Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 316

Table 8.2 MATLAB Functions for Bilinear Transformation Design

Lowpass to lowpass: HðsÞ¼H P ðsÞj s¼

s
u a

lp2lp(Bp, Ap, wa)
Lowpass to highpass:

[B, A]

>> [

HðsÞ¼H P ðsÞj s¼ u a
s

lp2hp(Bp, Ap, wa)
Lowpass to bandpass:

[B, A]

>> [

HðsÞ¼H P ðsÞj s¼ s 2 þu 0
sW

lp2bp(Bp, Ap, w0, W)
Lowpass to bandstop:

[B, A]

>> [

HðsÞ¼H P ðsÞj s¼ sW

s 2 þu 0

>> [B, A] [ lp2bs(Bp, Ap, w0, W)
Bilinear transformation to achieve the digital filter:
>> [b, a] [ bilinear(B, A, fs)
Plot of the magnitude and phase frequency responses of the digital filter:
>> freqz(b, a, 512, fs)
Definitions of design parameters:
Bp ¼ vector containing the numerator coefficients of the lowpass prototype

Ap ¼ vector containing the denominator coefficients of the lowpass prototype

wa ¼ cutoff frequency for the lowpass or highpass analog filter (rad/sec)

w0 ¼ center frequency for the bandpass or bandstop analog filter (rad/sec)

W ¼ bandwidth for the bandpass or bandstop analog filter (rad/sec)

B ¼ vector containing the numerator coefficients of the analog filter

A ¼ vector containing the denominator coefficients of the analog filter

b ¼ vector containing the numerator coefficients of the digital filter

a ¼ vector containing the denominator coefficients of the digital filter

fs ¼ sampling rate (samples/sec)

u d T
2

30p=90
2

u a ¼ T tan

2
1=90 tan

¼

that is, u a ¼ 180 tanðp=6Þ¼180 tanð30 Þ¼103:92 rad/sec.
2. Then perform the prototype transformation (lowpass to lowpass) as follows:

1 u a

s þ u a

HðsÞ¼H P ðsÞj s¼ s

u a ¼

u a þ 1 ¼

s

This yields an analog filter:

HðsÞ¼ 103:92
s þ 103:92

3. Apply the BLT, which yields

s¼ T z1
zþ1

HðzÞ¼ 103:92
s þ 103:92