Index Ranges for Zero-Phase Zero-Padding

Having looked at zero-phase zero-padding ``pictorially'' in matlab buffers, let's now specify the index-ranges mathematically. Denote the window length by $M$ (an odd integer) and the FFT length by $N>M$ (a power of 2). Then the windowed data will occupy indices 0 to $(M-1)/2$ (positive-time segment), and $N-(M-1)/2$ to $N-1$ (negative-time segment). Here we are assuming a 0-based indexing scheme as used in C or C++. We add 1 to all indices for matlab indexing to obtain 1:(M-1)/2+1 and N-(M-1)/2+1:N, respectively. The zero-padding zeros go in between these ranges, i.e., from $(M-1)/2 + 1$ to $N-(M-1)/2 - 1$ .