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03-JDSP-Z-Transforms

# 03-JDSP-Z-Transforms - CHAPTER 3 3 THE Z TRANSFORM 3.1...

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CHAPTER - 3 3. THE Z TRANSFORM 3.1. INTRODUCTION n this chapter, we review the z transform and its properties. We then use the z transform to define the transfer function of a discrete-time system. At the end of the chapter we give two sets of problems; theory problems on the z transform and a series of J-DSP computer exercises. Discrete-time signals are described in the transform domain with the z transform and the discrete- time Fourier transform (DTFT). These two transformations have similar roles as the Laplace transform and the CFT for analog signals respectively. The z transform is defined as ( ) ( ). n n X z x n z =−∞ = (3.1) where z is a complex variable. For a causal (right-handed) signal, i.e., if x(n)=0 for n<0 then the lower index on the sum starts from 0. The z transform describes the signal in the complex z domain and for finite-length causal signals the transform converges everywhere in z except for z=0 . Note that a unique z transform pair requires that the range of values of z for which the z transform exists, i.e. the region of I

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convergence (ROC), is specified. For causal signals the ROC extends outwards from the outermost pole of the z domain function. Example 3 . 1 : Consider the z transform of the unit impulse, δ (n) ( ) 1 0 0 n for n elsewhere δ = = = (3.2) ( ). 1 ( ) 1 n n n z hence n ROC z δ δ =−∞ = Example 3 . 2 : Consider a finite-length signal ( ) n x n 5 0 . 2 5 1 . 5 0 . 5 2 . 1 1 0 2 3 4 5 This signal is non-zero only between the discrete-time indexes –2 to 3. Mathematically this signal can be written in terms of unit impulses as follows ( ) .5 ( 2) 1.5 ( 1) 2 ( ) ( 1) .5 ( 2) 2.5 ( 3) x n n n n n n n δ δ δ δ δ δ = − + + + + + The z transform of this finite-length non-causal signal is given by 2 1 2 3 ( ) .5 1.5 2 .5 2.5 X z z z z z z = − + + + The z transform converges for all values of z except z=0 and z= , i.e., ROC: z 0, z .