scatter between Y(t) and N(t)
N(t
)
X(t)
Orthogonal (r=00) Non-orthogonal (r=069)
S(t) vs N(t) Y(t) vs N(t)
bull If two process are orthogonal for all delays in a give range the process are also orthogonal after linear transformation
IF
Then
bull Error ε(t)= Ŝ(t) - S(t)
bull Mean square error
E[|ε(t)|2]= E[| Ŝ(t) - S(t) |2]
E[|ε(t)|2]=E[| L[Y(ξ)] - S(t) |2]
E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]
0 1 2 3 4 5-5
0
5
Time (s)
Se(t
)
Se(t)
- 0 1 2 3 4 5
-5
0
5
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
e(t
)
e(t)=
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull The Minimum mean-square error is obtained when the error ε(t) is Orthogonal to the unfiltered signal Y(ξ) where ξ is in the range [ti tf]
E[Y(ξ) e(t)]=E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
bull That means thats the
filter is optimal when the
Correlation between Y(t+k)
and e(t) is low for all k
-5 -4 -3 -2 -1 0 1 2 3 4 5-08
-06
-04
-02
0
02
04
06
08
1
Y(t)
e(t
)
bull The mean-square error is mean when the correlation between the e(t) and the signal is zero that is the optimal filter
0 05 1 15 2 25 3 35 4 45 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Time (s)
Y(t) and e(t)
Y(t)
e(t)
The Minimum mean-square error can be estimated
Since E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
Is E[(S(t)- L[Ŝ(t)] ) L[Y(ξ)] ]=0
Thereby is the Minimum mean-square
em=E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]= E[(S(t)-L[Y(ξ)]) S(t)]
em=E[(S(t)-L[Y(ξ)]) S(t)]
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull FIR filters (M-order)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Or like
119884 119899 = 120596119896119883(119899 minus 119896)
119872minus1
119896=0
bull In linear system the order of multiplication and addition is irrelevant
bull Defined by super position
][][][][ 2121 nxTbnxTanxbnxaT
bull Linear algebra Two vectors is orthogonal if the dot product is zero
bull That corresponds zero correlation between two signals
0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
N(t
)N(t)
-2 -15 -1 -05 0 05 1 15 2-4
-3
-2
-1
0
1
2
3
4
S
N
Scatter between S(t) and N(t)
-5 -4 -3 -2 -1 0 1 2 3 4 5-4
-3
-2
-1
0
1
2
3
4Scatter between Y(t) and N(t)
N(t
)
X(t)
Orthogonal (r=00) Non-orthogonal (r=069)
S(t) vs N(t) Y(t) vs N(t)
bull If two process are orthogonal for all delays in a give range the process are also orthogonal after linear transformation
IF
Then
bull Error ε(t)= Ŝ(t) - S(t)
bull Mean square error
E[|ε(t)|2]= E[| Ŝ(t) - S(t) |2]
E[|ε(t)|2]=E[| L[Y(ξ)] - S(t) |2]
E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]
0 1 2 3 4 5-5
0
5
Time (s)
Se(t
)
Se(t)
- 0 1 2 3 4 5
-5
0
5
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
e(t
)
e(t)=
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull The Minimum mean-square error is obtained when the error ε(t) is Orthogonal to the unfiltered signal Y(ξ) where ξ is in the range [ti tf]
E[Y(ξ) e(t)]=E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
bull That means thats the
filter is optimal when the
Correlation between Y(t+k)
and e(t) is low for all k
-5 -4 -3 -2 -1 0 1 2 3 4 5-08
-06
-04
-02
0
02
04
06
08
1
Y(t)
e(t
)
bull The mean-square error is mean when the correlation between the e(t) and the signal is zero that is the optimal filter
0 05 1 15 2 25 3 35 4 45 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Time (s)
Y(t) and e(t)
Y(t)
e(t)
The Minimum mean-square error can be estimated
Since E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
Is E[(S(t)- L[Ŝ(t)] ) L[Y(ξ)] ]=0
Thereby is the Minimum mean-square
em=E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]= E[(S(t)-L[Y(ξ)]) S(t)]
em=E[(S(t)-L[Y(ξ)]) S(t)]
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull In linear system the order of multiplication and addition is irrelevant
bull Defined by super position
