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