Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 315

For the bandpass filter and bandstop filter :

tan u l 2 tan u h 2

2
T

2
T

u al ¼ ; u ah ¼ (8.19)

u al u a p , W ¼ u ah u al
2. Perform the prototype transformation using the lowpass prototype H P ðsÞ .

and u 0 ¼

From lowpass to lowpass : HðsÞ¼H P ðsÞj s ¼ u a (8.20)

From lowpass to highpass : HðsÞ¼H P ðsÞj s ¼ u a
s

(8.21)

From lowpass to bandpass : HðsÞ¼H P ðsÞj s ¼ s (8.22)

2 2

0

þu

sW

From lowpass to bandstop : HðsÞ¼H P ðsÞj s ¼ (8.23)

sW

2 0

s þu

3. Substitute the BLT to obtain the digital filter

HðzÞ¼HðsÞj s ¼ (8.24) 2

T

z 1
zþ 1

Table 8.2 lists MATLAB functions for the BLT design.
We illustrate the lowpass filter design procedure in Example (8.6). Other types of filters, such as
highpass, bandpass, and bandstop, will be illustrated in the next section.

EXAMPLE 8.6
The normalized lowpass filter with a cutoff frequency of 1 rad/sec is given as

H P ðsÞ¼ 1
s þ 1

a. Use the given H P ðsÞ and the BLT to design a corresponding digital IIR lowpass filter with a cutoff frequency of
15 Hz and a sampling rate of 90 Hz.
b. Use MATLAB to plot the magnitude response and phase response of HðzÞ.
Solution:
a. First, we obtain the digital frequency as

u d ¼ 2pf ¼ 2pð15Þ¼30p radians=second; and T ¼ 1=f s ¼ 1=90 sec

We then follow the design procedure:
1. First calculate the prewarped analog frequency as