Digital Signals and Systems - Digital Signal Processing: Fundamentals and Applications - page 68

3.3.2 System Representation Using Its Impulse Response
A linear time-invariant system can be completely described by its unit-impulse response, which is
defined as the system response due to the impulse input dðnÞ with zero initial conditions, depicted in
Figure 3.13 .
With the obtained unit-impulse response hðnÞ , we can represent the linear time-invariant system as
shown in Figure 3.14 .

() n h ()

Linear time-invariant system

FIGURE 3.13

Unit-impulse response of a linear time-invariant system.

x ( n ) y ( n )

h ( n )

FIGURE 3.14

Representation of a linear time-invariant system using the impulse response.

EXAMPLE 3.7
Assume we have a linear time-invariant system
yðnÞ¼0:5xðnÞþ0:25xðn 1Þ

with an initial condition xð1Þ¼0.
a. Determine the unit-impulse response hðnÞ.
b. Draw the system block diagram.
c. Write the output using the obtained impulse response.
Solution:
a. According to Figure 3.13 , let xðnÞ¼dðnÞ, then
hðnÞ¼yðnÞ¼0:5xðnÞþ0:25xðn 1Þ¼0:5dðnÞþ0:25dðn 1Þ

Thus, for this particular linear system, we have

<
:

0:5 n ¼ 0

hðnÞ¼ 0:25 n ¼ 1

0 elsewhere