Some results on biorthogonal-like functions for generalized Gabor expansions

Summary form only given. The decomposition and representation of signals in the joint time-frequency domain are increasingly useful in communication and information theory, and in signal processing. Given a countable collection of functions in the joint time-frequency domain, we may find another countable collection of joint time-frequency functions, such that any signal can be represented in terms of the two collections of functions. The associated technique is usually called the discrete Gabor transform or short-time Fourier transform, and is valid for both continuous-time and discrete-time signals. One of the collections of functions is defined by a synthesis function, and the other collection is defined by an analysis function. We show that: (1) the collection of biorthogonal-like functions are complete and, hence, the generalized Gabor expansion is well-defined; (2) the technique for addressing the existence of the collection of biorthogonal-like functions is simple, hence, more applicable in practice; and (3) the computation of analysis and synthesis functions is simple and very efficient for both continuous-time and discrete-time signals