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On the cross-biorthogonal representation

A novel cross term deleted Wigner representation can be obtained by expanding the Wigner distribution (WD) in terms of two complementary Gabor coefficients of the signal and a translated set of Wigner basis functions. Two such complementary Gabor coefficients of a signal can be obtained by reversing the role of the Gabor synthesis window h(t) and its biorthogonal function b(t). Such a representation is defined here, as the cross-biorthogonal representation (XBIO). Details of derivation of this new representation is provided in this paper. The choice of the synthesis functions and their corresponding biorthogonal functions with respect to (i) concentration/resolution capabilities, (ii) redundancy vs. minimum-dimension tradeoffs, (iii) noise reduction and (iv) basis set properties of the XBIO representation are also discussed. Simulation results are provided to substantiate the theoretical findings.