Signals and Systems - Electrical Engineering

9.4 One-Sided Z-Transform 535

n It is important to remember the relations

H(z)=Z[h[n]]=

Y(z)

X(z)

=

Z[y[n]]

Z[x[n]]

where H(z)is the transfer function and h[n]is the impulse response of the system, with x[n]as the input

and y[n]as the output.

nExample 9.10

Consider a discrete-time IIR system represented by the difference equation

y[n]=0.5y[n−1]+x[n] (9.25)

withx[n] as the input andy[n] as the output. Determine the transfer function of the system and

from it find the impulse and the unit-step responses. Determine under what conditions the system

is BIBO stable. If stable, determine the transient and steady-state responses of the system.

Solution

The system transfer function is given by

H(z)=

Y(z)

X(z)

=

1

1 −0.5z−^1

and its impulse response is

h[n]=Z−^1 [H(z)]=0.5nu[n]

The response of the system to any input can be easily obtained by the transfer function. If the input

isx[n]=u[n], we have

Y(z)=H(z)X(z)=

1

( 1 −0.5z−^1 )( 1 −z−^1 )

=

− 1

1 −0.5z−^1

+

2

1 −z−^1

so that the total solution is

y[n]=−0.5nu[n]+ 2 u[n]

From the transfer functionH(z)of the LTI system, we can test the stability of the system by finding

the location of its poles—very much like in the analog case. An LTI system is BIBO stable if and

only if the impulse response of the system is absolutely summable—that is,

∑

n

|h[n]|≤∞