][][][][ 2121 nxTbnxTanxbnxaT
bull Linear algebra Two vectors is orthogonal if the dot product is zero
bull That corresponds zero correlation between two signals
0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
N(t
)N(t)
-2 -15 -1 -05 0 05 1 15 2-4
-3
-2
-1
0
1
2
3
4
S
N
Scatter between S(t) and N(t)
-5 -4 -3 -2 -1 0 1 2 3 4 5-4
-3
-2
-1
0
1
2
3
4Scatter between Y(t) and N(t)
N(t
)
X(t)
Orthogonal (r=00) Non-orthogonal (r=069)
S(t) vs N(t) Y(t) vs N(t)
bull If two process are orthogonal for all delays in a give range the process are also orthogonal after linear transformation
IF
Then
bull Error ε(t)= Ŝ(t) - S(t)
bull Mean square error
E[|ε(t)|2]= E[| Ŝ(t) - S(t) |2]
E[|ε(t)|2]=E[| L[Y(ξ)] - S(t) |2]
E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]
0 1 2 3 4 5-5
0
5
Time (s)
Se(t
)
Se(t)
- 0 1 2 3 4 5
-5
0
5
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
e(t
)
e(t)=
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull The Minimum mean-square error is obtained when the error ε(t) is Orthogonal to the unfiltered signal Y(ξ) where ξ is in the range [ti tf]
E[Y(ξ) e(t)]=E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
bull That means thats the
filter is optimal when the
Correlation between Y(t+k)
and e(t) is low for all k
-5 -4 -3 -2 -1 0 1 2 3 4 5-08
-06
-04
-02
0
02
04
06
08
1
Y(t)
e(t
)
bull The mean-square error is mean when the correlation between the e(t) and the signal is zero that is the optimal filter
0 05 1 15 2 25 3 35 4 45 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Time (s)
Y(t) and e(t)
Y(t)
e(t)
The Minimum mean-square error can be estimated
Since E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
Is E[(S(t)- L[Ŝ(t)] ) L[Y(ξ)] ]=0
Thereby is the Minimum mean-square
em=E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]= E[(S(t)-L[Y(ξ)]) S(t)]
em=E[(S(t)-L[Y(ξ)]) S(t)]
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull Linear algebra Two vectors is orthogonal if the dot product is zero
bull That corresponds zero correlation between two signals
0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
N(t
)N(t)
-2 -15 -1 -05 0 05 1 15 2-4
-3
-2
-1
0
1
2
3
4
S
N
Scatter between S(t) and N(t)
-5 -4 -3 -2 -1 0 1 2 3 4 5-4
-3
-2
-1
0
1
2
3
4Scatter between Y(t) and N(t)
N(t
)
X(t)
Orthogonal (r=00) Non-orthogonal (r=069)
S(t) vs N(t) Y(t) vs N(t)
bull If two process are orthogonal for all delays in a give range the process are also orthogonal after linear transformation
IF
Then
bull Error ε(t)= Ŝ(t) - S(t)
bull Mean square error
E[|ε(t)|2]= E[| Ŝ(t) - S(t) |2]
E[|ε(t)|2]=E[| L[Y(ξ)] - S(t) |2]
E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]
0 1 2 3 4 5-5
0
5
Time (s)
Se(t
)
Se(t)
- 0 1 2 3 4 5
-5
0
5
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
e(t
)
e(t)=
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull The Minimum mean-square error is obtained when the error ε(t) is Orthogonal to the unfiltered signal Y(ξ) where ξ is in the range [ti tf]
E[Y(ξ) e(t)]=E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
bull That means thats the
filter is optimal when the
Correlation between Y(t+k)
and e(t) is low for all k
-5 -4 -3 -2 -1 0 1 2 3 4 5-08
-06
-04
-02
0
02
04
06
08
1
Y(t)
e(t
)
bull The mean-square error is mean when the correlation between the e(t) and the signal is zero that is the optimal filter
0 05 1 15 2 25 3 35 4 45 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Time (s)
Y(t) and e(t)
Y(t)
e(t)
The Minimum mean-square error can be estimated
Since E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
Is E[(S(t)- L[Ŝ(t)] ) L[Y(ξ)] ]=0
Thereby is the Minimum mean-square
em=E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]= E[(S(t)-L[Y(ξ)]) S(t)]
em=E[(S(t)-L[Y(ξ)]) S(t)]
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
-2 -15 -1 -05 0 05 1 15 2-4
-3
