Nonorthogonal signal representation by Gaussians and Gabor functions

This paper describes a novel approach for nonorthogonal representation of signals using Gaussians and an extension of this method for Gabor representation of signals, based on the equivalence of Gabor expansion to Gaussian expansion in the frequency domain. The Gaussian expansion scheme yields an efficient representation of signals for low number of bits per pixel and is better than the corresponding Discrete Cosine Transform (DCT) representation for very low bit rates. This advantage diminishes gracefully for higher bit rates where the residual approximation error signal to be represented is more random and less structured. It is proved in this paper that a finite number of Gaussians can theoretically approximate sinusoids in a bounded region with arbitrarily small error, and therefore any finite support L² (R) signal as well. Two methods for Gaussian representation of signals are outlined. The first, called the Max-Energy paradigm, involves successive extraction of the highest energy Gaussian that best “fits” the signal. The second is a parallel approach and uses an adaptive projection algorithm to first derive the Gaussian basis set to be used in parallel, and then optimizes the coefficients for minimum squared error