A complement to a derivation of discrete Gabor expansions

In the previous discrete Gabor expansion (DGE) presented by Wexler and Raz (1990), the ratio of the signal length L to the number of frequency channels N is restricted to be an integer. If L is a power of 2, then the oversampling rate is limited to 1, 2, 4, 8, etc. We derive a complementary condition to the derivation of the discrete Gabor expansion of Wexler et. al. and give a general pointwise biorthogonal relationship that relaxes the constraint on N. Consequently, the resulting Gabor expansion applies for integer as well as rational oversampling rates.<>