Two-Parameter Exponential and Rational Functions for Least-Square Approximations
A computationally efficient scheme for approximating system characteristics by sums of exponentials or by rational functions. Results are applicable to a broad class of system optimizations, such as those in network synthesis and radar filter design, and to the understanding of cross-correlation measurements, radioactive decay, gas absorption, and mass spectrograph and ultracentrifuge analysis curves. Two sets of two-parameter orthonormal elements are derived: one set constitutes a basis for exponential approximation and the other a basis for rational function approximation. The closure properties of the two orthonormal bases are examined and new expressions are developed for efficiently determining the orthonormal elements of each basis. Several relations are then deduced that connect important properties of each basis. Useful identities and numerical techniques involving the basis coefficients are derived that obviate storage of either the form or selected values of the orthonormal approximants. Finally, several numerical examples illustrate algorithms for both exponential and rational function approximation. Computer programs in ALTRAN and in the more efficient FORTRAN IV are appended.