-2
-1
0
1
2
3
4
S
N
Scatter between S(t) and N(t)
-5 -4 -3 -2 -1 0 1 2 3 4 5-4
-3
-2
-1
0
1
2
3
4Scatter between Y(t) and N(t)
N(t
)
X(t)
Orthogonal (r=00) Non-orthogonal (r=069)
S(t) vs N(t) Y(t) vs N(t)
bull If two process are orthogonal for all delays in a give range the process are also orthogonal after linear transformation
IF
Then
bull Error ε(t)= Ŝ(t) - S(t)
bull Mean square error
E[|ε(t)|2]= E[| Ŝ(t) - S(t) |2]
E[|ε(t)|2]=E[| L[Y(ξ)] - S(t) |2]
E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]
0 1 2 3 4 5-5
0
5
Time (s)
Se(t
)
Se(t)
- 0 1 2 3 4 5
-5
0
5
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
e(t
)
e(t)=
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull The Minimum mean-square error is obtained when the error ε(t) is Orthogonal to the unfiltered signal Y(ξ) where ξ is in the range [ti tf]
E[Y(ξ) e(t)]=E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
bull That means thats the
filter is optimal when the
Correlation between Y(t+k)
and e(t) is low for all k
-5 -4 -3 -2 -1 0 1 2 3 4 5-08
-06
-04
-02
0
02
04
06
08
1
Y(t)
e(t
)
bull The mean-square error is mean when the correlation between the e(t) and the signal is zero that is the optimal filter
0 05 1 15 2 25 3 35 4 45 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Time (s)
Y(t) and e(t)
Y(t)
e(t)
The Minimum mean-square error can be estimated
Since E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
Is E[(S(t)- L[Ŝ(t)] ) L[Y(ξ)] ]=0
Thereby is the Minimum mean-square
em=E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]= E[(S(t)-L[Y(ξ)]) S(t)]
em=E[(S(t)-L[Y(ξ)]) S(t)]
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull If two process are orthogonal for all delays in a give range the process are also orthogonal after linear transformation
IF
Then
bull Error ε(t)= Ŝ(t) - S(t)
bull Mean square error
E[|ε(t)|2]= E[| Ŝ(t) - S(t) |2]
E[|ε(t)|2]=E[| L[Y(ξ)] - S(t) |2]
E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]
0 1 2 3 4 5-5
0
5
Time (s)
Se(t
)
Se(t)
- 0 1 2 3 4 5
-5
0
5
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
e(t
)
e(t)=
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull The Minimum mean-square error is obtained when the error ε(t) is Orthogonal to the unfiltered signal Y(ξ) where ξ is in the range [ti tf]
E[Y(ξ) e(t)]=E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
bull That means thats the
filter is optimal when the
Correlation between Y(t+k)
and e(t) is low for all k
-5 -4 -3 -2 -1 0 1 2 3 4 5-08
-06
-04
-02
0
02
04
06
08
1
Y(t)
e(t
)
bull The mean-square error is mean when the correlation between the e(t) and the signal is zero that is the optimal filter
0 05 1 15 2 25 3 35 4 45 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Time (s)
Y(t) and e(t)
Y(t)
e(t)
The Minimum mean-square error can be estimated
Since E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
Is E[(S(t)- L[Ŝ(t)] ) L[Y(ξ)] ]=0
Thereby is the Minimum mean-square
em=E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]= E[(S(t)-L[Y(ξ)]) S(t)]
em=E[(S(t)-L[Y(ξ)]) S(t)]
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull Error ε(t)= Ŝ(t) - S(t)
bull Mean square error
E[|ε(t)|2]= E[| Ŝ(t) - S(t) |2]
E[|ε(t)|2]=E[| L[Y(ξ)] - S(t) |2]
E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]
0 1 2 3 4 5-5
0
5
Time (s)
Se(t
)
Se(t)
- 0 1 2 3 4 5
-5
0
5
Time (s)
S(t
)
S(t)
0 1 2 3 4 5-5
0
5
Time (s)
e(t
)
e(t)=
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull The Minimum mean-square error is obtained when the error ε(t) is Orthogonal to the unfiltered signal Y(ξ) where ξ is in the range [ti tf]
E[Y(ξ) e(t)]=E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
bull That means thats the
filter is optimal when the
Correlation between Y(t+k)
and e(t) is low for all k
-5 -4 -3 -2 -1 0 1 2 3 4 5-08
-06
-04
-02
0
02
04
06
08
1
Y(t)
e(t
)
