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**Z- Transform and Its Properties**

Dr. Wajiha Shah

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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

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

Solution:

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**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, . . .}

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**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

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**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

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**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

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**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:

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**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

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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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