Signal Representation by Prolate Spheroidal Wave Functions

The efficiency for signal representation of the angular prolate spheroidal wave function, particularly the two sets Sol(1, t) and Sol(8, t) is discussed Six signal waveforms are considered: rectangular, triangular, trapezoidal, exponential, Gaussian, and cosine-squared. For each, a representation is made in terms of the two sets above and also the Fourier cosine functions. As the number of terms of the representation increases, the approximation gets better. A measure of the ?goodness? of the approximation is the percentage of the total signal energy represented by the finite expansion, over a fixed, finite time interval. The angular prolate spheroidal wave functions are a very efficient orthogonal set in this sense. Their principal advantage over Fourier cosine functions occurs for cases whereby only a very few terms of the expansion are to be used to approximate a signal shape.