Optimal Band Filters

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

  1. there can be only one lowpass prototype filter in any modulated filter bank (such as the DFT filter bank),
  2. the prototype itself is the optimal Chebyshev lowpass filter of minimum bandwidth, and
  3. 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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Upsampling (Stretch) Operator
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Real Filter Bank Example