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Spectral Audio Signal Processing
Overlap-Add (OLA) STFT Processing
Time Varying OLA Modifications

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Time Varying OLA Modifications

In the preceding sections, we assumed that the spectral modification did not vary over time. We will now examine the implications of time-varying spectral modifications. The derivation below follows [10], except that we'll keep our previous notation:

$\begin{eqnarray*} X_m(\omega_k) &=& \hbox{sampled DTFT (FFT) of $m$th input fram... ...ame}\\ N &\ge& \hbox{ $M+L-1$\ to avoid time aliasing in $y_m$} \end{eqnarray*}$

Using in our OLA formulation with a hop size results in

$\begin{eqnarray*} y(n) &=& \sum_{m=-\infty}^\infty y_m(n) \\ &=& \sum_{m=-\inf... ...infty}^\infty x(l) \sum_{m=-\infty}^\infty w(l-m) h_m(n-l) \\ \end{eqnarray*}$

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

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

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

describes the time variation of the $r^{th}$ tap.
$\sum_{m=-\infty}^\infty h_m(r) w[(n-r)-m] = [h_{(\cdot)}(r) \ast w](n-r)$ is a filtered version of the $r^{th}$ tap . It is lowpass-filtered by w and delayed by samples.
Denote the th time-varying, lowpass-filtered, delayed-by- filter tap by ${\hat h}_{n-r}(r)$ . This can be interpreted as the weighting in the output at time of an impulse entering the time-varying filter at time .

Using this, we get

$\begin{eqnarray*} y(n) &=& \sum_{r=-\infty}^\infty x(n-r) {\hat h}_{n-r}(r) \\ ... ...+ x(n+1) {\hat h}_{n+1}(-1) + x(n+2) {\hat h}_{n+2}(-2) + \cdots \end{eqnarray*}$

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

Subsections

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Previous: Overlap-Save Method
Next: Block Diagram Interpretation of Time-Varying STFT Modifications

written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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