Digital Signals and Systems - Digital Signal Processing: Fundamentals and Applications - page 69

b. The block diagram of the linear time-invariant system is shown in Figure 3.15 .
c. The system output can be rewritten as

yðnÞ¼hð0ÞxðnÞþhð1Þxðn 1Þ

xn ( )

yn ()

hn ( ) 0.5 ( ) 0.25 ( n n 1)

FIGURE 3.15

The system block diagram for Example 3.7.

From the result Example 3.7, it is noted that the difference equation does not have the past output
terms, yðn 1 Þ , . , yðn NÞ , that is, the corresponding coefficients a 1 , . , a N , are zeros, and the impulse
response hðnÞ has a finite number of terms. We call this system a finite impulse response (FIR) system. In
general, Equation (3.12) contains the past output terms and the resulting impulse response h ( n ) has an
infinite number of terms. We can express the output sequence of a linear time-invariant system using its
impulse response and inputs as

yðnÞ¼ / þ hð 1 Þxðn þ 1 Þþhð 0 ÞxðnÞþhð 1 Þxðn 1 Þþhð 2 Þxðn 2 Þþ / (3.15)

Equation (3.15) is called the digital convolution sum , which will be explored in a later section. We can
verify Equation (3.15) by substituting the impulse sequence xðnÞ¼dðnÞ to get the impulse response
hðnÞ¼ / þ hð 1 Þdðn þ 1 Þþhð 0 ÞdðnÞþhð 1 Þdðn 1 Þþhð 2 Þdðn 2 Þþ /

where . hð 1 Þ , hð 0 Þ , hð 1 Þ , hð 2 Þ . are the amplitudes of the impulse response at the corresponding
time indices. Now let us look at another example.

EXAMPLE 3.8
Consider the difference equation
yðnÞ¼0:25yðn 1ÞþxðnÞ for n 0 and yð1Þ¼0

a. Determine the unit-impulse response hðnÞ.
b. Draw the system block diagram.
c. Write the output using the obtained impulse response.
d. For a step input xðnÞ¼uðnÞ, verify and compare the output responses for the first three output samples using
the difference equation and digital convolution sum (Equation ( 3.15 )).
Solution:
a. Let xðnÞ¼dðnÞ, then

hðnÞ¼0:25hðn 1ÞþdðnÞ

To solve for hðnÞ, we evaluate
hð0Þ¼0:25hð1Þþdð0Þ¼0:25 0 þ 1 ¼ 1
hð1Þ¼0:25hð0Þþdð1Þ¼0:25 1 þ 0 ¼ 0:25