polytechnic university inverse systemsdtft, filter design
TRANSCRIPT
EE3054, S08 Yao Wang, Polytechnic University 2
Discrete Time Fourier Transform
� Recall � h[n] <-> H(e^jw) = H(z)|z=e^jw
� Can be applied to any discrete time signal� x[n] <-> X(e^jw) = X(z)|z=e^jw
� More generally can be applied to signals starting before 0
� When x[n] has infinite duration, converge only when� \sum |x[n]| < \infty� x[n] has finite energy
nj
n
j enheH ωω ˆ
0
ˆ ][)( −∞
=∑=
nj
n
j enxeX ωω ˆ
0
ˆ ][)( −∞
=∑=
nj
n
j enxeX ωω ˆˆ ][)( −∞
−∞=∑=
EE3054, S08 Yao Wang, Polytechnic University 3
Properties of DTFT
� Periodic with period =2 \pi� Only need to show in the range of (-pi,pi)
� x[n] real -> X(e^-jw)=X*(e^jw)� Magnitude of X is symmetric� Phase is antisymmetric
� Delay property � x[n-n0] <-> e^-jwn0 X(e^jw)
� Convolution � x[n]*y[n] <-> X(e^jw) Y(e^jw)
EE3054, S08 Yao Wang, Polytechnic University 5
Example
� x[n]=rectangular pulse
( )( )2/ˆsin
)21(ˆsin)(
]1[][,0,1
][
ˆ
ωωω +=
−−−+= <=
=
MeX
MnuMnuotherwise
Mnnx
j
EE3054, S08 Yao Wang, Polytechnic University 6
Example
( ) ( ))ˆˆ()ˆˆ()(ˆcos][
ˆ,ˆ),ˆˆ(2)(][
00ˆ
0
00ˆˆ 0
wwwweXnwnx
wwwweXenxj
jnwj
++−=⇔=
<<−=⇔=
δδπ
πππδω
ω
EE3054, S08 Yao Wang, Polytechnic University 7
Inverse DTFT
� If x[n] has finite duration: identify from coefficients associated with z^-n in X(z) or with e^{-jwn} from X(e^jw)
� What if not?� IDTFT� Proof difficult, after we
learn FT and FT of sampled signals
( ) wdeeXnx
enxeX
nwjwj
nj
n
j
ˆ21][
(IDTFT) transformInverse
][)(
(DTFT) transformForward
ˆˆ
ˆˆ
∫
∑
−
−∞
−∞=
=
=
π
π
ωω
π
EE3054, S08 Yao Wang, Polytechnic University 8
What does DTFT X(e^jw) represent?
( ) ( )
( )( ) (spectrum) x[n]!ofon distributifrequency theshows
.ˆfrequency with sinusoid theof amplitude theis
. ˆˆ sfrequenciewith sinusoidmany of sum a as considered becan ][
ˆ21ˆ
21][
(IDTFT) transformInverse
ˆ
ˆ
ˆˆˆˆ
k
k
wjk
wjk
nwjk
k
wjknwjwj
eX
weX
wkwnx
weeXwdeeXnx
∆=
∆≈= ∆∞
−∞=
∆
−∑∫ ππ
π
π
EE3054, S08 Yao Wang, Polytechnic University 9
Filter design
� The desired frequency response (low-pass,high-pass,etc, and cutoff freq.) is determined by the underlying application
� Ideal freq. response with sharp cutoff is not realizable
� Must be modified to have non-zero transition band and variations (ripples in pass band and stop band).
� Show figure.
EE3054, S08 Yao Wang, Polytechnic University 11
FIR or IIR?
� FIR: can have linear phase, always stable� Weighted average (positive coeff.): low pass� Difference of neighboring samples: high pass
� IIR: can realize similar freq. resp. (equal in transition bandwidth and ripple) with lower order
EE3054, S08 Yao Wang, Polytechnic University 12
Ideal Low Pass Filter
� Show desired freq. response� Ideal low pass <-> Sinc function in time! (Show using
IDTFT)
EE3054, S08 Yao Wang, Polytechnic University 13
Truncated Sinc Filter (FIR)
� Truncated sinc function <-> non-ideal low pass� Much better than averaging filter of same length! (Show
using MATLAB)
EE3054, S08 Yao Wang, Polytechnic University 14
FIR filter design
� Given the desired response and the order of filter, can determine the coefficients by minimizing the difference between the desired response and the resulting one � Least square� Mini-max (resulting in equal ripple) -> Parks-McClellen algorithm
� MATLAB implementation:� B = FIR1(N,Wn,'high')� B = FIR2(N,F,A)� B=FIRLS(N,F,A): linear phase (symmetric), least square� B=FIRPM(N,F,A): lienar phase, equal ripple
EE3054, S08 Yao Wang, Polytechnic University 15
IIR Filter
� Butterworth filters� Maximally flat in pass and stop band� [B,A] = BUTTER(N,Wn,’low’)
� Chebychev filters� Equal ripple in stop (or pass) band, flat in pass (or
stop) band� [B,A] = CHEBY1(N,R,Wn,'high')
� Elliptic filters� Equal ripple in both pass and stop band� [B,A] = ELLIP(N,Rp,Rs,Wn,'stop')
EE3054, S08 Yao Wang, Polytechnic University 16
Inverse system
� Example: telephone system, echo problem
� Model: y[n]=x[n]+A x[n-n0]� Equalizer: obtain x[n] from y[n] (inverse)� How?
EE3054, S08 Yao Wang, Polytechnic University 17
Using Z-domain analysis
� Y(z)= H(z) X(z)� X(z)=Y(z)/H(z)� Let W(z)= Y(z)*G(z)
� With G(z)=1/H(z), then W(z)=X(z)� Previous example:
� H(z)=1+A z^-n0� G(z)= 1/(1+A z^ -n0)
� Implementation with difference equation� w[n]= - A w[n-n0] + y[n]
� Draw block diagram of general inverse system
EE3054, S08 Yao Wang, Polytechnic University 19
Any problem with previous design?
� Is the inverse system G(z) stable? � If all the poles of G(z) (zeros of H(z) are inside unit
circle
For a system to be stable, all its poles must be inside unit circle
For a system to have stable inverse, all its zeros must be also be inside unit circle
� For the previous example, this requires |A|<1
EE3054, S08 Yao Wang, Polytechnic University 20
Stable Inverse Systems
� When the inverse system is not stable, there are non-causal versions which are stable � See Selesnick’s notes on stable inverse
systems� Optional reading only
EE3054, S08 Yao Wang, Polytechnic University 21
Other applications
� Debluring of video captured while camera/objects in motion
� Equalization of received signals in a cell phone, which are sum of signals going through multiple paths with different delays (multipath fading)
� Etc.
Summary
� DTFT and IDTFT� X(e^jw) represents the energy of x[n] in freq. w� Computation and properties
� Filter design� Freq. response spec: cutoff freq. transition band, ripples� FIR vs. IIR� Matlab functions
� Inverse systems:� Determine original signal from an altered one due to
communication or other processing� G(z)=1/H(z)� Conditions for stable inverse
READING ASSIGNMENTS
� This Lecture:� DTFT: Chapter 12-3.5� Filter design:
� Oppenheim and Wilsky, Signals and Systems, Chap 6.
� Also see Lab6 note� Inverse systems:
� Selesnick’s note on inverse systems: http:eeweb.poly.edu/~yao/EE3054/AddLabNotes.pdf
� Finding stable but non-causal inverse is not required.