Abstract
The most widely used methods of signal representation are the time function and the frequency function or spectrum representations. This work is concerned with the development of a representation which is a combination of these two. Two previous attempts at defining this type of signal representation, which is referred to as two dimensional representation, have been made and a summary and evaluation of these attempts is presented. The primary objective of the work reported here was to develop a practical two dimensional representation which has the desired two dimensional conceptual properties as well as mathematical convenience. The representations defined are based on the angular prolate spheroidal functions. These functions have a number of desirable properties among which are the followings they are orthogonal over both a finite and the infinite interval, they are bandlimited, and they have certain properties concerning their maximal proximity to being timelimited. The procedure used in defining the first two dimensional representation is to make an orthogonal expansion, using the prolate spheroidal functions, of each timelimited portion of each bandlimited portion of the signal to be represented. The second two dimensional representation is defined from an orthogonal expansion of each bandlimited portion of each timelimited portion of the signal to be represented. For both of these, the summation over all time intervals and all frequency intervals results in the complete representation of the signal. It is seen from this that since it is not possible to timelimit and bandlimit simultaneously, these limiting processes have been carried out serially. Due to the peculiar properties of the prolate spheroidal functions, as the number of orthogonal function terms is increased, the representation of a timelimited function converges first in a certain bandwidth, and the representation of a band- limited function converges first in a certain time interval. It is demonstrated that both series representations will converge to either a timelimited, or a bandlimited portion of the represented signal upon inclusion of the proper terms. Following this, several applications of the representations are presented. First, it is shown that the result of the convolution of 2 two dimensionally represented functions may be determined at discrete values of time from the expansion coefficients alone. The spectrum of the product of two functions may be determined in a similar manner at discrete values of frequency. As a result, it is possible to determine the contribution made to the output of a linear system at any time due to the portion of the input in any time and frequency interval. A technique is also developed for the solution of this same problem for the more general time variable linear system with the output being determined in continuous form rather than only at discrete values. It is somewhat more difficult to calculate the coefficients in this case, however. Another application demonstrated is a method by which the value of Woodward's ambiguity function may be calculated for discrete values of the time and frequency variables. The two dimensional nature of the representation is demonstrated by two numerical examples using very elementary time functions. A further numerical example is provided for the case of the determination of the output of a linear system at discrete values of time. This work is concluded by a brief listing of further problems which seem amenable to solution as a result of this type of analysis. This list includes such problems as biological system signal analysis, signal design, and random process representation.
Date of this Version
1-1-1962