INTRODUCTION TO
SIGNAL PROCESSING
Iasonas Kokkinos
Ecole Centrale Paris
Lecture 6
Analysis & Design of
Discrete-time Systems
th
6
Lecture Layout
•  Z-transform, Continued
•  Poles, Zeros & Frequency response
•
Pole and Zero placement
Lecture 5: Z-transform
•  Complex valued, complex function
Zeros
•  DTFT: Restriction of this function
on the unit circle
Poles
ROC for finite duration sequences
•  If x[n] is a finite duration sequence: ROC is the
whole z-plane (except possibly zero and infinity)
•
Example I:
•
•
Pole at 0, zero at -1
Example II:
•
Zero at -1, diverges as
ROC & Finite Sequences
ROC, finite duration signals
Z-transform Example-I
•  Sequence:
•  Z-transform:
•  ROC:
Z-transform Example I cont’d
•  Consider system with impulse response
•
Is it stable? (argue based on its z-transform)
Unit circle is not contained in the ROC
•  FT does not converge
•
•
System is unstable
Z-transform Example-II
•  Sequence:
•  Z-transform:
•  ROC:
•  Same formula, different ROC: different sequence
Z-transform examples I & II
Z-transform Example III
•  Sequence:
•  Z-transform
•  Conditions for convergence:
•  If both conditions hold:
ROC & Right-Sided Sequences
ROC, Infinite duration signals
Single real pole
•  From the examples discussed so far:
•  Pole at z = a
•  Examine time and z-domain relation:
Exponential sequence in time and z-plane
Pair of complex conjugate poles
•  Consider sequence:
•  Use
•  From linearity
•  Consider
•  From previous slides
•  Therefore
•  Similar:
Sinusoidal sequence in time and z-plane
•  |a|<1
•  |a|=1
•  |a|>1
Pair of conjugate complex poles
ROC & poles
•  If x[n] is right-sided sequence: ROC extends
outwards from outermost finite pole to infinity
•  If x[n] is left-sided sequence: ROC extends inwards
from innermost nonzero pole to zero
Properties of the ROC
•  ROC: disk in the z-plane, centered at the origin
•  Fourier Transform of x[n] converges if and only if |
z|=1 is in the ROC
•  ROC cannot contain any poles
•
A pole at |z|=1 means:
•
•
Fourier Transform does not converge
System is not stable
Stability, Causality and ROC
•  Consider system with Z-transform
•  For which ROC will it be stable?
•  For which ROC will it be causal?
•  Can it be both stable and causal?
Poles, causality and stability
•  A system is causal: h[n] is left handed
•
ROC will be extending outwards from largest
pole to infinity
•  A system is stable: |z|=1 is in the ROC
•  Consequence:
•
A stable and causal system has all poles
inside the unit circle
th
6
Lecture Layout
•  Z-transform, Continued
•  Poles, Zeros & Frequency response
•
Pole and Zero placement
THREE DOMAINS
Why use the
Z-domain ?
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Finite Impulse Response systems
•  FIR: special case of DE:
for
•  Impulse response:
•  If we ignore the time shift:
•
Box average, Gaussian smoothing, differentiator…
FIR systems: z-transform
•  Impulse response:
•  System function:
•  Rational function:
•  Stable: all poles within unit circle (at zero)
•  Zero placement -> Frequency response
Design Example: FIR filter
•  FIR Filter:
•  Design a Highpass FIR filter
•  Reject completely DTFT frequencies 0, π/3, and –π/3.
•  Estimate minimum filter length. How many bk ?
ANGLE is FREQUENCY
PLOT ZEROS in z-DOMAIN
−1
H(z) = (1− z )(1− e
jπ /3 −1
z )(1− e
− jπ /3 −1
z )
UNIT
CIRCLE
3 ZEROS
H(z) = 0
3 POLES
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FIR Frequency Response
ZEROS of H(z) and H(ω)
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NULLING PROPERTY of H(z)
•  When H(z)=0 on the unit circle.
•
Find inputs x[n] that give zero output
−1
−2
−3
H(z) = 1− 2z + 2z − z
jωˆ
− jωˆ
− j 2 ωˆ
− j3 ωˆ
H(e ) = 1− 2e + 2e
−e
x[n]
x[n] = e
j( π / 3)n
H(z)
y[n]
H(e
y[n] = H(e
jπ / 3
j ( π /3)
) =?
)⋅e
30
j (π / 3)n
FIR System example II
•  Impulse response:
•  Z-transform:
•  Poles & Zeros:
L-pt RUNNING SUM H(z)
L−1
H(z) = ∑ z
−k
k=0
z −1 = 0 ⇒ z = 1 = e
zk = e
ZEROS on
UNIT CIRCLE
L
1− z
z −1
=
−1 = L−1
(1 − z ) z (z −1)
L
L
−L
j(2 π /L)k
j2πk
(k an integer)
for k = 1, 2,K L -1
(z-1) in
denominator
cancels k=0 term
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11-pt RUNNING SUM H(z)
zk = e
j(2 π /11)k
(4 π / 11)
NO zero at z=1
FILTER DESIGN: CHANGE L
(4 π / 10)
Passband
Narrower
for L bigger
(4π / 20)
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Box Average filter: Z & DTFT
th
6
Lecture Layout
•  Z-transform, Continued
•  Poles, Zeros & Frequency response
•
Pole and Zero placement
Digital Resonators
•  Consider moving poles closer to unit circle
•  Impulse response for r=1?
Sinusoidal at frequency
•  Digital oscillators
•
•  Building blocks for digital signal synthesis
•  Choose frequency by changing location of pole
Low- and High- pass
•  Rotate location of poles (and zeros)
Poles at 0, zeros at π: Low-pass
•  Poles at π, zeros at 0: High-pass
•
rd
3
order system
•  System function
•  Poles & zeros
•  Log magnitude:
1st order system
•  Magnitude
nd
2
order system
•  Log magnitude
3rd order system
•  System function
•  Poles & zeros
System order
•  FIR System:
IIR System:
Which order is good enough?
•  Continuous Time Butterworth filter:
Filter specifications
•  Ideal filters: non-causal, infinite impulse
response
•  Actual specifications:
•  Minimal-order system that satisfies
specifications