A theory of time series for waveform-transmission systems

Abstract
Although many communication systems have to transmit waveforms with reasonable freedom from distortion, the use of actual waveforms is inconvenient in calculating the effect of combining systems, because the appropriate mathematical processes involve the evaluation of integrals and the solution of integral equations. Several writers have dealt with the use of time series to replace waveforms, with a consequent simplification of calculation.A theoretical study of such time series, suitable for application to low-pass systems with a fixed upper limit to the frequency band transmitted, is presented, and several topics are treated which have not been dealt with adequately before. In particular, it is pointed out that different theoretical bases for time series, appropriate to different applications, are possible. For “division” and “inversion” of time series, algebraic division (which previous writers have used) is not, in general, legitimate for time series based on bandwidth limitation, although it is legitimate for certain other forms of time series. More general methods of inversion are dealt with, and an iterative method, which can always be used, is discussed in some detail, a practical computing scheme being given.Other new topics include the notion of the “most compact series” associated with a given waveform, and various definitions and manipulations associated with an algebra of time series. Numerical tables of certain series that have been used are given.

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