Filter Banks Equivalent to STFTs

We now turn to various practical examples of perfect reconstruction filter banks, with emphasis on those using the FFT in their implementation (i.e., various STFT filter banks).

Figure 11.29 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:

From Chapter 9, we recognize this expression as the sliding-window STFT, where is the flip of a sliding window ``centered'' at time , and is the th DFT bin at time . We also know from that discussion that remodulating the DFT channel outputs and summing gives perfect reconstruction of whenever is Nyquist(N) (the defining condition for Portnoff windows [196] in §9.7.

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.

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Subsections

• Polyphase Analysis of Portnoff STFT

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.