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, [253,302]. 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).
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).
Upsampling (Stretch) Operator
Real Filter Bank Example