Decoding Kotlin - Your guide to solving the mysterious in Kotlin.pptx
Z-transform and Its Inverse.ppt
1. Chapter Five
Z-Transform and Its Inverse
Addis Ababa Science and Technology University
College of Electrical & Mechanical Engineering
Electromechanical Engineering Department
Signals and Systems Analysis ( EME3262)
2. Z-transform and Its Inverse
Outline
The Z-transform
Properties of the Z-transform
Transfer function of Discrete-time LTI Systems
Transform Domain Analysis using the Z-transform
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3. The Z-transform
The Z-transform of a discrete-time signal x(n), denoted by X(z),
is defined as:
The Z-transform is a mapping (transformation) from a sequence
to a power series.
We say that x(n) and X(z) are Z-transform pairs and denote this
relationship as:
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n
n
z
n
x
z
X )
(
)
(
)
certain
(with
)
(
)
( ROC
z
X
n
x Z
4. The Z-Plane
The variable z is complex and can be viewed in the z-plane.
The Z-transform of a discrete-time signal x(n) is a function X(z)
defined on the z-plane.
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5. Region of Convergence (ROC)
The region of convergence (ROC) is defined as the set of all
values of z for which X(z) has a finite values.
Every time we cite a Z-transform, we should indicate its ROC.
Example:
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6. Region of Convergence (ROC)……
Exercise:
1. Find the Z-transform of the following discrete-time signals and
state the ROC.
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7. Region of Convergence (ROC)……
2. Find the Z-transform of the following discrete-time signals and
state the ROC.
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11. Properties of the ROC……
In general, the ROC has the following properties.
i. The ROC can not contain any poles inside it.
ii. If x(n) is left-sided signal, then:
iii.If x(n) is right-sided signal, then:
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pole
outermost
the
is
:
,
: 2
2 r
r
z
ROC
pole
innermost
the
is
:
,
: 1
1 r
r
z
ROC
12. Properties of the ROC……
iv. If x(n) is two-sided signal, then:
v. If x(n) is a finite length signal, then ROC is the entire z-plane
except possibly at
vi. The DTFT of x(n) exists if and only if the ROC of x(n)
includes the unit circle.
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: 1
2 r
z
r
ROC
.
or
0
z
z
13. Properties of the ROC……
Exercise:
The Z-transform of a discrete-time signal x(n) is given by:
Determine:
a. all the possible ROCs
b. the corresponding discrete-time signal x(n) for each of the
above ROCs
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z
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z
X
20. Rational Z-transforms
The most important and most commonly used Z-transforms are
those for which X(z) is a rational function of the form:
The roots of the numerator N(z) are known as the zeros of X(z).
The roots of the denominator D(z) are known as the poles of
X(z).
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N
M
M
M
N
k
k
k
M
k
k
k
z
a
z
a
a
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a
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b
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D
z
N
z
X
.....
.....
)
(
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( 1
1
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0
0
0
21. Rational Z-transforms……
The above rational Z-transform contains:
M zeros at z1, z2, ……, zM
N poles at p1, p2, ……, pM
If M<N, then there are N-M additional zeros at the origin z=0.
If M>N, then there are M-N additional poles at the origin z=0.
If M=N, then X(z) has exactly the same number of poles and
zeros.
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22. Rational Z-transforms……
Exercise:
Find the Z-transform and sketch the pole-zero plots of the
following discrete-time signals.
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26. Inverse Z-transform
Inverting by Inspection:
The simplest inversion method is by inspection, or by comparing
with the table of common Z-transform pairs.
Exercise:
Find the inverse of the following Z-transforms by inspection.
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0
:
,
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1
1
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,
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(
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1
1
z
ROC
z
z
X
b
z
ROC
z
z
X
a
27. Inverse Z-transform……
Inverting by Partial Fractional Expansion:
This is a method of writing complex rational Z-transforms as a
sum of simple terms.
After expressing the complex rational Z-transform as a sum of
simple terms, each term can be inverted by inspection.
Exercise:
Find the inverse Z-transform by partial fractional expansion method.
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5
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0
:
,
5
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0
1
25
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0
1
1
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( 1
1
z
ROC
z
z
z
X
28. Transfer Function of Discrete-time LTI Systems
The Z-transform of the impulse response h(n) is known as the
transfer function of the system.
Mathematically:
We say that h(n) and H(z) are Z-transform pairs and denote this
relationship as:
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n
n
z
n
h
z
H )
(
)
(
)
(
)
( z
H
n
h Z
29. Transfer Function of Discrete-time LTI Systems……
The output y(n) of a discrete-time LTI system equals the
convolution of the input x(n) with the impulse response h(n),
i.e.,
Taking the Z-transform of both sides of the above equation by
applying the convolution property, we obtain:
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(
*
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(
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( n
h
n
x
n
y
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(
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(
z
X
z
Y
z
H
z
H
z
X
z
Y
30. Transfer Function of Discrete-time LTI Systems……
i. Causal LTI Systems
A discrete-time LTI system is causal if h(n)=0, n<0. In other
words, h(n) is right-sided signal.
Therefore, ROC of H(z) is an exterior region starting from the
outermost pole.
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31. Transfer Function of Discrete-time LTI Systems……
ii. Anti-causal LTI Systems
A discrete-time LTI system is anti-causal if h(n)=0, n>0. In
other words, h(n) is left-sided signal.
Therefore, ROC of H(z) is an interior region starting from the
innermost pole.
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32. Transfer Function of Discrete-time LTI Systems……
iii. BIBO Stable LTI Systems
A discrete-time LTI system is BIBO stable if h(n) is absolutely
summable, i.e. ,
Therefore, ROC of H(z) always contains the unit circle.
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n
n
h )
(
34. Transfer Function of Discrete-time LTI Systems……
iv. Causal & BIBO stable LTI Systems
The ROC of H(z) must be an exterior region starting from the
outermost pole and contains the unit circle.
In other words, all poles must be inside the unit circle.
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35. Transfer Function of Discrete-time LTI Systems……
Exercise:
1. The transfer function of a discrete-time LTI system is given by:
a. Find the poles and zeros of H(z).
b. Sketch the pole-zero plot.
c. Find the impulse response h(n) if the system is known to be:
i. causal iii. BIBO stable
ii. anti-causal
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H
36. Transfer Function of Discrete-time LTI Systems……
2. Plot the ROC of H(z) for discrete-time LTI systems that are:
a. causal & BIBO stable
b. causal & unstable
c. anti-causal & BIBO stable
d. anti-causal & unstable
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37. Transform Domain Analysis using the Z-transform
The procedure for evaluating the output y(n) of a discrete-time
LTI system using the Z-transform consists of the following four
steps.
1. Calculate the Z-transform X(z) of the input signal x(n).
2. Calculate the Z-transform H(z) of the impulse response h(n) of
the discrete-time LTI system.
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38. Transform Domain Analysis using the Z-transform….
3. Based on the convolution property, the Z-transform of the
output y(n) is given by Y(z) = H(z)X(z).
4. The output y(n) in the time domain is obtained by calculating
the inverse Z-transform of Y(z) obtained in step (3).
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39. Exercise
1. Find the Z-transform of the following discrete-time signals.
2. Find the inverse Z-transform of:
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1
(
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1
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1
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,
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1
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4
1
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1
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(
1
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z
ROC
z
z
z
X
40. Exercise……
3. The input to a causal discrete-time LTI system is given by:
The Z-transform of the output of this system is:
a. Determine the impulse response h(n) of the system.
b. Find the output y(n) of the system.
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Y
