EE3054 Signals and Systems

DTFT, Filter Design, Inverse Systems

Yao WangPolytechnic University

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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 ][)( −∞

−∞=∑=

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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)

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Example

� x[n]=a^n u[n]� Special case x[n]=u[n]

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Example

� x[n]=rectangular pulse

( )( )2/ˆsin

)21(ˆsin)(

]1[][,0,1

][

ˆ

ωωω +=

−−−+= <=

=

MeX

MnuMnuotherwise

Mnnx

j

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Example

( ) ( ))ˆˆ()ˆˆ()(ˆcos][

ˆ,ˆ),ˆˆ(2)(][

00ˆ

0

00ˆˆ 0

wwwweXnwnx

wwwweXenxj

jnwj

++−=⇔=

<<−=⇔=

δδπ

πππδω

ω

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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

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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

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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.

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Filter Design Specification(Desired Freq Response)

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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

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Ideal Low Pass Filter

� Show desired freq. response� Ideal low pass <-> Sinc function in time! (Show using

IDTFT)

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Truncated Sinc Filter (FIR)

� Truncated sinc function <-> non-ideal low pass� Much better than averaging filter of same length! (Show

using MATLAB)

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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

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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')

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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?

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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

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Block diagram of general inverse system

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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

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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

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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.