Fast algorithms for generalized discrete Gabor expansion

Abstract

Abstract Several methods for efficient implementing the generalized Gabor expansion are presented. The continuous-time Gabor expansion is converted to a finite discrete-time form suitable for numerical implementation. The results are obtained for an arbitrary choice of synthesis window function and the removal of Gabor's original constraint ωT = 2 π . This paper improves the traditional biorthogonal-function-based method by providing an algorithm of complexity O( L log 2 L ) for computing the biorthorgonal sequence for both the critical and oversampled cases, and an algorithm for computing the Gabor coefficients from the biorthogonal function with the efficiency of the 2D-FFT. Moreover, using the discrete Zak transform, both critical and oversampled Gabor expansions are directly with an algorithm whose complexity is of the order of the FFT. The results obtained extend the results of Janssen (1988) and Orr (1991) to the oversampling case. The frame concept is also used to define the discrete Gabor expansion, and an efficient algorithm to compute the Gabor coefficients directly is presented via the derivation of the dual frame. It is shown that the mother sequence of the dual frame is actually the minimum-energy biorthogonal sequence by demonstrating that their respective DZTs are equivalent. This leads to an intuitive interpretation of the role of the dual frame in the Gabor expansion of discrete-time signals.