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This section gives an extended explanation of how to use M atlab to design a bandpass filter to fit a specified frequency response with a flat passband. The same procedure(with suitable modification) also works for the design of other basic filter types.
A bandpass filter is intended to scale, but not distort, signals with frequencies that fall within the passband, and to rejectsignals with frequencies in the stopband. An ideal, distortionless response for the passbandwould be perfectly flat in magnitude, and would have linear phase (corresponding to a delay).The transition band from the passband to the stopband should be as narrow as possible.In the stopband, the frequency response magnitude should be sufficiently small and the phase is of no concern.These objectives are captured in [link] . Recall (from [link] ) for a real $w\left(t\right)$ that $\left|W\right(f\left)\right|$ is even and $\angle W\left(f\right)$ is odd, as illustrated in [link] .
M
atlab has several commands that carry out filter design.
The
firpm
command
Some early versions of
M
atlab use
remez
instead of
firpm
. provides a linear phase impulse response
(with real, symmetric coefficients
$h\left[k\right]$ ) that has the best
approximation of a specified (piecewise flat) frequencyresponse.
There are many possible meanings of the word “best”;
for the
firpm
algorithm, “best” is defined in terms of maintaining
an equal ripple in the flat portions. The syntax of the
firpm
command for the design of a
bandpass filter as in
[link] is
b = firpm(fl,fbe,damps)
which has inputs
fl
,
fbe
, and
damps
, and output
b
.
fl
specifies (one less than) the number of terms in the
impulse response of the desired filter. Generally, more is betterin terms of meeting the design
specifications. However, larger
fl
are
also more costly in terms of computation and in terms ofthe total throughput delay, so a compromise is usually made.fbe
is a vector of frequency band edge values as a fraction
of the prevailing Nyquist frequency.For example, the filter specified in
[link] needs six values: the bottom of the
stopband (presumably zero), the top edge of the lower stopband(which is also the lower edge of the lower transition band),
the lower edge of the passband, the upper edge of the passband,the lower edge of the upper stopband, and the upper edge of the
upper stopband (generally the last value will be 1).The transition bands must have some nonzero width
(the upper edge of the lower stopband cannot equalthe lower passband edge) or M
atlab produces
an error message.damps
is the vector of desired amplitudes of the frequency
response at each band edge. The length of
damps
must match the length of
fbe
.b
is the output vector containing the impulse response of
the specified filter.The following M
atlab script designs a filter that mimics
[link] :
fbe=[0 0.24 0.26 0.74 0.76 1]; % frequency band edges as a% fraction of the Nyquist frequency
damps=[0 0 1 1 0 0]; % desired amplitudes at band edges
fl=30; % filter sizeb=firpm(fl,fbe,damps); % b is the designed impulse response
freqz(b) % plot frequency response to check design
bandex.m
design a bandpass filter and plot frequency response
(download file)
[link] .
Observe that the stopband is about 14 dB lower than thepassband, a marginal improvement over the naive lowpass
filter of
[link] , but the design is
much flatter in the pass band. The “equiripple” nature of this filteris apparent in the slow undulations of the magnitude
in the passband.
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