Generalized Gabor expansions of discrete-time signals in l2(Z) via biorthogonal-like sequences

In this paper, a biorthogonal-like sequences (BLS) theory and its application to the generalized Gabor expansions (equivalently, the generalized short-time Fourier transform/filterbank summation) are presented. A pair of BLS's are defined to be two sequences satisfying a biorthogonal-like condition (BLC), which is a moment equation and equivalent to a linear difference equation. We show that two collections in a Hilbert space generated by a pair of BLS's in the joint time-frequency domain are complete, either can be used as an analysis filter, and the other can be used as a synthesis filter for a generalized Gabor expansion of discrete-time signals. A sufficient and necessary condition on the existence of BLS's based on the moment equation is presented, which is simpler to use than frame theory. Given a filter generating a frame, its BLS's also generate frames. The dual frame is one of them. Given a FIR analysis/synthesis filter, there is a FIR synthesis/analysis filter if BLS's exist. The algorithm to compute FIR analysis and synthesis filters based on the linear difference equation is presented in this paper, which is simpler than frame operator