Infinite Impulse Response Filter Design - Digital Signal Processing: Fundamentals and Applications - page 304

(2) performing analog filter design, and (3) applying bilinear transformation (which will be introduced
in the next section) and verifying the frequency response.

8.2.1 Analog Filters Using Lowpass Prototype Transformation
Before we begin to develop the BLT design, let us review analog filter design using lowpass prototype
transformation . This method converts an analog lowpass filter with a cutoff frequency of 1 radians per
second, called the lowpass prototype, into practical analog lowpass, highpass, banspass, and bandstop
filters with specified frequencies.
Letting H P ðsÞ be a transfer function of the lowpass prototype, the transformation of the lowpass
prototype into a lowpass filter is given in Figure 8.2 .
As shown in Figure 8.2 , H LP ðsÞ designates the analog lowpass filter with a cutoff frequency u c
radians/second. The lowpass prototype to lowpass filter transformation substitutes s in the lowpass
prototype function H P ðsÞ with s=u c ,where v is the normalized frequency of the lowpass prototype and u c
is the cutoff frequency of the lowpass filter. Let us consider the following first-order lowpass prototype:

1
s þ 1

H P ðsÞ¼

(8.1)

Its frequency response is obtained by substituting s ¼ jv into Equation (8.1) , that is,

1
jv þ 1

H P ðjnÞ¼

and the magnitude gain is

1
1 þ v

jH P ðjvÞj ¼ p (8.2)

2

We compute the gains at v ¼ 0, v ¼ 1, v ¼ 100, v ¼ 10 ; 000 to obtain 1 ; 1 = p ,0 : 0995, and 0.01,

respectively. The cutoff frequency gain at v ¼ 1 equals 1 = p , which is equivalent to 3 dB, and

the direct-current (DC) gain is 1. The gain approaches zero when the frequency goes to v ¼þ N .
This verifies that the lowpass prototype is a normalized lowpass filter with a normalized cutoff
frequency of 1. Applying the prototype transformation s ¼ s=u c

in Figure 8.2 , we get an analog

lowpass filter with a cutoff frequency of u c :

1
s=u c þ 1 ¼

u c
s þ u c

HðsÞ¼ (8.3)

Hj

LP ()

Hv

P ()

s

s

c

v

0 1 0

c

HsHs
LP

() () /

P

ss c

FIGURE 8.2