bull The mean-square error is mean when the correlation between the e(t) and the signal is zero that is the optimal filter
0 05 1 15 2 25 3 35 4 45 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Time (s)
Y(t) and e(t)
Y(t)
e(t)
The Minimum mean-square error can be estimated
Since E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
Is E[(S(t)- L[Ŝ(t)] ) L[Y(ξ)] ]=0
Thereby is the Minimum mean-square
em=E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]= E[(S(t)-L[Y(ξ)]) S(t)]
em=E[(S(t)-L[Y(ξ)]) S(t)]
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull The Minimum mean-square error is obtained when the error ε(t) is Orthogonal to the unfiltered signal Y(ξ) where ξ is in the range [ti tf]
E[Y(ξ) e(t)]=E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
bull That means thats the
filter is optimal when the
Correlation between Y(t+k)
and e(t) is low for all k
-5 -4 -3 -2 -1 0 1 2 3 4 5-08
-06
-04
-02
0
02
04
06
08
1
Y(t)
e(t
)
bull The mean-square error is mean when the correlation between the e(t) and the signal is zero that is the optimal filter
0 05 1 15 2 25 3 35 4 45 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Time (s)
Y(t) and e(t)
Y(t)
e(t)
The Minimum mean-square error can be estimated
Since E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
Is E[(S(t)- L[Ŝ(t)] ) L[Y(ξ)] ]=0
Thereby is the Minimum mean-square
em=E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]= E[(S(t)-L[Y(ξ)]) S(t)]
em=E[(S(t)-L[Y(ξ)]) S(t)]
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull The Minimum mean-square error is obtained when the error ε(t) is Orthogonal to the unfiltered signal Y(ξ) where ξ is in the range [ti tf]
E[Y(ξ) e(t)]=E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
bull That means thats the
filter is optimal when the
Correlation between Y(t+k)
and e(t) is low for all k
-5 -4 -3 -2 -1 0 1 2 3 4 5-08
-06
-04
-02
0
02
04
06
08
1
Y(t)
e(t
)
bull The mean-square error is mean when the correlation between the e(t) and the signal is zero that is the optimal filter
0 05 1 15 2 25 3 35 4 45 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Time (s)
Y(t) and e(t)
Y(t)
e(t)
The Minimum mean-square error can be estimated
Since E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
Is E[(S(t)- L[Ŝ(t)] ) L[Y(ξ)] ]=0
Thereby is the Minimum mean-square
em=E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]= E[(S(t)-L[Y(ξ)]) S(t)]
em=E[(S(t)-L[Y(ξ)]) S(t)]
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull The mean-square error is mean when the correlation between the e(t) and the signal is zero that is the optimal filter
0 05 1 15 2 25 3 35 4 45 5-5
-4
-3
-2
-1
0
1
2
3
4
5
Time (s)
Y(t) and e(t)
Y(t)
e(t)
The Minimum mean-square error can be estimated
Since E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
Is E[(S(t)- L[Ŝ(t)] ) L[Y(ξ)] ]=0
Thereby is the Minimum mean-square
em=E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]= E[(S(t)-L[Y(ξ)]) S(t)]
em=E[(S(t)-L[Y(ξ)]) S(t)]
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
The Minimum mean-square error can be estimated
Since E[(S(t)- Ŝ(t) ) Y(ξ) ]=0
Is E[(S(t)- L[Ŝ(t)] ) L[Y(ξ)] ]=0
Thereby is the Minimum mean-square
em=E[|ε(t)|2]=E[S(t) 2 - 2 S(t)L[Y(ξ)] + L[Y(ξ)]2]= E[(S(t)-L[Y(ξ)]) S(t)]
em=E[(S(t)-L[Y(ξ)]) S(t)]
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
18 19 2 21 22 23 24 25 26 27
-3
-2
-1
0
1
2
3
4
Time (s)
Y(t)
e(t)
S(t)
Se(t)
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull Determine the impulse response h(ξ) of the filter
bull Use the Minimum mean-square error principle which requires the error S(t)-Ŝ(t) to be Orthogonal (correlation=0) to Y (ξ)
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull If we assume stationarity and time invariance
bull Thereby we ldquojustrdquo need to solve for h(ξ)
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
