Transcript Fourier (1) - Petra Christian University

Z Transform (1)
Hany Ferdinando
Dept. of Electrical Eng.
Petra Christian University
Overview
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Introduction
Basic calculation
RoC
Inverse Z Transform
Properties of Z transform
Exercise
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Introduction
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For discrete-time, we have not only
Fourier analysis, but also Z transform
This is special for discrete-time only
The main idea is to transform
signal/system from time-domain to zdomain  it means there is no time
variable in the z-domain
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Introduction
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One important consequence of
transform-domain description of LTI
system is that the convolution
operation in the time domain is
converted to a multiplication
operation in the transform-domain
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Introduction
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It simplifies the study of LTI system by:
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Providing intuition that is not evident in
the time-domain solution
Including initial conditions in the solution
process automatically
Reducing the solution process of many
problems to a simple table look up, much
as one did for logarithm before the
advent of hand calculators
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Basic Calculation
X(z) 

k
x
z
 k
k  
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or
X(z) 

n
x(n)z

n  
They are general formula:



Index ‘k’ or ‘n’ refer to time variable
If k > 0 then k is from 1 to infinity
Solve those equation with the geometrics
series
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Basic Calculation
Calculate:
xk 
0, k  0
2 ,k  0
k
z
X(z) 
z2
xk 
-2 ,k  0
k
0, k  0
z
X(z) 
z2
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Basic Calculation
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Different signals can have the same
transform in the z-domain  strange
The problem is when we got the
representation in z-domain, how we
can know the original signal in the time
domain…
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Region of Convergence (RoC)
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Geometrics series for infinite sum has
special rule in order to solve it
This is the ratio between adjacent
values
For those who forget this rule, please
refer to geometrics series
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Region of Convergence (RoC)
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Region of Convergence (RoC)
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Region of Convergence (RoC)
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RoC Properties
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RoC of X(z) consists of a ring in the zplane centered about the origin
RoC does not contain any poles
If x(n) is of finite duration then the RoC
is the entire z-plane except possibly z
= 0 and/or z = ∞
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RoC Properties
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If x(n) is right-sided sequence and if |z|
= ro is in the RoC, then all finite values
of z for which |z| > ro will also be in the
RoC
If x(n) is left-sided sequence and if |z|
= ro is in the RoC, then all values for
which 0 < |z| < ro will also be in the
RoC
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RoC Properties
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If x(n) is two-sided and if |z| = ro is in
the RoC, then the RoC will consists of
a ring in the z-plane which includes the
|z| = ro
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Inverse Z Transform
Use RoC
information
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Direct division
Partial expansion
Alternative partial expansion
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Direct Division
a
X(z) 
za
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If the RoC is less than ‘a’, then
expand it to positive power of z
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a is divided by (–a+z)
If the RoC is greater than ‘a’, then
expand it to negative power of z

a is divided by (z-a)
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Partial Expansion
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If the z is in the power of two or more,
then use partial expansion to reduce
its order
a 0 z m  a1z m1  a 2 z m2  ...  a m
A1
A2
An


 ...
(z  p1 )(z  p 2 )...(z p n )
z  p1 z  p 2
z  pn
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Then solve them with direct division
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Properties of Z Transform
General term and condition:
 For every x(n) in time domain, there is
X(z) in z domain with R as RoC
 n is always from –∞ to ∞
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Linearity
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a x1(n) + b x2(n) ↔ a X1(z) + b X2(z)
RoC is R1∩R2
If a X1(z) + b X2(z) consist of all poles of X1(z)
and X2(z) (there is no pole-zero cancellation),
the RoC is exactly equal to the overlap of the
individual RoC. Otherwise, it will be larger
anu(n) and anu(n-1) has the same RoC, i.e.
|z|>|a|, but the RoC of [anu(n) – anu(n-1)] or
d(n) is the entire z-plane
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Time Shifting
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x(n-m) ↔ z-mX(z)
RoC of z-mX(z) is R, except for the
possible addition or deletion of the origin
of infinity
For m>0, it introduces pole at z = 0 and
the RoC may not include the origin
For m<0, it introduces zero at z = 0 and
the RoC may include the origin
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Frequency Shifting
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ej(Wo)nx(n) ↔ X(ej(Wo)z)
RoC is R
The poles and zeros is rotated by the
angle of Wo, therefore if X(z) has
complex conjugate poles/zeros, they
will have no symmetry at all
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Time Reversal
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x(-n) ↔ X(1/z)
RoC is 1/R
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Convolution Property
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x1(n)*x2(n) ↔ X1(z)X2(z)
RoC is R1∩R2
The behavior of RoC is similar to the
linearity property
It says that when two polynomial or power
series of X1(z) and X2(z) are multiplied, the
coefficient of representing the product are
convolution of the coefficient of X1(z) and
X2(z)
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Differentiation
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dX(z)
nx(n)  z
dz
RoC is R
One can use this property as a tool to
simplify the problem, but the whole
concept of z transform must be
understood first…
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Next…
For the next class, students have to read Z
transform:
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Signals and Systems by A. V. Oppeneim ch
10, or
Signals and Linear Systems by Robert A.
Gabel ch 4, or
Sinyal & Sistem (terj) ch 10
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