Signals and Systems - Electrical Engineering

11.6 Realization of Discrete Filters 691

Direct Form I
Thedirect form Iis the implementation of the above difference Equation (11.63) as is, by means of
delays, constant multipliers, and adders. Assuming the inputx[n] is available, thenM−1 delays are
needed to generate the delayed inputs{x[n−k]}fork=1,...,M−1. Likewise, the output compo-
nents require additionalN−1 delays. Thus, a direct form I realization requiresM+N−2 delays for
an(N− 1 )th-order difference equation. In terms of number of delays, the direct form I is the least
efficient realization.

nExample 11.14

Use the direct form I to realize the transfer function

H(z)=

1 +1.5z−^1

1 +0.1z−^1

of a discrete filter.

Solution

The transfer function corresponds to a system with a first-order difference equation

y[n]=x[n]+1.5x[n−1]−0.1y[n−1]

soM=N=2 and this equation can be realized as shown in Figure 11.26 withM+N− 2 = 2

delays.

The difference equation, and thus the transfer function, for this filter can be easily obtained from

the realization. The above realization is nonminimal since it uses two delays to represent a first-

order system.

The output of the above realization is seen to be

y[n]=−0.1y[n−1]+x[n]+1.5x[n−1]

so that the transfer function is

H(z)=

Y(z)

X(z)

=

1 +1.5z−^1

1 +0.1z−^1

by letting a delay be represented byz−^1 in thez-domain. n

FIGURE 11.26
Direct form I realization of
H(z)=( 1 +1.5z−^1 )/( 1 +0.1z−^1 ).

+ +

−

x[n] 1.5 y[n]

0.1

z−^1

z−^1