Fourier transform
Therefor H(f) is the cross correlation between S(t) and Y(t) dived with the power spectrum of Y(t)
cross correlation between S(t) and Y(t)
power spectrum of Y(t)
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull If S(t) and N(t) are statistically independent and zero mean
Therefore H(f) can be determined from the power spectrums of noise and the desired signal
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
100
102
-80
-60
-40
-20
0
Frequency (Hz)
H(f
) dB
Hz
PSD H(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
Sss(f
)+S
nn(f
) dB
Hz
PSD Sss
(f)+Snn
(f)
100
102
-80
-60
-40
-20
0
20
Frequency (Hz)
S(f
) dB
Hz
PSD S(f)
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull The minimum mean-square error can be calculated
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull The h(t) is not causal and thereby is the filter not realizable in real-time
bull There solutions based on spectral factorizations
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
119878119899119899 119891 = 1198730
2120575 120591 119890minus1198952120587119891120591119889120591 =
1198730
2
infin
minusinfin
119878119904119904 119891 = 119890minus120572|120591|119890minus1198952120587119891120591119889120591 =2120572
1205722 + 412058721198912
infin
minusinfin
Wolfram
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull Inver Fourier transform
119867 119891 =
2120572
1205722+412058721198912
2120572
1205722+412058721198912+11987302
=4120572
1198730
41205721198730 +1205722+412058721198912 =
119886
1198872+412058721198912
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull Different methods for estimation of filter coefficients
ndash Spectrum based bull (Estimate impulse response h(n))
ndash Mean square error bull (Estimate filter coefficients ω(n) )
ω(n)
Y(n)=ω0 X(n)+ ω1 X(n-1)+ ω2 X(n-2)hellip ωM-1 X(n-M+1)
Y(n)=h(0) X(n)+ h(1) ω1 X(n-1)+ h(2) X(n-2)hellip h(M-1)X(n-M+1)
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull As in the continues domain the transfer function of H(z) is the cross spectrum between S(t) and Y(t) dived by the PSD of Y(t)
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
(Real cases)
bull The discrete filter
bull Determine the coefficients 120596 which minimize the Mean square error
119890 119899 = 119878 119899 minus 119878(119899)
119862 120596 = 119864 119890 119899 119890 119899 = 119864 119890 119899 2 = 119864[(119878 119899 minus 119878 119899 )2] Mean square error
Error
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull Denote the filter coefficients
120596 = 1205960 1205961 1205962hellip 120596119872minus1
119879
bull Denote the signal
119884(119899) = 119884 119899 119884 119899 minus 1 119884 119899 minus 2 hellip 119884(119899 minus 119872) 119879
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull Thereby filter equations can be described using the dot product of the filter coefficients and the signal
119878 119899 = 120596119879119884 119899 = 1205960 1205961 1205962hellip 120596119872minus1
119884 119899119884 119899 minus 1119884 119899 minus 2119884(119899 minus 119872)
Ŝ(n)=ω0 Y(n)+ ω1 Y(n-1)+ ω2 Y(n-2)hellip ωM-1 Y(n-M+1)
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
(continued)
bull Since
119878 119899 = 120596119879119884 119899 =119884 119899 119879120596
bull Mean square error is
bull Which can be expanded to
119862 120596 = 119864 119890 119899 119890 119899 = 119864[ 119878 119899 minus 120596119879119884 119899 (119878 119899 minus 119884 119899 119879120596)]
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
(continued)
bull Expanded Mean square error explained
bull Compact form
119862 120596 = 119864 119878 119899 119878 119899 minus 120596119879119864[119884 119899 119878(119899)] minus 119864[119878 119899 119884(119899)119879]120596+ 120596119879E[119884 119899 119884(119899)119879]120596
