Length L FIR Frame Filters

To avoid time aliasing, we restrict the filter length to a maximum of $L$ samples. Since $H_m(\omega_k)$ is an arbitrary multiplicative weighting of the $m$th spectral frame, the frame filter need not be causal. For odd $L$, the filter impulse response indices may run from $-L_h$ to $L_h$, where

$$L_h \isdef \frac{L-1}{2}$$

This gives

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

This is the general length $L$ time-varying FIR filter convolution sum for time $n$, when $L$ is odd.