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Optimal Equiripple Design Technique 359
For linear-phase FIR filters, it is possible to derive a set of conditions
for which it can be proved that the design solution is optimal in the sense
of minimizing the maximum approximation error (sometimes called the
minimax or the Chebyshev error). Filters that have this property are called
equiripple filters because the approximation error is uniformly distributed
in both the passband and the stopband. This results in lower-order
filters.
In the following we first formulate a minimax optimal FIR design
problem and discuss the total number of maxima and minima (collectively
called extrema) that one can obtain in the amplitude response of
a linear-phase FIR filter. Using this, we then discuss a general equiripple
FIR filter design algorithm, which uses polynomial interpolation for its solution.
This algorithm is known as the Parks-McClellan algorithm, and it
incorporates the Remez exchange algorithm for polynomial solution. This
algorithm is available as a subroutine on many computing platforms. In
this section we will use MATLAB to design equiripple FIR filters.
7.5.1 DEVELOPMENT OF THE MINIMAX PROBLEM
Earlier in this chapter we showed that the frequency response of the four
cases of linear-phase FIR filters can be written in the form
H(e jω )=e jβ M−1
−j
e 2 ω H r (w)
where the values for β and the expressions for H r (ω) are given in
Table 7.2.
TABLE 7.2
Amplitude response and β-values for linear-phase FIR filters
Linear-phase FIR Filter Type β H r(e jω )
Type-1: M odd, symmetric h(n) 0
Type-2: M even, symmetric h(n) 0
M/2
∑
1
(M−1)/2
∑
0
a(n) cos ωn
b(n) cos [ω(n − 1/2)]
Type-3: M odd, antisymmetric h(n)
Type-4: M even, antisymmetric h(n)
π
2
π
2
M/2
∑
1
(M−1)/2
∑
1
c(n) sin ωn
d(n) sin [ω(n − 1/2)]
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