Z Transform (1) - Hany Ferdinando 2
Overview
Introduction Basic calculation RoC Inverse Z Transform Properties of Z transform Exercise
Z Transform (1) - Hany Ferdinando 3
Introduction
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 z-domain it means there is no time variable in the z-domain
Z Transform (1) - Hany Ferdinando 4
Introduction
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
It simplifies the study of LTI system by: 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
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
k
kkzxX(z)
n
nx(n)zX(z)or
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Basic Calculation
0k,2
0k0,k
kx
0k0,
0k,2- k
kx
Calculate:
2z
zX(z)
2z
zX(z)
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Basic Calculation
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)
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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RoC Properties
RoC of X(z) consists of a ring in the z-plane 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
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
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
Direct division Partial expansion Alternative partial expansion
Use RoCinformation
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Direct Division
If the RoC is less than ‘a’, then expand it to positive power of z 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)
az
aX(z)
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Partial Expansion
If the z is in the power of two or more, then use partial expansion to reduce its order
Then solve them with direct division
n
n
2
2
1
1
n21
m2m
21m
1m
0
pz
A...
pz
A
pz
A
)p)...(zp)(zp(z
a...zazaza
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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
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 (n) is the entire z-plane
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Time Shifting
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
ej(o)nx(n) ↔ X(ej(o)z) RoC is R The poles and zeros is rotated by the
angle of o, therefore if X(z) has complex conjugate poles/zeros, they will have no symmetry at all
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Convolution Property
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
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…
dz
dX(z)znx(n)
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