Signal variance σs2 Cross correlation vector
between Y(n) and S(n) Rys =[rys(0) rys(1) helliphelliprys(M-1) ]
rys(τ)=E[Y(n+τ)S(n)] τ=012hellipM-1
Transformed version of Rys
Autocorrelation matrix of Y(n)
Ryy=
ryy(0) ryy(1) ryy(2)
ryy(minus1) ryy(0) ryy(1)
ryy(minus2) ryy(minus1) ryy(0)
ryy(τ)=E[Y(n+τ)Y(n)]
It an Toeplitz matrix
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Page 432
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
(continued)
bull The Mean square error is a second order equation
bull Thereby does the equation has a global minimum which can be identified by differentiation according to 120596 and solving for zero
bull The filter coefficients is then easily solved
Page 433-35
119941119914(120654)
119941120654=
119941
119941120654(σs
2 minus 120596119879Rys minus Rys119879
120596+ 120596119879Ryy120596) = minus2Rys+120784Ryy120596 = 0
119862 120596 = σs2 minus 120596119879Rys minus Rys
119879
120596+ 120596119879Ryy120596
Differentiating equation components
119889
119889120596(120596119879Rys)= Rys
119889
119889120596(Rys
119879
120596)= Rys
119889
119889120596(120596119879Ryy120596)= 120784Ryy120596
120654 = Ryy minus120783Rys Wiener-Hopf equation
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull Error e(n)=S(n)-YT(n)ω
bull Minimum mean square error
119890119898 = 1205901199042 minus Rys
119879120596 = 1205901199042 minus Rys
119879 Ryy minus120783Rys
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull Introduction to Wiener filters
bull Linear filters and orthogonality
bull The time continues Wiener filter
bull The time discrete Wiener filter
bull Example
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull Construct a 4th order wiener filter which estimates S(t)
Wiener filter
L[Y(t)] 0 1 2 3 4 5-2
0
2
Time (s)
S(t
)
S(t)
asymp 0 1 2 3 4 5
-5
0
5
Time (s)
Y(t
)
Y(t)
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
-5 0 54000
6000
8000
10000
Ryy
()
ryy(τ)=[9970 5089 4993 5046 4984] τ=[0 hellip4]
Make Autocorrelation matrix
Autocorrelation of Y(t) (M=4) xcorr(Y4)
toeplitz(ryy)
9970 5089 4993 5046 4984 5089 9970 5089 4993 5046 4993 5089 9970 5089 4993 5046 4993 5089 9970 5089 4984 5046 4993 5089 9970
Ryy=
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
Rys(τ)= [5116 5115 5114 5113 5111] τ=[0 hellip4]
Cross correlation of Y(t) and S(t) (M=4) xcorr(SY4)
-5 0 55110
5115
5120
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
ω
bull The filter coefficients is estimated using the Wiener-Hopf equation
120654 = Ryy minus120783Rys
ω =
01683 minus 00359 minus 00323 minus 00342 minus 00323 minus00359 01697 minus 00356 minus 00312 minus 00342 minus00323 minus 00356 01690 minus 00356 minus 00323 minus00342 minus 00312 minus 00356 01697 minus 00359 minus00323 minus 00342 minus 00323 minus 00359 01683
51165115
51145113
5111
=
0172001680016990167701710
Se(n)= 0172 Y(n)+ 01680Y(n-1)+ 01699 Y(n-2)+ 01677 Y(n-3)+ 0171Y(n-4)
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
Ŝ
0 05 1 15 2 25 3 35 4 45 5-3
-2
-1
0
1
2
3
Time (s)
Se(t
)
Se(t)
Se(t)
S(t)
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull When
ndash Both noise and desired signal are stationary processes
ndash You know the desired signal or at least the ideal characteristics of the desired signal
bull In these case the wiener filter is the optimal filter
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull Exercise 78
bull Matlab Exercise
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat
bull A blood pressure signal is contaminated by white
noise (Variable name y) The first 8 seconds of
the recording is available in a Noise free version
(Variable name s)
bull Make a FIR filter for filtering of the whole signal
bull Select an appropriate filter order
bull httppersonhstaaudksschmidtSTBPmat