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Time Varying Modifications in FBS

Consider now applying a time varying modification.

$\displaystyle Y_m(\omega_k) = X_m(\omega_k)H_m(\omega_k) \qquad \hbox{($R=1$)}$ (10.32)

where

$\displaystyle H_m(\omega_k) \;\longleftrightarrow\;h_m(n) = \frac{1}{N} \sum_{k=0}^{N-1} H_m(\omega_k) e^{j\omega_kn}$ (10.33)

$ h_m(n)$ refers to the $ n^{th}$ tap of the FIR filter at time $ m$ .

\begin{eqnarray*} y(m) &=& \frac{1}{N} \sum_{k=0}^{N-1} Y_m(\omega_k) e^{j\omega_k m} \\ &=& \frac{1}{N} \sum_{k=0}^{N-1} X_m(\omega_k)H_m(\omega_k) e^{j\omega_k m} \\ &=& \frac{1}{N} \sum_{k=0}^{N-1} \left\{ \sum_{n=-\infty}^\infty x(n)w(n-m)e^{-j\omega_kn} \right\} H_m(\omega_k) e^{j\omega_k m} \\ &=& \frac{1}{N} \sum_{n=-\infty}^\infty x(n)w(n-m) \sum_{k=0}^{N-1} H_m(\omega_k) e^{j\omega_k(m-n)} \\ &=& \sum_{n=-\infty}^\infty x(n) [ w(n-m) h_m(m-n)] \\ &=& \sum_{n=-\infty}^\infty x(n) [\tilde{w}(m-n)h_m(m-n)] \\ &=& (x*[\tilde{w} \cdot h_m])(m) \\ \end{eqnarray*}

Hence, the result is the convolution of $ x$ with the windowed $ h_m$ .


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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
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