# UCCS ECE 2610 - Z-Transforms (28 pages)

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## Z-Transforms

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## Z-Transforms

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Lecture Notes

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Ece 2610 - Introduction to Signals and Systems
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Chapter z Transforms 7 In the study of discrete time signal and systems we have thus far considered the time domain and the frequency domain The zdomain gives us a third representation All three domains are related to each other A special feature of the z transform is that for the signals and system of interest to us all of the analysis will be in terms of ratios of polynomials Working with these polynomials is relatively straight forward Definition of the z Transform Given a finite length signal x n the z transform is defined as X z N k 0 x k z k N 1 k x k z 7 1 k 0 where the sequence support interval is 0 N and z is any complex number This transformation produces a new representation of x n denoted X z Returning to the original sequence inverse z transform x n requires finding the coefficient associated with the nth power 1 of z ECE 2610 Signal and Systems 7 1 Definition of the z Transform Formally transforming from the time sequence n domain to the z domain is represented as z n Domain z Domain x n N z x k n k X z k 0 N x k z k k 0 A sequence and its z transform are said to form a z transform pair and are denoted z x n X z 7 2 In the sequence or n domain the independent variable is n In the z domain the independent variable is z Example x n n n 0 Using the definition X z N x k z k 0 k N k n 0 z k z n0 k 0 Thus z n n0 z ECE 2610 Signals and Systems n0 7 2 The z Transform and Linear Systems Example x n 2 n 3 n 1 5 n 2 2 n 3 By inspection we find that X z 2 3z Example X z 4 5z 2 z 1 3 5z 2z 2 2z 3 4 By inspection we find that x n 4 n 5 n 2 n 3 2 n 4 What can we do with the z transform that is useful The z Transform and Linear Systems The z transform is particularly useful in the analysis and design of LTI systems The z Transform of an FIR Filter We know that for any LTI system with input x n and impulse response h n the output is y n x n h n 7 3 We are interested in the z transform of h n where for an FIR filter h n M bk n k 7 4 k 0 ECE 2610 Signals and Systems 7 3

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