Signals and Systems - Electrical Engineering

11.4 IIR Filter Design 673

n Design a prototype low-pass discrete filter and then transform it into the desired discrete filter
[1,55].

The first approach has the advantage that the analog frequency transformations (See Chapter 6) are
available and well understood. Its drawback appears when applying the bilinear transformation as it
may cause undesirable warping in the higher frequencies. So the second approach will be used.

Given a prototype low-pass filterH`p(Z), we wish to transform it into a desired filterH(z), which is
typically another low-pass, band-pass, high-pass, or stopband filter. The transformation

G(z−^1 )=Z−^1 (11.41)

should preserve the rationality and the stability of the low-pass prototype. Accordingly,G(z−^1 )
should

n Be rational to preserve the rationality.
n Map the inside of the unit circle in the Z-plane into the inside of the unit circle in thez-plane to
preserve stability.
n Map the unit circle|Z|=1 into the unit circe|z|=1.

IfZ=Rejθandz=rejω, the third condition onG(z−^1 )corresponds to

G(e−jω)=|G(e−jω)|ej∠(G(e

−jω))

= (^1) ︸︷︷e−jθ︸
unit circle in Z-plane

(11.42)

indicating that the frequency transformationG(z−^1 )has the characteristics of an all-pass filter, with
magnitude|G(e−jω)|=1 and phase∠G(e−jω)=−θ.

Using the general form of the transfer function of an all-pass filter (ratio of two equal-order polyno-
mials with poles and zeros being the conjugate inverse of each other), we obtain the general form of
the rational transformation as

Z−^1 =G(z−^1 )=K

∏

k

z−^1 −αk

1 −α∗kz−^1

(11.43)

where|αk|<1 andKis±1. The values ofKand{αk}are obtained from the prototype and the desired
filters.

Low-Pass to Low-Pass Transformation
We wish to obtain the transformationZ−^1 =G(z−^1 )to convert a prototype low-pass filter into a
different low-pass filter. The all-pass transformation should be able to expand or contract the fre-
quency support of the prototype low-pass filter but keep its order. Thus, it should be a ratio of linear
transformations,

Z−^1 =K

z−^1 −α

1 −αz−^1

(11.44)

for some parametersKandα. Since the zero frequency in the Z-plane is to be mapped into the zero
frequency in thez-plane, we letZ=z=1 in the transformation to getK=1. To obtainα, we let