Signal Representation by Prolate Spheroidal Wave Functions

Abstract

The efficiency for signal representation of the angular prolate spheroidal wave function, particularly the two sets S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ol</sub> (1, t) and S <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ol</sub> (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.