The application of correlation functions in the detection of small signals in noise

The first part of this paper presents the general concept of the correlation functions. A mathematical reinterpretation of the correlation function is given in terms of the sum of the products obtained by representing the correlation function as a discrete time series. An electronic correlator capable of performing the necessary mathematical operations to obtain correlation functions is described and a description of the application of the correlator to the detection of small signals is presented. The advantage of cross-correlation in the detection of a signal of known frequency is demonstrated. *This report was presented at the I.R.E. National Convention in New York, March 7-10, 1949. THE APPLICATION OF CORRELATION FUNCTIONS IN THE DETECTION OF SMALL SIGNALS IN NOISE I. Some General Properties of Correlation Functions Mathematically, the autocorrelation function of a time function fl(t) is defined as: T ¢11(T) = lim A f rl(t)fl(t + r)dt (1) T c -T In an analogous fashion, the cross-correlation functions between two time functions fl(t) and f2(t) are defined as T *1 2 () lim fl(t)f 2(t + T)dt (2) and rT 21(T) im f2(t)fl(t + )dt . (3) Quite simply, correlation functions are a measure of the mean relationship existing between the product of all pairs of points of the time series involved, separated in time by a delay T. The autocorrelation function and similarly the cross-correlation function are equally well-defined for both periodic and random time functions. It can be shown that the autocorrelation function 1 1 (T) is a maximum at T = O, being equal to the mean of the square of the time function. In the absence of a periodic component, 1 1 (T) approaches the square of the mean of the time function as T increases. These general properties allow us to sketch a rough picture of the bounds or envelope of autocorrelation functions for random time series. However, if a periodic component such as a sine wave were present in fl(t), the autocorrelation function would appear roughly as it does in Fig. 2, since component parts of correlation functions add linearly, and the correlation function of a sinusoid is a cosine function of the same frequency. If the random portion of fl(t) is normally distributed noise, its mean value is zero and therefore its correlation function approaches zero for large values of T. The correlation for the sinusoid, however, is again periodic and of constant amplitude. It is pointed out, therefore, that theoretically an infinite signal-to-noise ratio can be obtained in the detection of a periodic signal (no matter how small) in noise through the property of autocorrelation alone. One should remember, however, that "theoretically"