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358 Chapter 7 FIR FILTER DESIGN
Note that the fir2 does not implement the classic optimum frequency
sampling method. By incorporating window design, fir2 has found an alternative
(and somewhat clever) approach to do away with the optimum
transition band values and the associated tables. By densely sampling
values in the entire band, interpolation errors are reduced (but not minimized),
and stopband attenuation is increased to an acceptable level. However,
the basic design is contaminated by the window operation; hence,
the frequency response does not go through the original sampled values. It
is more suitable for designing FIR filters with arbitrary shaped frequency
responses.
The type of frequency sampling filter that we considered is called a
Type-A filter, in which the sampled frequencies are
ω k = 2π
M k, 0 ≤ k ≤ M − 1
There is a second set of uniformly spaced samples given by
ω k = 2π ( )
k + 1 2
, 0 ≤ k ≤ M − 1
M
This is called a Type-B filter, for which a frequency sampling structure is
also available. The expressions for the magnitude response H(e jω ) and the
impulse response h(n) are somewhat more complicated and are available
in Proakis and Manolakis [23]. Their design can also be done in MATLAB
using the approach discussed in this section.
7.5 OPTIMAL EQUIRIPPLE DESIGN TECHNIQUE
The last two techniques—namely, the window design and the frequency
sampling design—were easy to understand and implement. However, they
have some disadvantages. First, we cannot specify the band frequencies
ω p and ω s precisely in the design; that is, we have to accept whatever
values we obtain after the design. Second, we cannot specify both δ 1 and
δ 2 ripple factors simultaneously. Either we have δ 1 = δ 2 in the window
design method, or we can optimize only δ 2 in the frequency sampling
method. Finally, the approximation error—that is, the difference between
the ideal response and the actual response—is not uniformly distributed
over the band intervals. It is higher near the band edges and smaller in
the regions away from band edges. By distributing the error uniformly,
we can obtain a lower-order filter satisfying the same specifications. Fortunately,
a technique exists that can eliminate these three problems. This
technique is somewhat difficult to understand and requires a computer
for its implementation.
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