Chapter 2 The z-transform and Fourier Transforms The Z Transform The Inverse of Z Transform The Prosperity of Z Transform System Function System Function

The Z Transform Introduction: Why do we introduce the z transform? The reason is that the Fourier transform does not converge of all sequence, and it is useful to have a generalization of the Fourier transform that encompasses a broader class of signals. A second advantage is that z transform allows us to bring the power of complex variable theory to bear on problems of discrete-time signals and systems.

The Z Transform

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The Inverse z-transform In general, there are three method about the inverse z-transform 1.The contour integration method 2.The partial fraction expansion method 3.The power series expansion

The contour integration method 1. The formal inverse z-transform expression If X(z) and ROC are given, the x(n) can be obtained by follow evaluating Where C is a counterclockwise contour that encircle the origin in the convergence region of X(z)

The contour integration method 2. The residue theorem

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Partial Fraction Expansion Let X(z) is expressed as a ratio of polynomials in z -1 : i.e

Power Series Expansion

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Properties of the z transform Many of the properties of the z-transform are particularly useful in studying discrete-time signal and system. In this section we sate some of the more important z-transform properties.

Properties of the z transform

Solution

Properties of the z transform

Solution

Properties of the z transform

Solution

Properties of the z transform

System function As we have seen in chapter 1, a discrete-time linear system can be characterized by a difference equation, In this section, we show how the z-transform can be used to solve difference equations, and therefore characterize linear system.

System function 1.Definition:

System function 2. System function & linear constant-coefficient difference equation: If we assume that the system is causal, the difference equation can be used to compute the output recursively. If the auxiliary conditions correspond to initial rest condition, the system will be causal, linear and time-invariant. i.e. linearity, time-invariant and causal of the system will depend on the auxiliary conditions If an additional condition Is that initial rest condition, the system will be linearity, time- invariant and causal.

Stability and causality in the z domain 2. Causal system If discrete-time LTI system is causal, its impulse response h(n) must be satisfied the following relation(necessary and sufficient condition h(n)=0 n<0 That is, h(n) is a causal sequence, so, the ROC of H(z) is form as Thus, for any causal stable system, The convergence region of system function H(z) must include the unit circle, or That is, all the poles of a causal stable system must be in the unit circle.

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System function 3. The Method for Obtaining the All Poles of System Function