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Applications of the Z-transform
March 19, 2006
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Power series method. Partial fraction expansion method.
Residue method.
1. Inverse Z-transform
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H(z) =
n=
h(n)zn System Function
ROC at least the intersection of the ROCs of H(z)
and X(z). Can be larger if there are pole/zero
cancellation. e.g.
H(z) =1
z a, z > a
x(z) = z a, z =
Y(z) = 1, ROC : all z
Y(z) = H(z)X(z),
h(n)x(n) y(n) = x(n)*h(n)
2. Convolution Property and System Functions
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A discrete-time linear time-invariant system function H(z) is causal when,
and only when the ROC of H(z) is the exterior of a circle and includes
z =
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
When h(n) right-sided, then ROC is the exterior of a circle:
H(z) =
n=N1
h(n)zn.
If N1 < 0, then h(N1)zN1 at z = . ROC outside a circle, butdoes not include .
Causal N1 0
3. Causality
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Delay
Multiplication
Addition
Branch
x(n)
x(n)
x(n)
x(n)
x(n1)
x(n)y(n)
x(n)+y(n)
x(n)
T
y(n)
y(n)
x(n)
4. Structure of a Digital System
4.1. Symbols for Digital Operations
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z1
1n+1
n
x(n) y(n)
y(n) =n
n + 1y(n 1) +
1
n + 1x(n)
We want a system that calculates:
y(n) =1
n + 1
nk=0
x(k)
4.2. Example - Cumulative Averaging System
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5. DT LTI systems described by LCCDEs
Nk=0
aky(n k) =
Mk=0
bkx(n k)
Using the time-shift property:
N
k=0
akzkY(z) =
M
k=0
bkzkX(z)
Y(z) = H(z)X(z)
H(z) =Mk=0 bkzk
Nk=0 akz
k
ROC: Depends on boundary conditions, left-, right-, or two-sided. For
causal systems - ROC is outside the outermost pole.
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6. Rational Z-Transform
X(z
) =
N(z)
D(z) =
b0 + b1z1 + + bMz
M
a0 + a1z1 + + aNzN
X(z) =b0
a0zNM
(z z1)(z z2) (z zM)
(z p1)(z p2) (z pN)
The transform has M finite zeros at z = z1, . . . zM, N infinite poles at
z = p1 . . . zN and N-M zeros (if N > M) or poles (if N < M) at the
origin z = 0.
A DT LTI system is causal if the ROC is the exterior of a circle outsideoutermost pole include . Thus
N M, for a causal system
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1
0
1
1
0
1
0
0.5
1
1.5
2
2.5
3
3.5
4
Re
Im
|X(z)|
7. Poles - Zero Description of Discrete-Time Systems
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0 1
zplane x(n)
0 1
zplane x(n)
0 1
zplane x(n)
0 1
zplane x(n)
0 1
zplane x(n)
0 1
zplane x(n)
7.1. Time-domain behavior - Single real-pole causal signal
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0 1
zplane x(n)
0 1
zplane x(n)
0 1
zplane x(n)
7.2. Time-domain behavior - Complex-conjugate poles
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Geometric Evaluation of a Rational z-Transform
Example #1:
Example #3:
Example #2:
All same as
in s-plane
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Geometric Evaluation of DT Frequency Responses
First-Order System one real pole
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Second-Order System
Two poles that are a complex conjugate pair (z1= rej=z2
*)
Clearly, |H| peaks near =
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Definition:
X(z) =
n=0
x(n)zn
Characteristics:
1. No information about x(n) for n < 0.
2. Unique only for causal signals.3. Identical to the two-sided z-transform of the signal x(n)u(n).
9. The one-sided z-transform
9.1. Definition and characteristics
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All properties are like for the two-sided Z-transform except for:
Shifting property:
x(n k)z
zk
X(z) +
k
n=1x(n)zn
, k > 0
x(n + k)z
zk
X(z) k1n=0
x(n)zn
, k > 0
Final value theorem
limn
x(n) = limn1
(z 1)X(z)
9.2. Properties
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Example taken from Digital Signal Processing, Principles, Algorithms and Applications by Proakisand Manolakis
Example
The well known Fibonacci sequence of integer numbers is obtained by com-
puting each term as a sum of the two previous ones. The first few terms of
the sequence are:
1, 1, 2, 3, 5, 8, . . .
Determine a closed-form expression for the nth term of the Fibonacci se-
quence.
10. Solution of Difference Equations