A Unified Approach to the Synthesis of Orthonormal Exponential Functions Useful in Systems Analysis

A unified approach is presented for the construction of sets of orthonormal exponential functions whose elements have the properties such that: 1) their asymptotic order may be chosen arbitrarily, and 2) their poles may be real, complex, or some real and others complex. The main results are summarized as theorems and propositions in which the sets of exponentials are derived as transfer functions in the s domain. These theorems and propositions supplement the more conventional Gram-Schmidt procedure which is useful for the orthonormalization of functions in the time domain. Examples are given which illustrate applications of the main results. In addition, a generalized spectrum analyzer, which can be synthesized on an analog computer, is developed for use in the automatic evaluation of Fourier coefficients.