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116 Chapter 4 Digital Filters

be tolerated. FIR filters require shorter data word length than the corresponding
IIR filters. However, they require much higher orders than IIR filters for the same
magnitude specification and they sometimes introduce large delays that make them
unsuitable for many applications.
The transfer function of an FIR filter of order M is

where h(ri) is the impulse response. Instead of using the order of the filter to
describe an FIR filter, it is customary to use the length of the impulse response,
which is N = M + 1.
The poles of the transfer function are at the origin of the z-plane for nonrecur-
sive algorithms. The zeros can be placed anywhere in the z-plane, but most are
located on the unit circle in order to provide better attenuation. FIR filters cannot
realize allpass filters except for the trivial allpass filter that is a pure delay.
Some of the poles may be located outside the origin in recursive FIR algo-
rithms, but these poles must always be canceled by zeros. We will not discuss
recursive FIR algorithms further since they generally have poor properties and
are not used in practice.

4.2.1 Linear-Phase FIR Filters
The most interesting FIR filters are filters with linear phase. The impulse
response of linear-phase filters exhibits symmetry or antisymmetry [3, 5, 15—18,
27], Linear-phase response, i.e., constant group delay, implies a pure delay of the
signal. Linear-phase filters are useful in applications where frequency dispersion
effects must be minimized—for example, in data transmission systems. The group
delay of a linear-phase FIR filter is

FIR filters with nonlinear-phase response are rarely used in practice although
the filter order required to satisfy a magnitude specification may be up to 50%
lower compared with a linear-phase FIR filter [1]. The required number of arith-
metic operations for the two filter types are, however, of about the same order.
To simplify the design of linear-phase FIR filters with symmetric impulse
response, it is convenient to use a real function HR which is defined by

where fRcJT) = c - Tg(afT) with c = 0 and c = n/2 for symmetric and antisymmetric
0
impulse responses, respectively. H^eJ ^) is referred to as the zero-phase response.
For an FIR filter with symmetric impulse response, HR can be written