Discrete and Digital Signals and Systems - Digital Signal Processing: An Introduction with MATLAB and Applications - page 100

A. Design of FIR Digital Filters by Frequency-Sampling
A intuitively appealing approach to designing a FIR filter is to simply take the
desired frequency transfer function and then inverse Fourier transform to find the
filter's impulse response. Within a digital computer, however, one cannot specify
the Fourier transform, only the samples of the desired Fourier transform. In other
words one can define the samples of a desired frequency response H d ð e jxT s Þ¼
H d ð e jX Þ; then use the IDFT to find the impulse response h(n). Often in digital
filters it is the transition band which is particularly critical and so it is necessary
to specify this portion of the transfer function most accurately. This implies that
one needs to have one or more samples within the transition band if one is to
achieve good filter designs. It is also important to note that if one seeks to
specify too sharp a transition band one will be confronted by the Gibbs Phe-
nomenon-substantial ripple will then manifest in both the pass-band and the stop-
band.
Example The goal is to design a 33rd order LPF with cutoff frequency f c = f s /4
(or, X c ¼ 2p = 4 ¼ p = 2). With the frequency sampling method it is necessary to use
a 33-point DFT to define the samples of the desired frequency response H d ð e jxT s Þ:
This desired response is the ideal rectangular response over the principal domain
0 \ f \ f s which is sampled to yield the set of samples f H d ð k Þj k ¼ 0 ; 1 ; ... ; 32 g:
The frequency sampling method is illustrated in Fig. ( 2.25 ) for two scenarios, The
first scenario is where no samples are specified in the transition band, and the
second is where one sample is specified in the transition band. Once the samples
have been specified the IDFT can be taken to recover the filter's impulse response.
Figure ( 2.25 ) shows the DTFT magnitudes of the impulse response so obtained. It
is seen in Fig. ( 2.25 ) that when no transition band sample is specified the ripple in
both the pass and stop bands is significantly greater [Note that H d (9) and H d (24)
are the transition samples, chosen to be 0.5].
B. Optimal Frequency-Sampling FIR Filter Design
The frequency-sampling method described above is straightforward but it is an ad
hoc| method. It is possible to design filters which are optimal in the sense of
minimizing the maximum error between the desired frequency response and the
actual frequency response. These types of filters are referred to as equiripple, and
can be designed using the. Parks-McClellan algorithm. The latter utilizes Remez
and Chebychev approximation theories to design optimal equiripple linear-phase
FIR filters.
MATLAB the Parks-McClellan algorithm is available on MATLAB as:
h ¼ remez ð N 1 ; Fd ; Ad Þ: