The probability distributions of sinusoidal oscillations combined in random phase

摘要

This paper discusses a fundamental problem involved in the theory of multi-channel transmission. In modern carrier telephone systems a number of channels pass through common line amplifiers. The loading of these amplifiers therefore fluctuates in accordance with the sum of the speech voltages applied to the individual speech channels in the carrier group, and the maximum loading depends upon whether there can be a simultaneous occurrence of peak values in all the speech channels. In general, this is an extremely rare occurrence. As an alternative to providing speech circuits, some of the channels may, however, be sub-divided to provide voice-frequency telegraph channels, the speech being then replaced by a number of single-frequency tones. When the telegraph channels are idle, these tones are continuous and the maximum loading of any common amplifier may be increased thereby. Some modern automatic signalling systems also use continuous tones or long periods of tone for the transmission of signals, so that possible overloading of the amplifiers must again be considered. It becomes important, therefore, to be able to determine the probabilities associated with the combination of sinusoidal signals from a number of different channels, and in particular the probabilities that the instantaneous value and the length of the resultant vector will exceed certain limits. The instantaneous value of the resulting signal is, of course, the sum of the values in all the combined channels. Under suitable conditions the peaks of the instantaneous value trace out an envelope for the resulting signal, and the probabilities associated with this envelope are, in general, the same as those associated with the resultant vector. The paper discusses theoretically the probabilities associated with the instantaneous value and the length of the resultant vector obtained by combining n cosine oscillations of equal amplitude and random phase relationship. It is mainly concerned with very small v-nalues of n (=2, 3, 4? 12), and gives, for the first time, a complete set of curves showing the probabilities of the instantaneous value and the length of the resultant vector exceeding any given limits. For large values of n one aspect of the problem has been discussed by Rayleigh, and solutions for very large values of n have been given by Landon and more recently by Rice in connection with fluctuation noise. For the small values of n discussed here no solution has hitherto been given, but the empirical combination of speech-voltage distributions for a small number of channels has been studied in the Post Office Engineering Research Station and has been discussed in application to the design of apparatus by a number of authors. The mathematical theory used to derive the results is given in Section 6. It is hoped that this Section will be of general interest, because the underlying theories are of fundamental importance and may be capable of application to other physical problems, as, for example, when the amplitudes as well as the phases of the combined oscillations are random. While some of the formulae used are not original, as the list of references shows, they are, for convenience, collated here from the scattered memoirs in which they appear and which contain much other mathematical material extraneous to the present subject.