Z Transform Analysis of DT Systems

5.5 Z TRANSFORM ANALYSIS OF DT SYSTEMS

5.5.1 Pole Zero Plots

Z transform of discrete time signal x(n) can be expressed in rational form as

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Poles are the values of z for which X(z) = ∞

Zeros are the values of z for which X(z) = 0.

Poles does not lie in Roc of X(z), because poles are the values where X(z) = ∞.

Zeroes lie in Roc of X(z), because zeros are the values where X(z) = 0.

Rational form of X(z) i.e., (1) can be written as

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5.5.2 Transfer Function of the Linear Time Invariant (LTI) System

Output of LTI system is given as

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O/P y(n) is convolution of input x(n) and impulse response h(n)

Taking z transform of (1)

Y(z) = z{x(n) * h(n)}

By convolution property of z transform, the above equation can be written as

Y(z) = X(z) H(z)

H(z) = Y(z) / X(z)

Here H(z) is called the transfer function (or) system function.

5.5.3 Causality and Stability

Causality:

Condition for LTI system to be causal is given as

h(n) = 0, n < 0

LTI system is causal if and only if the Roc of the system function is exterior of a circle of radius r < ∞.

Stability:

Condition for stable system is given as

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Transfer function of LTI system is given as

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Taking magnitude of both the sides.

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If H (z) is evaluated on unit circle, |z ^-n| = |z| = 1. Then the above equation becomes,

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If the system is BIBO stable, then eeeeeeeeeee

Hence

eeeeeeeeee Evaluated on unit circle.

But this condition requires that the unit circle should present in the Roc of H(z)

LTI system is BIBO stable if and only if the Roc of the system function includes the unit circle.