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In the filter-bank literature, one class of filter banks is called
``cosine modulated'' filter banks. DFT filter banks are similar. The
lowpass-filter prototype in such filter banks can be used in place of
the Dolph-Chebyshev window used here. Therefore, any result on
optimal design of cosine-modulated filter banks can be adapted to this
context. See, for example, [253,303]. Note,
however, that in principle a separate optimization is needed for each
different channel bandwidth. An optimal lowpass prototype only
optimizes channels having a one-bin pass-band, since the prototype
frequency-response is merely shifted in frequency (cosine-modulated in
time) to create the channel frequency response. Wider channels are
made by summing such channel responses, which alters the stop-bands.
In practice, the Dolph-Chebyshev window, used in the examples of this
section, typically yields a filter bank magnitude frequency response that is
optimal in the Chebyshev sense, when at least one channel is minimum width, because
- there can be only one lowpass prototype filter in any modulated
filter bank (such as the DFT filter bank),
- the prototype itself is the optimal Chebyshev lowpass filter of
minimum bandwidth, and
- summing shifted copies of the prototype frequency response
(to synthesize a wider pass-band)
generally improves the stop-band rejection over that of the
prototype, thereby meeting the Chebyshev optimality requirement for the
filter-bank as a whole (keeping below the worst-case deviation of the prototype).
All channel bands, whatever their width, are constructed by some
linear combination of shifted copies of the lowpass prototype frequency
response. The Dolph-Chebyshev window is precisely optimal (in the Chebyshev sense)
for any pass-band that is one bin wide. Summing shifts of the window transform
to synthesize wider bands has been observed to invariably improve the
stop-band rejection significantly. The examples shown above illustrate
the margin beyond 80 dB stop-band rejection achieved for the octave
filter bank.
The Dolph-Chebyshev window has faint impulsive endpoints on the order
of the side-lobe level (about 50 dB down in the 80-dB-SBA examples
above), and in some applications, this could be considered an
undesirable time-domain characteristic. To eliminate them, an optimal
Chebyshev window may be designed by means of linear programming with a
time-domain monotonicity constraint (§3.13).
Alternatively, of course, other windows can be used, such as the
Kaiser, or three-term Blackman-Harris window, to name just two
(Chapter 3).
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