![]() The z-transform provides the framework for this mathematics. A series of mathematical conversions are then used to obtain the desired digital filter. Recursive digital filters are often designed by starting with one of the classic analog filters, such as the Butterworth, Chebyshev, or elliptic. 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. Correspondingly, the z-transform deals with difference equations, the z-domain, and the z-plane. ![]() The Laplace transform deals with differential equations, the s-domain, and the s-plane. 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. Just as analog filters are designed using the Laplace transform, recursive digital filters are developed with a parallel technique called the z-transform.
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