Delay Approximations for Correlation Measurements Using Analog Computers

In the most direct method for measuring correlation functions with an analog computer a delay must be simulated. Since a lumped parameter system is used, we can only hope to approximate this delay. We intentionally exclude such storage systems as magnetic tapes because of their cost, and confine ourselves to the use of commonly available analog computer components. Extensive work has been carried out toward finding the best delay approximation according to different criteria, based mainly on transient or frequency response considerations. In this paper a new point of view is adopted. The overall effect of the delay approximation on the measured value of the autocorrelation function is taken into account. The best approximation is then chosen as the one that produces the closest agreement between the theoretically measured value and the exact value of the autocorrelation. It is seen that this method for selecting the delay approximation sometimes leads to very different results from those formerly obtained. For example, it is shown that for measuring the autocorrelation of noise generated by filtering white noise with a filter with real poles, the best approximation is one whose value for a real argument (instead of jω) best approaches the exponential function for a real argument. The treatment of the subject in this paper deals mainly with the autocorrelation function, but is later extended to the case of cross-correlations.