Time statistics of noise

The measurements made on a system containing noise are usually time averages of the signals, or of quantities defined in terms of the signals. Such measurements are called time statistics. The object of this paper is to develop the theory of time statistics and in turn to give methods for calculating them. For the most part the time statistics are formulated in terms of ensemble statistics which are usually provided by statistical mechanics. If a process consists of, say, all physically realizable models of a system containing noisy resistors, there is no practical way to identify which model one has available for "testing." Thus, a time statistic measured with the available model will not be predictable unless this statistic is the same for almost all the models; when this is the case, the process is called uniform
¹
for this statistic. A dual property is in common use for ensemble statistics. The process is called stationary for an ensemble statistic, provided it is the same at all times. Though some discussion of stationarity is given in this paper, the emphasis is on not requiring stationarity. In particular, special attention is given to nonstationarity introduced by determinate signals. While stationarity plays only a minor role in the theory of the time statistics of noise, uniformity plays a crucial role. Given only uniformity, Theorem 1 formulates time statistics as the time average of the corresponding ensemble statistics. The additional condition of stationarity merely simplifies the calculation by rendering the "ergodic hypothesis" satisfied, i.e., by rendering equality of time and ensemble statistics. With Theorem 1 as a nucleus, the remainder of the paper attempts to develop an understanding of what makes a process uniform. There is no attempt to give detailed proofs, but there is an effort to maintain a clear distinction between physical motivations, the definitions, and the theorems. Some elementary sample calculations of practical interest are included; these serve to illustrate several parts of the theory. Though calculations involving such problems as the evaluation of difficult integrals do arise in some applications of the theory, simple samples have been used here, since they are adequate as an aid to understanding the theory.