While other types of filters are often of interest, this note focuses on the
lowpass linear phase filter. Even though it is not immediately obvious, virtually
all of the analytical results developed in this note apply to the other types
as well. This fact is amplified in the module Extension to Nonlowpass Filters.
It is known that the ParksMcClellan filter synthesis software package produces
“optimal" filters in the sense that the best possible filter performance is
attained for the number of “filter taps" allowed by the designer. “Optimal"
can be defined various ways. The ParksMcClellan package uses the Remez
exchange algorithm to optimize the filter design by selecting the impulse
response of given length, termed here NN, which minimizes the peak ripple in
the passband and stopband. It can be shown, though not here, that minimizing
the peak, or maximum, ripple is equivalent to making all of the local peaks
in the ripple equal to each other. This fact leads to three different names for
essentially the same filter design. They are commonly called “equalripple"
filters, because the local peaks are equal in deviation from the desired filter
response. Because the maximum ripple deviation is minimized in this optimization
procedure, they are also termed “minimax" filters. Finally, since the Russian
Chebyshev is usually associated with minimax designs
,
these filters are often given his name.
The design template for an equalripple lowpass filter is shown in
Figure 1.
The passband extends from 0 Hz to the cutoff frequency denoted fcfc.
The gain in the passband is assumed to be unity. Any other gain is
attained by scaling the whole impulse response appropriately. The stopband
begins at the frequency denoted fstfst and ends at the socalled Nyquist
or “folding" frequency, denoted by fs2fs2, where fsfs is the
sampling frequency of the data entering the digital filter. In some
references, [1] for example, the sampling rate fsfs is assumed to be
normalized to unity just as the passband gain has here. The dependence on
the sampling frequency is kept explicit in this note, however, so that its
impact on design parameters can be kept visible.
The optimal synthesis algorithm is assumed here to produce an impulse response
whose associated frequency response has ripples in both the passband
and the stopband. The peak deviation in the passband is denoted δ1δ1
and the peak deviation in the stopband is denoted δ2δ2. It is commonly
thought that an “equalripple" design forces δ1δ1 to equal δ2δ2.
In fact this is not true. The local ripple peaks in the passband will all
equal δ1δ1 and those in the stopband will all equal δ2δ2. For
a given filter specification the two are linked together by a weight denoted
WW, so that δ1=Wδ2δ1=Wδ2. In fact the ParksMcClellan routines
insure the design of weighted equalripple filters. The choice of WW is
discussed shortly.
An important design parameter is the transition band, denoted ΔfΔf,
and defined as the difference between the stopband edge fstfst and the
passband edge fcfc. Thus,
Δ
f
=
f
s
t

f
c
.
Δ
f
=
f
s
t

f
c
.
(1)In theory the required filter order NN is a function of all of the
design parameters defined so far, that is, fs,fc,fst,δ1fs,fc,fst,δ1, and δ2δ2. The central point of this technical note is
that under a large range of practical circumstances the required value of
NN can be estimated using only fs,Δffs,Δf, and the smaller of
δ1δ1 and δ2δ2.