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

$$L_h \isdef \frac{L-1}{2}$$ (9.43)

This gives

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

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