z-transform

1. Date : 30-7-2020
ECEN341 Signals and Systems
Chapter-3
Z-Transform and System Representation using Difference Equations for DT systems
Chapter-3

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Z-Transform
What is the use of z-transform
• Z-transform is a mapping of discrete-time sequences from the time domain
into a complex frequency variable domain
• Very useful in the analysis of discrete-time signals and systems

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Z-Transform Definition
• Definition of Z-Transform
The Z-transform of a sequence x[n], denoted X(z), is defined as
Chapter-3

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Z-Transform of Elementary Signals
Chapter-3
Z-Transform of an Impulse Sequence
Z-Transform of a Unit Step Sequence
Note that for an infinite geometric series, we have
1 + 𝑎 + 𝑎2
+ 𝑎3
+ 𝑎4
+ ⋯ =
1
1−𝑎
when |a| < 1
X(z) =
1
1−𝑧−1 =
1
1−
1
𝑧
=
𝑧
𝑧− 1

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Chapter-3

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Chapter-3
Problem: Find the Z-Transform of the causal sequence

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Chapter-3
Problem: Find the Z-Transform of the DT sequence

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Chapter-3
Problem: Find the Z-Transform of the two sided DT sequence

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Problem: Find the Z-Transform of the sequence x(n) = u(n) – u(n-3).
Chapter-3

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Representation of DT systems using Difference Equation
The input /output relationship of a LTI discrete-time system is given by the linear
constant-coefficient difference equation (LCCDE).
Example:
y n = 2x n + 5x n − 1 − 3x n − 2 + 0.5y n − 1 − 4y[n − 2]
System function:
If 𝑥[𝑛]
𝑍.𝑇
𝑋(𝑧)
𝑦[𝑛]
𝑍.𝑇
𝑌(𝑧)
ℎ[𝑛]
𝑍.𝑇
𝐻(𝑧)
then
𝑆𝑦𝑠𝑡𝑒𝑚 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝐻 𝑧 =
𝑌(𝑧)
𝑋(𝑧)

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The general form of a Linear Constant-Coefficient Difference Equation (LCCDE)
a0 y[n] + a1 y[n-1] + a2 y[n-2] + … aN y[n-N] = b0 x[n] + b1 x[n-1] + b2 x[n-2] +… bM x[n-M]

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Chapter-2 Time Domain Analysis of systems
Problem : A causal discrete-time LTI system is described by 𝑦 𝑛 −
3
4
𝑦 𝑛 − 1 +
1
8
𝑦 𝑛 − 2 = 𝑥 𝑛
where x(n) and y(n) are the input and the output of the system respectively. Determine the system function H(z).

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Problem :
Given a transfer function determine the system difference equation
H(z)=
(𝑧+1)2
(𝑧−
1
2
)(𝑧+
3
4
)

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14
References
• “Signals and Systems” by Oppenhehim, Willsky and Hamid,,
2nd edition, Pearson Education India, 2015.
• Signals and Systems :Principles and Applications by
Shaila Dinkar Apte , Cambridge University Press
Chapter-1