Time Varying Modifications

In FFT convolution using the STFT, the filter can be changed each frame ($h\to h_m$ ):

• $X_m(\omega_k) =$ sampled DTFT (FFT) of $m^{th}$ input frame
• $H_m(\omega_k) =$ time varying spectral modification
• $Y_m(\omega_k) = X_m(\omega_k) H_m(\omega_k) = m$ th output frame
• $\omega_k = 2\pi k / N$ = $k$ th spectral sample
• $N =$ FFT length
• $M =$ window $w$ length: $x_m(n) = x(n)w(n-m)$
• $L =$ max length of FIR filter $h_m$ applied to each frame
• $N >= M+L-1$ to avoid time aliasing in $y_m$

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

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

Letting $r \mathrel{\stackrel{\Delta}{=}}n-l \Rightarrow l = n-r$ results in:

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

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

• $h_m(r)$ describes the time variation of the $r^{th}$ tap of the time-varying FIR filter
• $\sum_{m=-\infty}^\infty h_m(r) w[(n-r)-m] = [h_{(\cdot)}(r) \ast w](n-r)$ 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 $\hat{h}^w_{n-r}(r)$ . 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

$$\begin{eqnarray*} y(n) &=& \sum_{r=-\infty}^\infty x(n-r) \hat{h}^w_{n-r}(r) \qquad\mbox{( $=x\ast \hat{h}^w$\ if LTI)}\\ &=& x(n) \hat{h}^w_n(0) \\ & & + x(n-1) \hat{h}^w_{n-1}(1) + x(n-2) \hat{h}^w_{n-2}(2) + \cdots \\ & & + x(n+1) \hat{h}^w_{n+1}(-1) + x(n+2) \hat{h}^w_{n+2}(-2) + \cdots \end{eqnarray*}$$

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