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605 CHAPTER33X(s)'m4t'&4x(t)e&stdtThe z-TransformJust as analog filters are designed using the Laplace transform , recursive Digital filters aredeveloped with a parallel technique called the z- transform . The overall strategy of these twotransforms is the same: probe the impulse response with sinusoids and exponentials to find thesystem'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-planeis arranged in a rectangular coordinate system, while the z-plane uses a polar format. Recursivedigital filters are often designed by starting with one of the classic analog filters, such as theButterworth, Chebyshev, or elliptic. A series of mathematical conversions are then used to obtainthe desired Digital filter.
608 The Scientist and Engineer's Guide to Digital Signal Processing z ’re &jT EQUATION 33-1 The z-transform. The z-transform defines the …
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