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:

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

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

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

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

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

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

• $h_m(r)$ 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 $h_m(r)$ . It is lowpass-filtered by w and delayed by $r$ samples.
• Denote the $r$ th time-varying, lowpass-filtered, delayed-by-$r$ filter tap by ${\hat h}_{n-r}(r)$ . This can be interpreted as the weighting in the output at time $r$ of an impulse entering the time-varying filter at time $n-r$ .

Using this, we get

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

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