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Time Varying OLA Modifications

In the preceding sections, we assumed that the spectral modification $ H$ did not vary over time. We will now examine the implications of time-varying spectral modifications. The derivation below follows [9], except that we'll keep our previous notation:

X_m(\omega_k) &=& \hbox{sampled DTFT (FFT) of $m$th input frame, $k=0,1,\ldots,N-1$}\\
H_m(\omega_k) &=& \hbox{time varying spectral modification (new each frame)}\\
Y_m(\omega_k) &=& \hbox{$X_m(\omega_k) H_m(\omega_k) = m$th output spectrum}\\
\omega_k &=&\hbox{ $2\pi k / N$\ = $k$th spectral sample}\\
N &=& \hbox{FFT length}\\
M &=& \hbox{window $w$\ length: $x_m(n) = x(n)w(n-m)$}\\
L &=& \hbox{\emph{maximum} length of FIR filter $h_m$\ applied to each frame}\\
N &\ge& \hbox{ $M+L-1$\ to avoid time aliasing in $y_m$}

Using $ H_m$ in our OLA formulation with a hop size $ R=1$ results in

y(n) &=& \sum_{m=-\infty}^\infty y_m(n) \\
&=& \sum_{m=-\infty}^\infty \frac{1}{N}\sum_{k=0}^{N-1} X_m(\omega_k) H_m(\omega_k) e^{j\omega_kn} \\
&=& \sum_{m=-\infty}^\infty \frac{1}{N}\sum_{k=0}^{N-1}
\left[ \sum_{l=-\infty}^\infty x(l) w(l-m)e^{-j\omega_kl} \right]
H_m(\omega_k) e^{j\omega_kn} \\
&=& \sum_{l=-\infty}^\infty x(l) \sum_{m=-\infty}^\infty w(l-m)
\frac{1}{N}\sum_{k=0}^{N-1} H_m(\omega_k)
e^{j\omega_k(n-l)} \\
&=& \sum_{l=-\infty}^\infty x(l)
\sum_{m=-\infty}^\infty w(l-m) h_m(n-l) \\

Define $ r \mathrel{\stackrel{\Delta}{=}}n-l \;\Rightarrow\; l = n-r$ to get

$\displaystyle y(n)=\sum_{r=-\infty}^\infty x(n-r) \sum_{m=-\infty}^\infty h_m(r) w(n-r-m).$ (9.42)

Let's examine the term $ \displaystyle\sum_{m=-\infty}^\infty h_m(r) w(
n-r-m )$ in more detail: Using this, we get

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

This is a superposition sum for an arbitrary linear, time-varying filter $ {\hat h}_{n-r}(r) = [h_{(\cdot)}(r) \ast w](n-r)$ .

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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