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Uniform Running-Sum Filter Banks

Using a length $ N$ running-sum filter, let's make $ N$ bandpass filters tuned to center frequencies

$\displaystyle \omega_k\isdef k\frac{2\pi}{N}, \quad k=0,1,2,\ldots,N-1.$ (10.11)

Since the bandwidths, as defined, are $ 4\pi/N$ , the filter pass-bands overlap by 50%. A superposition of the bandpass frequency responses for $ N=5$ is shown in Fig.9.14. Also shown is the frequency-response sum, which we will show to be exactly constant and equal to $ N$ . This gives our filter bank the perfect reconstruction property. We can simply add the outputs of the filters in the filter bank to recreate our input signal exactly. This is the source of the name Filter-Bank Summation (FBS).

Figure: Example filter-bank channel frequency responses for $ N=5$

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2022-02-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University