3.1.3 Impulse Response of Digital Systems
If the input signal to the DSP system is the unit-impulse sequence dn defined in (3.1.1),
then the output signal, h(n), is called the impulse response of the system. The impulse
response plays a very important role in the study of DSP systems. For example, consider
a digital system with the I/O equation
yn b
0
xn b
1
xn À 1 b
2
xn À 2:
3:1:15
The impulse response of the system can be obtained by applying the unit-impulse
sequence dn to the input of the system. The outputs are the impulse response coeffi-
cients computed as follows:
h0 y0 b
0
Á 1 b
1
Á 0 b
2
Á 0 b
0
h1 y1 b
0
Á 0 b
1
Á 1 b
2
Á 0 b
1
h2 y2 b
0
Á 0 b
1
Á 0 b
2
Á 1 b
2
h3 y3 b
0
Á 0 b
1
Á 0 b
2
Á 0 0
. . .
Therefore the impulse response of the system defined in (3.1.15) is fb
0
, b
1
, b
2
, 0, 0, . . .g.
The I/O equation given in (3.1.15) can be generalized as the difference equation with
L parameters, expressed as
yn b
0
xn b
1
xn À 1 Á Á Á b
LÀ1
xn À L 1
LÀ1
l0
b
l
xn À l :
3:1:16
Substituting xn dn into (3.1.16), the output is the impulse response expressed
as
À 1
hn
LÀ1
l0
b
l
dn À l
b
n
n = 0 , 1, ..., L
0
otherwise.
&
3:1:17
Therefore the length of the impulse response is L for the difference equation defined in
(3.1.16). Such a system is called a finite impulse response (FIR) system (or filter). The
impulse response coefficients, b
l
, l 0, 1, . . . , L À 1, are called filter coefficients
(weights or taps). The FIR filter coefficients are identical to the impulse response
coefficients. Table 3.2 shows the relationship of the FIR filter impulse response h(n)
and its coefficients b
l
.
3.2 Introduction to Digital Filters
As shown in (3.1.17), the system described in (3.1.16) has a finite number of non-zero
impulse response coefficients b
l
, l 0, 1, . . . , L À 1. The signal-flow diagram of the
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