Infinite Impulse Response Filters - Introduction to Digital Signal Processing and Filter Design - page 225

10

0

− 10

− 20

− 30

− 40

− 50

− 60

− 70

− 80
10 − 1

10 0 10 1

Frequency in rad./sec

Figure 4.20 Magnitude response of the analog prototype lowpass filter in Example 4.8.

The transfer function of the prototype lowpass filter is therefore given by

2 . 5317

H( p ) = (4.90)

p 4

+ 3 . 296 p 3 + 5 . 4325 p 2 + 5 . 24475 p + 2 . 5317

The magnitude response of this lowpass filter is plotted in Figure 4.20.
Next we substitute p =

( 1 / 0 . 547 ) [ (s 2

1 . 705 2 )/s ], in (4.90) and after sim-

+

plifying, the resulting transfer function is

0 . 2267 s 4
D(s)

H(s) =

where D(s) is given by
(s 8

+ 1 . 8030 s 7 + 13 . 2535 s 6 + 16 . 5824 s 5 + 60 . 3813 s 4

(4.91)

+ 48 . 205 s 3 + 112 . 0006 s 2

+ 44 . 2926 s + 71 . 4135 )

+ 1 ) ]wechosein
this example, on this H(s) , and simplify the transfer function H(z) of the digital
filter to

Now we apply the bilinear transformation s = 2[ (z − 1 )/(z

3 . 6272 z 8 − 14 . 5088 z 6 + 21 . 7632 z 4 − 14 . 5088 z 2

+ 3 . 6272

( 3825 z 8

H(z) =

− 4221 z 7 + 13127 z 6 − 9857 z 5 + 15753 z 4 − 7615 z 3

+ 7849 z 2 − 1934 z + 1354 )

(4.92)