Signals and Systems - Electrical Engineering

538 C H A P T E R 9: The Z-Transform

H 1 (z) H 2 (z) = =

x[n] y[n]

H 1 (z)H 2 (z)

x[n] y[n]

H 1 (z)

x[n] y[n]

H 2 (z)

x[n] y[n]

H 1 (z)+H 2 (z)

H 1 (z)

H 2 (z)

+

x[n] y[n]

=

(a)

(b)

(c)

=

H 1 (z) x[n] y[n]

H 2 (z)

x[n] y[n]

+

−

e[n]

w[n]

H 1 (z)

1+H 1 (z)H 2 (z)

FIGURE 9.7

Connections of LTI systems: (a) cascade, (b) parallel, and (c) negative feedback.

showing that there is no effect on the overall system if we interchange the two systems (see

Figure 9.7(a)). Recall that such a property is only valid for LTI systems. In the parallel system, as

in Figure 9.7(b), both systems have the same input and the output is the sum of the output of the

subsystems. The overall transfer function is

H(z)=H 1 (z)+H 2 (z) (9.27)

Finally, the negative feedback connection of the two systems shown in Figure 9.7(c) gives in the

feedforward path

Y(z)=H 1 (z)E(z) (9.28)

where Y(z)=Z[y[n]] is the Z-transform of the output y[n] and E(z)=X(z)−W(z) is the

Z-transform of the error functione[n]=x[n]−w[n]. The feedback path gives that

W(z)=Z[w[n]]=H 2 (z)Y(z)

ReplacingW(z)inE(z), and then replacingE(z)in Equation (9.28), we obtain the overall transfer

function

H(z)=

Y(z)

X(z)

=

H 1 (z)

1 +H 1 (z)H 2 (z)

(9.29)