11.6 Realization of Discrete Filters 699
FIGURE 11.32
Parallel realization forH(z)=( 3 +3.6z−^1 +
0.6z−^2 )/( 1 +0.1z−^1 −0.2z−^2 ).
x[n] y[n]
3
7
0.5
0.4
+
−
+
+
− +
−
z−^1
z−^1
FIGURE 11.33
Direct form realization of FIR filter.
++
+ +
z−^1 z−^1 z−^1
b 0 b 1 b 2 b 3
x[n]
y[n]
11.6.2 Realization of FIR Filters
The realization of FIR filters can be done using direct and cascade forms. Since these filters are
nonrecursive, there are no different direct forms and there is no way to implement FIR filters in
parallel.
The direct realization of an FIR filter consists in realizing the input–output equation using delays,
constant multipliers, and summers. For instance, if the transfer function of an FIR filter is given by
H(z)=
∑M
k= 0
bkz−k (11.75)
the Z-transform of the filter output can be written as
Y(z)=H(z)X(z)
whereX(z)is the Z-transform of the filter input. In the time domain we have
y[n]=
∑M
k= 0
bkx(n−k)
which can be realized as shown in Figure 11.33 in the case ofM=3.
Notice thatMis the number of delays needed and that there areM+1 taps, which has given the
name oftapped filtersto FIR filters realized this way.
