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.
CHAPTER
The z-Transform
33

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