1 bien425 – lecture 8 by the end of the lecture, you should be able to: –compute cross-...
TRANSCRIPT
1
BIEN425 – Lecture 8
• By the end of the lecture, you should be able to:– Compute cross- /auto-correlation using matrix
multiplication– Compute cross- /auto-correlation using DFT and Matlab– Relate auto-correlation of white noise with its average
power– Extract periodic signal from noise
2
Recall: Convolution
• Is a filter operation that computes the zero-state output corresponding to an input x(k)
• Matlab code: conv.m
• As denoted by *• If h(i) is a L-point signal, and x(k) is a M-point
signal then the convolution has length L+M-1
k
i
ikxihkxkhky0
)()()(*)()(
3
Circular convolution
• Let h(k) and x(k) be N-point signals and xp(k) be the periodic extension of x(k). Circular convolution is defined as
• Very simply yc(k) = C(x)h, where C(x) is defined as:
1
0
)()()()(N
ip ikxihkxkh
)0()1()2(
)1()0()1(
)2()1()0(
)(xxx
Nxxx
NxNxx
xC
4
• Note that circular convolution is more efficient (faster)
• However, it cannot be used directly to compute zero-state response
• Also needs to assume h(k) and x(k) have the same length.
• Another problem is with periodicity
• We can solve this using zero-padding.
)()( kyNky cc
But the zero-state response is not periodic
5
Zero padding
• Pad both h(k) and x(k) to be length N = L+M-1, now denoted as hz(k) and xz(k)
• Now, y = C(xz)hz
Tz
Tz
Mxxkx
Lhhkh
],0,0),1()0([)(
],0,0),1()0([)(
y is now M+L-1 in length
6
Example
• Matlab example
7
Fast convolution
• Convolution in the time domain maps into multiplication in the frequency domain using Z-transform.
• We may substitute circular for linear convolution; DFT for Z-transform
)()()}({ zXzHkyZ
)()()}()({ iXiHkxkhDFT
8
• Now if enough zeros are padded to h(k) and x(k) so that they both have length N≥L+M-1 where N = 2r
• This is called fast linear convolution
We get Hz(i) and Xz(i)
Zero pad
Zero pad
FFT
FFT
IFFT
h(k)
x(k)
y(k)
xz(k)
hz(k)
)}()({)( iXiHIFFTky zz
9
Cross-correlation
• Measures the degree to which a pattern of L-point signal y(k) is similar to the pattern of an M-point signal x(k)
• This definition works fine for deterministic signals.
• To generalize it to include stochastic signals:
For example: on next slide
1
0
)()(1
)(L
ixy kiyix
Lkr
)]()([)( kiyixkrxy
10
11
• Again, it can be computed using matrix formulation:
• Example 4.8, the index corresponding to the largest rxy(k) represents the optimal delay.
)0(00
00)1()2()1()0(0
00)1()1()0(
1)(
)(
y
MyMyyy
Myyy
LyC
xyDrxy
12This shows optimal delay for y is 3 samples
13
Normalized linear cross-correlation
• So that it takes on values [-1, 1]
)0()0(
)()(
yyxxLM
xyxy rr
krkP
This allows us to measure the amount of “significance” that is independent on the scaling or amplitude of the signal
14We can have a fixed threshold to determine if two signalsare significantly similar.
15
Fast correlation
• Again, similar to fast convolution, we could utilize properties of FFT to achieve a fast version of correlation
• Block diagram:
Zero pad
Zero pad
FFT
FFT *
IFFT /L
x(k)
y(k)
rxy(k)
yz(k)
xz(k)
LiYiXIFFT
kr zzxy
)}()({)(
*
16
Auto-correlation
• For special case where y(k) = x(k)
• It can be expressed in terms of expected values
• For the special case of white noise v(k),
1
0
)()(1
)(N
ixx kixix
Nkr
xxx
xx
Pr
kixixkr
)0(
)]()([)(
0,
0,0)(
kP
kkr
vvv
17
Relating auto-correlation and PSD
)()()(1
)}({
)}()({)}({
*
1
iSiXiXN
kcDFT
kxkxDFTkcDFT
Nxx
Nxx
If we apply the same principle of circular convolution to correlation, circular-correlation cxy(k) can be obtained
Notice here (–k)
18
Extracting periodic signals from noise
• Let’s say y(k) = x(k) + v(k), 0≤ k <N and the period M<N.
• Therefore, for k≠0, cyy(k) = cxx(k)
Motivation?
x(k) is periodic M (but we didn’t know that from the beginning
)()()(
)}()({)(
kPkckc
kiyiykc
vxxyy
pyy
Because correlation of noise with any other signal is 0!
19
• Auto-correlation averages the noise out!
• Also note that cxx(k) is periodic
• Therefore, the period of x(k) can be estimated by examining the peaks and valleys in cyy(k)
)()( kcMkc xxxx
20
• Once the period of a noise-corrupted signal is determined, we can estimate the original periodic signal
• Let L = floor (N/M)• We define delta function of period M
1
0
)()(L
iM iMkk 0 ≤ k < M
1
0
1
0
)(1
)(
)()(1
)(
)()(
L
qpy
N
ipMy
yy
kqMyN
kc
kiyiN
kc
kckc
M
M
MM
By definition
Because L = floor of (N/M)
21
• Now the second term approaches 0 for large L• Therefore
1
0
1
0
1
0
1
0
1
0
)(1
)()(
)(1
)(1
)(
)(1
)(1
)(
L
qpy
L
qp
L
qy
L
qp
L
qpy
kqMvN
kxNL
kc
kqMvN
kxN
kc
kqMvN
kqMxN
kc
M
M
M
Because x is period M
)()(ˆ kcLN
kxMy
22
Block diagram
Circularcross
correlation
y(k)x(k)
v(k)
(k)
)(kcMy )(ˆ kxN/L