[picture of book]

The Scientist and Engineer's
Guide to Digital Signal Processing

by Steven W. Smith
California Technical Publishing

ISBN 0-9660176-3-3 (1997)



Chapter 33. The z-Transform
  • The Nature of the z-Domain
  • Analysis of Recursive Systems
  • Cascade and Parallel Stages
  • Spectral Inversion
  • Gain Changes
  • Chebyshev-Butterworth Filter Design
  • Summary of the key concepts
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.



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