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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: (a) {0, 1, 2, 4, 0, 0,...} (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 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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