14-05-2012, 04:47 PM
Practical FIR Filter Design in MATLAB
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FIR filter design specifications
Both the passband/stopband ripples and the transition
width are undesirable but unavoidable deviations from the
response of an ideal lowpass filter when approximating
with a finite impulse response. Practical FIR designs typically
consist of filters that meet certain design specifications,
i.e., that have a transition width and maximum
passband/stopband ripples that do not exceed allowable
values.
FIR lowpass filters
Because the impulse response required to implement the
ideal lowpass filter is infinitely long, it is impossible to
design an ideal FIR lowpass filter.
Finite length approximations to the ideal impulse response
lead to the presence of ripples in both the passband
(w < wc) and the stopband (w > wc) of the filter, as well
as to a nonzero transition width between the passband and
stopband of the filter (see Figure 1).
Linear-phase designs
A filter with linear-phase response is desirable in many
applications, notably image processing and data transmission.
One of the desirable characteristics of FIR filters is
that they can be designed very easily to have linear phase.
It is well known [3] that linear-phase FIR filters will have
impulse responses that are either symmetric or antisymmetric.
For these types of filters, the zero-phase response
can be determined analytically [3], and the filter design
problem becomes a well behaved mathematical approximation
problem [4]: Determine the best approximation
to a given function - the ideal lowpass filter’s frequency
response - by means of a polynomial - the FIR filter - of
given order -the filter order -. By “best” it is meant the one
which minimizes the difference between them - EW(w) -
according to a given measure.
Least-squares filters
Equiripple designs may not be desirable if we want to
minimize the energy of the error (between ideal and actual
filter) in the passband/stopband. Consequently, if we
want to reduce the energy of a signal as much as possible
in a certain frequency band, least-squares designs are
preferable.
Minimum-phase designs
If one is able to relax the linear-phase constraint (i.e. if the
application at hand does not require a linear-phase characteristic),
it is possible to design minimum-phase equiripple
filters that are superior to optimal equiripple linearphase
designs based on a technique described in [8].