Biomedical Signal Processing - Computational Intelligence in Biomedical Engineering - page 51

s ( n )

b 0

x ( n )

y ( n )

+ +

z − 1

− a 0

b 1

− a 1

b 2

− a N

b N

S N

FIGURE 2.15
A canonical IIR digital filter configuration.

with internal states, s i ( n )= s n−i for all i =0 , 1 ,...,N . The states themselves
are updated via the equation s + i ( n +1)= s i− 1 ( n ) for all i =0 , 1 ,...,N .An
example of a canonical IIR filter construction is depicted in Figure 2.15.

2.6.1.3 Cascaded IIR Filter
In the cascaded IIR filter, a number of smaller IIR filters are combined or
cascaded to form a single filter designed by first decomposing the direct IIR
Equation 2.130 into products or partial fractions. The cascaded IIR filter
transfer function can be written as a product of the smaller filter transfer
functions, that is,

K

H ( z )= H k ( z )= H 1 ( z ) H 2 ( z ) ...H K ( z )

k =1

The cascaded IIR filter in Figure 2.16, for example, has transfer function

1+ z − 1

H ( z )=

(1 − (1 / 2) z − 1 )(1 − z − 1 )

2.6.2 Design of IIR Digital Filters
The digital IIR filter can be obtained from the corresponding analog filter
that has been designed to specification. There are two common methods to
convert the transfer function of the analog IIR filter to their digital counter-
parts, namely, the impulse-invariant transformation and the bilinear trans-
formation (Tompkins, 1993). These methods convert the filter pole and zero
coecients to the corresponding poles and zeroes on the unit z -circle.