Signal-Flow Graphs and Random Signals

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Abstract
A compact and meaningful description of a signal is offered by the flow-graph representation of the signal generator. A large class of important signal generators may be represented by a "trigger generator" which delivers a sequence of one or more impulses to various waveform generators. By describing the "trigger generator" by a Markoff process, by identifying the transition probability-densities between states of the Markoff process with impulse responses, and by interpreting the final response as an expectation-density (or average over the ensemble of possible signals), it is possible to obtain, by methods which are very similar to those used in computing the transfer function of an ordinary circuit, expressions for the power spectrum and correlation functions of signals produced by such sources. The important relationships between stochastic processes and familiar circuit concepts are first illustrated by calculating the probabilities associated with four different coin-tossing experiments of increasing complexity. As a fifth example, these relations are used to develop the well-known Poisson distribution and to introduce the expectation-density of occurrence of a recurrent event. General formulas for the correlation functions and power spectra of signals produced by Markoffian sources are then obtained for: a random telegraph message; a series of identical pulses having time jitter, both for the free-running and clock-synchronized cases; and, finally, a series of identical pulses of alternating polarity but with random spacing.

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