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# ztrans - ECE 421 Sum 2010 Notes Set 3 Z-Transforms 1...

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ECE 421 - Sum 2010 Notes Set 3 : Z-Transforms 1 INTRODUCTION The Z -transform is the “frequency-domain” analytical tool for digital (discrete-time) sys- tems. It serves a purpose for digital systems similar to the purpose served by the Laplace transform for analog (continuous time) systems. The theory of Z -transforms is very broad and we will only discuss here some concepts needed for Digital Signal Processing. How- ever, we will see that even the concepts covered in this Notes set are very useful for many applications. If future work, we will see that the z -plane and the Z -transform allow us to develop rigorous methods for designing digital filters. This is similar to using the s -plane and the Laplace transform for designing analog filters. Some of these methods map existing analog filters onto digital structures, while other methods are based entirely in the digital domain. We will also study digital applications other than digital filters. One is called a Digital Equal- izer, which is the topic of Notes Set 3. The present Notes Set gives us a foundation for understanding that application.

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ECE 421 - Sum 2010 Notes Set 3 : Z-Transforms 2 DEFINITION The Z -transform definition arises naturally in considering the computation of the output of a digitale linear system. Let a digital linear system (like a digital filter) have input x m and output y m . A block diagram of this system is shown below, where the system is quantified by its impulse response h m : h m y m [ ] [ ] x m [ ] We know from our previous work that the output is computed by the convolution sum: y m h m x m n h n x m n 1 Suppose T s is the time sampling interval and x m is the time-domain complex exponential given by x m e smT s 2 where the variable s is a general complex value, like the s which was used in Laplace transforms. With x m defined in (2), we see that x m n would become x m n e s m n T s e smT s e snT s 3 Substituting (3) into (1) therefore produces y m n h n e smT s e snT s e smT s n h n e snT s 4 Now define the new variable z as z e sT s 5 Using the definition in (5), equation (4) can then be simplified as y m z m H z 6 where H z is the Z -transform of the impulse repsonse h m : H z m h m z m 7 It does not matter that m is the variable of summation in (7) and that n is the variable of summation in (4). In each case, the variable of summation “sums out”, leaving a function of the variable z .
ECE 421 - Sum 2010 Notes Set 3 : Z-Transforms 3 GEOMETRIC SERIES Many computations of z -transforms can be performed by using the Geometric Series sum- mation. As an example, let’s compute the z -transform of g m a m u m 8 In (8), u m is the unit step and a is a constant having 0 a 1. The z -transform of g m , denoted by G z or Z g m , is defined as G z Z g m m g m z m 9 In (9), z is a continuous, complex-valued variable. We sometimes say z spans the entire complex plane. To compute G z , substitute (8) into (9), giving G z m a m u m z m m 0 a m z m m 0 az 1 m 10

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