# Z- Transform and Its Properties

## Presentation on theme: "Z- Transform and Its Properties"— Presentation transcript:

Z- Transform and Its Properties
Dr. Wajiha Shah

The z-Transform Given the causal sequence {x[1], x[2], …., x[k],….}, its z-transform is defined as The variable z may be regarded as a time delay operator

Example Find the z-transform of the sequence x = {1, 1, 3, 2, 0, 4}
Solution:

Exercise Find the z-transform of the following sequences:
(b) {0, 0, 0, 1, 1, 1, 0, 0, 0, . . .} (c) {0, 2−0.5, 1, 2−0.5, 0, 0, 0, . . .}

Z-Transform of Standard Discrete-Time Signals
Unit Impulse Function A discrete time impulse Shown in the Fig. is defined as The z-Transform of δ(k) is X(z) = 1

Sampled Step Sampled Step A sampled step function is shown in the Fig.
Mathematically, it is defined as The z-Transform of u(k) is computed as

Exponential Exponential:
A sampled exponential function is shown in the Fig. Mathematically, it is defined as The z-Transform of x(k) is computed as

Properties of z-Transform
Linearity Let x1(k) ,x2(k) , .... be discrete time sequences and a1, a2, …. be constants, then according to this property Example: Find z-Transform of x(k) = 2u(k) + 4δ(k), where u(k) is a unit step sequence and k = 0, 1, 2, …. Solution:

Contd… • Time Delay z[x(k D)]= z X(z) where D denotes time delay.
Proof: By definition Let m = k-D or k = m+D

Contd… Example Find the z-transform of the causal sequence Solution:

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Inverse of the z-Transform

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Convolution Sum

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Transfer Function

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