A theory of signals
Robert E. Wernikoff · 1958
requirements for the degree of Doctor of Science. An experiment is presented in which an attempt is made to arrive at a mathematical description of physical signals that embodies, more realistic-ally than the usual functional representation, our limitations in performing measurements. The object is to achieve a closer relation between the struc-ture of the mathematical description and the finite-resolution properties of the detector that characterizes any real measurement process. An algebra of signals is obtained, appropriate to the model in which essen-tially frequency-limited signals interact with linear, time-invariant systems, and observations are made by means of a linear, finite-resolution oscilloscope. The properties of this algebra are studied, and a metric that indicates which operators give physically indistinguishable outputs is defined. The algebra is used to study problems in uniform and nonuniform sampling, the discrim-ination of two events from one in noisy, radar-like systems, and the condi-tions under which a signal is indistinguishable from its short-time average. A general procedure for linear, least-peak-error prediction is obtained. In the limit as the detector resolution becomes perfect, the present model is shown to tend smoothly to the usual functional model.