We now turn to various practical examples of perfect reconstruction filter banks, with emphasis on those using FFTs in their implementation (i.e., various STFT filter banks).
Figure 11.28 illustrates a generic filter bank with channels, much like we derived in §9.3. The analysis filters , are bandpass filters derived from a lowpass prototype by modulation (e.g., ), as shown in the right portion of the figure. The channel signals are given by the convolution of the input signal with the th channel impulse response:
Suppose the analysis window (flip of the baseband-filter impulse response ) is length . Then in the context of overlap-add processors (Chapter 8), is a Portnoff window, and implementing the window with a length FFT requires that the windowed data frame be time-aliased down to length prior to taking a length FFT (see §9.7). We can obtain this same result via polyphase analysis, as elaborated in the next section.