# CH33 - CHAPTER 33 The z-Transform Just as analog filters...

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605 CHAPTER 33 X ( s ) m 4 t ’& 4 x ( t ) e & st dt The z-Transform Just as analog filters are designed using the Laplace transform, recursive digital filters are developed with a parallel technique called the z-transform. The overall strategy of these two transforms is the same: probe the impulse response with sinusoids and exponentials to find the system's poles and zeros. The Laplace transform deals with differential equations, the s-domain, and the s-plane. Correspondingly, the z-transform deals with difference equations, the z-domain, and the z-plane. However, the two techniques are not a mirror image of each other; the s-plane is arranged in a rectangular coordinate system, while the z-plane uses a polar format. Recursive digital filters are often designed by starting with one of the classic analog filters, such as the Butterworth, Chebyshev, or elliptic. A series of mathematical conversions are then used to obtain the desired digital filter. The z-transform provides the framework for this mathematics. The Chebyshev filter design program presented in Chapter 20 uses this approach, and is discussed in detail in this chapter. The Nature of the z-Domain To reinforce that the Laplace and z-transforms are parallel techniques, we will start with the Laplace transform and show how it can be changed into the z- transform. From the last chapter, the Laplace transform is defined by the relationship between the time domain and s-domain signals: where and are the time domain and s-domain representation of the x ( t ) X ( s ) signal, respectively. As discussed in the last chapter, this equation analyzes the time domain signal in terms of sine and cosine waves that have an exponentially changing amplitude. This can be understood by replacing the

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The Scientist and Engineer's Guide to Digital Signal Processing 606 X ( F , T ) m 4 t ’& 4 x ( t ) e & F t e & j T t dt X ( F , T ) j 4 n 4 x [ n ] e & F n e & j T n y [ n ] e & F n y [ n ] r -n or complex variable, s , with its equivalent expression, . Using this alternate F % j T notation, the Laplace transform becomes: If we are only concerned with real time domain signals (the usual case), the top and bottom halves of the s-plane are mirror images of each other, and the term, , reduces to simple cosine and sine waves. This equation identifies each e & j T t location in the s-plane by the two parameters, F and T . The value at each location is a complex number, consisting of a real part and an imaginary part. To find the real part, the time domain signal is multiplied by a cosine wave with a frequency of T , and an amplitude that changes exponentially according to the decay parameter, F . The value of the real part of X ( F , T ) is then equal to the integral of the resulting waveform. The value of the imaginary part of is found in a similar way, except using a sine X ( F , T ) wave. If this doesn't sound very familiar, you need to review the previous chapter before continuing.
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CH33 - CHAPTER 33 The z-Transform Just as analog filters...

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