Best least-squares representation of signals by exponentials

In this paper the approximation of a given real time function over
(0, \infty)
by a linear combination of a given number
n
of exponentials is considered, such that the integrated squared error is minimized over both the
n
coefficients of the linear combination and the
n
exponents used. The usual necessary condition for stationarity of the integrated squared error leads to a set of
2n
simultaneous equations, nonlinear in the exponents. This condition is interpreted in the geometric language of abstract vector spaces, and an equivalent condition involving only the exponents, with the coefficients suppressed, is developed. It is next indicated how this latter condition can be applied to signals which are not known analytically, but only, for example, as voltages recorded on magnetic tape, or as a table of sampled values. The condition still in effect requires solution of nonlinear algebraic equations, and a linear iterative method is proposed for this purpose. Finally, the procedure is illustrated with a simple example.