Signals and Systems - Electrical Engineering

698 C H A P T E R 11: Introduction to the Design of Discrete Filters

FIGURE 11.31

(a) Cascade and (b) parallel realizations of IIR

filters.

Hi(z) first- or second-order direct form II realization

x[n] H 1 (z) H 2 (z) ··· HN(z) y[n]

···

x[n] + y[n]

G 1 (z)

G 2 (z)

GM(z)

Gi(z) first- or second-order direct form II realization

(b)

(a)

Solution

The transfer functionH(z)is not proper rational, in either positive or negative powers ofz, and the

poles arez=−0.5 andz=0.4. Thus, the transfer function can be expanded as

H(z)=A 1 +

A 2

1 +0.5z−^1

+

A 3

1 −0.4z−^1

In this case we needA 1 because the numerator, in positive as well as in negative powers ofz, is of

the same order as the denominator. We then have

A 1 =H(z)|z= 0 =− 3

A 2 =H(z)( 1 +0.5z−^1 )|z− (^1) =− 2 =− 1
A 3 =H(z)( 1 −0.4z−^1 )|z− (^1) =2.5= 7
Letting
H 1 (z)=

− 1

1 +0.5z−^1

H 2 (z)=

7

1 −0.4z−^1

we obtain the parallel realization forH(z)shown in Figure 11.32. n