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Block Diagram Interpretation of Time-Varying STFT Modifications

Assuming $ {\hat h}$ is causal gives

y(n) &=& \sum_{r=0}^\infty x(n-r) {\hat h}_{n-r}(r) \\
&=& x(n) {\hat h}_n(0) + x(n-1) {\hat h}_{n-1}(1) + x(n-2) {\hat h}_{n-2}(2) + \cdots

This is depicted in Fig.8.17.

% latex2html id marker 23815\psfrag{zm1}{\large $z^{-1}$\ }\psfrag{h(0,n)}{\large$ h_n(0) $}\psfrag{h(1,n)}{\large$ h_{n-1}(1) $}\psfrag{h(2,n)}{\large$ h_{n-L+1}(L-1) $}\psfrag{+}{\large$\Sigma$}\psfrag{w(n)}{\large$ w $}\psfrag{y(n)}{\large$ y(n) $}\begin{figure}[htbp]
\caption{System diagram giving
an interpretation of the bandlimited time-varying filter coefficients
in the overlap-add STFT processor with a new filter each frame.}

The term $ h_n(k)$ can be interpreted as the FIR filter tap $ k$ at time $ n$ . Note how each tap is lowpass filtered by the FFT window $ w$ . The window thus enforces bandlimiting each filter tap to the bandwidth of the window's main lobe. For an $ L$ -term length-$ M$ Blackman-Harris window, for example, the main-lobe reaches zero at frequency $ L\Omega_M=2\pi L/M$ (see Table 5.2 in §5.5.2 for other examples). This bandlimiting places a limit on the bandwidth expansion caused by time-variation of the filter coefficients, which in turn places a limit on the maximum STFT hop-size that can be used without frequency-domain aliasing. See Allen and Rabiner 1977 [9] for further details on the bandlimiting property.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2020-01-24 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University