FBS Window Constraints for R=1

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Spectral Audio Signal Processing

Filter Bank View of the STFT

Inverse DFT and the DFT Filter Bank Sum

Nyquist(N) Windows

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Spectral Audio Signal Processing

Filter Bank View of the STFT

Inverse DFT and the DFT Filter Bank Sum

Nyquist(N) Windows

FBS Window Constraints for R=1

Recall that in overlap-add (Chapter 8), perfect reconstruction required only that the analysis window meet a constant overlap-add (COLA) constraint:

$\displaystyle \sum_{m=-\infty}^\infty w(n-mR) = c$

(10.18)

where $c\neq 0$ is any constant (always true for ).^10.4

The Filter Bank Summation (FBS) is interpreted as a demodulation (frequency shift by $-\omega_k$ ) and subsequent lowpass filtering by . Therefore, to resynthesize our original signal, we need to remodulate each baseband signal and sum up the channels. For (no downsampling), this sum is given by [9]

$\displaystyle \hat{x}(m)$	$\displaystyle =$	$\displaystyle \sum_{k=0}^{N-1} X_m(\omega_k)e^{j\omega_km}$
	$\displaystyle =$	$\displaystyle \sum_{k=0}^{N-1} \left[ \sum_{n=-\infty}^\infty x(n)w(n-m)e^{-j\omega_kn} \right] e^{j\omega_km}$
	$\displaystyle =$	$\displaystyle \sum_{n=-\infty}^\infty x(n)w(n-m) \sum_{k=0}^{N-1} e^{j\omega_k(m-n)}$
	$\displaystyle =$	$\displaystyle \sum_{n=-\infty}^\infty x(n)w(\underbrace{n-m}_{-rN}) N\sum_{r=-\infty}^\infty \delta(m-n-rN)$
	$\displaystyle =$	$\displaystyle N\sum_{r=-\infty}^\infty {\tilde w}(rN)x(m-rN)$
	$\displaystyle =$	$\displaystyle Nw(0) x(m) \qquad \hbox{if $w(rN)=0,\,\forall r\neq 0$} \protect$	(10.19)

We have thus derived the following sufficient condition for perfect reconstruction [213]:

$\textstyle \parbox{0.8\textwidth}{Perfect reconstruction is assured in the sliding STFT remodulated filter-bank sum provided the analysis window is zero at all nonzero multiples of $N$.}$

Since normally our windows are shorter than , this condition holds trivially. In the overlap-add context, we also had guaranteed perfect reconstruction in this case ( ), because every window overlap-adds to a constant at displacement .

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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

FBS Window Constraints for R=1

``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

``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