Discrete Gabor expansion of discrete-time signals in [formula omitted] via frame theory

Abstract

This paper presents the Gabor representation for discrete-time signals in l 2( Z) . The frame theory for discrete-time signals is introduced first. In this discrete Gabor expansion, the time-shift parameter N and the frequency-shift parameter M are taken to be integers with M ⩾ N for stable reconstruction. With the assumption that the Gabor analysis function has finite length, many useful formulae can be derived from frame theory. In particular, we discuss discrete-time signal decomposition and reconstruction and an algorithm for computing the dual frame by means of frame theory and time-frequency analysis. Important features of these results is that the decomposition and reconstruction can be easily computed, and perfect reconstruction is achieved. In particular: (1) we develop a frame theory for the discrete-time Gabor decomposition and reconstruction problem in l 2( Z) and derive formulae that can be directly implemented via DSP methods (these results cannot be obtained simply from digitizing the equivalent continuous-time formulation of Daubechies);(2) there are no constraints on the time-shift parameter N and frequency-shift parameter M except M ⩾ N for stable reconstruction, i.e., the oversampling ratio R = M/ N is a rational number, while in previous work that also dealt with discrete-time functions, the oversampling ratio can only be chosen from a few numbers dependent on the signal length; (3) the computation of the dual sequence is easy and fast in terms of Eqs. (1.9) and (1.12) (moreover, the matrix equation (1.12) for solving the dual can be separated into submatrix equations (1.15) thereby significantly reducing the computing time by a factor of M); and (4) examples are provided that illustrate the computation of dual frames (analysis sequences), the relationship between the dual frame and parameters M, N, and Q, comparisons with existing results, and the achievement of perfect reconstruction.