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EE3054 Signals and Systems
DTFT, Filter Design, Inverse Systems
Yao WangPolytechnic University
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 4
Example
� x[n]=a^n u[n]� Special case x[n]=u[n]
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 10
Filter Design Specification(Desired Freq Response)
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 18
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.