Double series representation of bounded signals

Series representations of the form f(f)-IE 2 a,,kv(t-n)e2n'k' for bounded signals f(t) are studied, as are conditions on the unit function v(r), such that the coefficients a,, k reveal the energy content of f(t) in the time interval n-(l/2) < t I n +(1/2) and frequency interval 2m(k-(l/2)) I w I 2n(k +(1/2)). These conditions turn out to he 1) orthogonal&y, i.e., if n=k=O otherwise and 2) integrability, v(r) EL!(R) V(w) E L'(R). Based on these conditions a number of properties of the expansion are derived, including smmnability of the double series and energy and power estimations. A unit function u(t) is constructed which is optimal, within a restricted class, with respect to the duration of V(w), and also a unit function which is optimal with respect to the duration of u(t). Finally, some examples of the expansion are presented. T HE PROBLEM we consider is closely related to aspects of information theory and signal analysis. We shall briefly sketch its origins. Many real signals that exhibit essentially time-varying properties nonetheless have properties that are described best in the frequency domain. Important examples are speech and signals obtained from pulse or carrier modulation schemes. Several authors have suggested representations which use both time and frequency as independent variables. In Gabor's historic attempt to construct a theory of information [l], a representation of the form f(t) = fJ f a,,,u(t-n)e2nikt (1.1) with appears as an essential step in the definition of a unit of information. From the point of view of information theory , this definition suffered primarily from the lack of appreciation of noise as a limiting factor. From a signal analysis point of view, the major defect in (1.1) was that the basis functions u(t-n)e2*jkt were not mutually orthogonal. Gabor correctly pointed out the significance of the uncertainty relations as a limitation on the resolution of time-frequency representations. Shannon [2] gave the correct definition of the unit of information, but in his analysis of continuous channels he relied on the sampling expansion. This approach amounts to using sin at u(t) =-p in (1.1). Although this makes the system of basis functions orthogonal, it is unsatisfactory in that the sampling expansion has poor convergence properties. Lerner [3] gave a valuable discussion of time-frequency series expansions and suggested a method for orthogonalizing a given system of functions. However, this approach does not lead to a function u(t) with the required properties. …