Signals and Systems - Electrical Engineering

11.6 Realization of Discrete Filters 697

x[n] w[n]

z−^1

−

+

+

z−^1

y[n]

z−^1

+

−

+

1

1

1

−0.4

1

0.2

y 1 [n] v[n]

FIGURE 11.30
Cascade realization ofH(z)=[( 1 +z−^1 )/( 1 +z−^2 )] [( 1 +0.2z−^1 )/( 1 −0.4z−^1 )].

Parallel Realization
In this case the given transfer functionH(z)is represented as a partial fraction expansion,

H(z)=

B(z)

A(z)

=C+

∑r

i= 1

Hi(z) (11.74)

whereCis a constant and therfiltersHi(z)are first- or second-order systems with real coefficients
that are implemented with the direct form II.

The constantCin the expansion is needed when the numerator (in positive powers ofz) is of larger
or equal order than the denominator. If the numerator is of larger order than the denominator, the
filter is noncausal. To illustrate this, consider a first-order filter with a transfer function where the
numerator is of second order (in terms of positive powers ofz)

H(z)=

Y(z)

X(z)

=

b 0 z^2 +b 1 z+b 2

z+a 1

=

b 0 z+b 1 +b 2 z−^1

1 +a 1 z−^1

The difference equation representing this system is

y[n]=−a 1 y[n−1]+b 0 x[n+1]+b 1 x[n]+b 2 x[n−1]

requiring a future inputx[n+1] to compute the presenty[n] (i.e., corresponding to a noncausal
filter).

The cascade and parallel realizations are shown in Figure 11.31.

nExample 11.18

Let

H(z)=

3 +3.6z−^1 +0.6z−^2

1 +0.1z−^1 −0.2z−^2

=

3 z^2 +3.6z+0.6

z^2 +0.1z−0.2

Obtain a parallel realization.