Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 91

2.6.3 Classification of Digital Filters

Digital filters can be classified into two major categories according to their impulse
response length:

1. A finite impulse response (FIR) digital filter: has an impulse response with a
finite number of non-zero samples.
2. An infinite impulse response (IIR) digital filter: has an impulse response with an
infinite number of non-zero samples.

2.6.4 FIR Digital Filters

2.6.4.1 Structure and Implementation of FIR Filters

A causal FIR filter can be specified mathematically by the following difference
equation [see Fig. ( 2.16 )]:

y ð n Þ¼ h ð n Þ x ð n Þ;
¼ X

N 1

h ð k Þ x ð n k Þ

k ¼ 0

¼ h 0 x ð n Þþ h 1 x ð n 1 Þþþ h N 1 x ½ n ð N 1 Þ

where h k is used in place of h(k) for notational simplicity. Taking the z-transform
of both sides yields:

Y ð z Þ¼ h 0 X ð z Þþ h 1 z 1 X ð z Þþþ h N 1 z ð N 1 Þ X ð z Þ½ Using Tables :

Hence, the transfer function is given by:

H ð z Þ¼ Y ð z Þ= X ð z Þ¼ h 0 þ h 1 z 1 þþ h N 1 z ð N 1 Þ ð 2 : 21 Þ
Remembering that a z -1 corresponds to a single sample delay, one can imple-
ment the causal FIR filter using delay elements and digital multipliers as shown in
Fig. ( 2.16 ).

2.6.4.2 Software Implementation of FIR Filters

The I/O relation in ( 2.6 ) could easily be implemented using a software program on
a DSP chip or a digital computer. However, implementation according to the
definition in ( 2.6 ) would require O(N h N x ) operations. As suggested in
Sect. 2.4.1.5 , a more efficient implementation is possible in the frequency domain
with the use of FFTs. One uses the relations: