A procedure for calculating the best exponents for signal representation with digital computation

Given a signal over (o, ∞), if it can be expressed as a linear combination of exponential functions as x(t) = ∑kck exp(skt), then the expression in the frequency domain is X(s) = ∑k ck/(s-sk) or equivalently its z-transform is ∑k ck/(1-exp(skT)z-1) where T is the corresponding sampling interval. As shown by Young and Huggins [1], the construction of a set of discrete orthonormal exponentials is chosen in such a way that the poles in the z-domain are corresponding to the frequency domain poles, the zeros are chosen in such a way that as the sampling interval T approaches zero, the chosen discrete basis will approach the frequency domain orthonormal exponentials. As in the analog computation, applying Mc Donough and Huggins' criterion [2] and adaptive control principle, we will use digital computer with discrete orthonormal filter to find the best exponents for the signal representation in the sense that the sum of the squared error is minimized over the coefficients of representation and the discrete exponents used. The operation is just searching for the values of the filter parameter such that the discrete complementery signal is orthogonal to the subspace spanned by the discrete orthonormal exponentials. A simple example has been exploited to illustrate the procedure for the searching and implementation of the best exponents and the corresponding best coefficients.