Signals and Systems - Electrical Engineering

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

The cascade realization of an FIR filter is based on the representation ofH(z)in Equation (11.75) as

a cascade of first- and second-order filters—that is, we let

H(z)=

∏r

i= 1

Hi(z)

where

Hi(z)=boi+b 1 iz−^1 or

Hi(z)=boi+b 1 iz−^1 +b 2 iz−^2

nExample 11.19

Provide the cascade realization of an FIR filter with transfer function

H(z)= 1 + 3 z−^1 + 3 z−^2 +z−^3

Solution

The transfer function is factored as

H(z)=( 1 + 2 z−^1 +z−^2 )( 1 +z−^1 )

which can be realized as the cascade of two FIR filters,

y 1 [n]=x[n]+x[n−1]

y[n]=y 1 [n]+ 2 y 1 [n−1]+y 1 [n−2]

which are realized as shown in Figure 11.34.

FIGURE 11.34

Cascade realization of FIR filter.

+

+

z−^1

z−^1 z−^1

x[n] x[n−1]

y 1 [n] y 1 [n−1] y 1 [n−2]

y[n]

2

n