Optimum vs. correlation methods in tracking random signals in background noise
I. Summary. A method of target tracking used exclusively in some applications is that of space diversity reception. We consider a two-dimensional problem with two receivers, although in principle the techniques developed are applicable to a threedimensional problem utilizing three or more receivers. With N receivers the computational difficulties increase as N2. The tracking problem considered here is formulated as follows. During a time interval 0 < t < T one observes random processes xi(t) = s(t) + nx (/) and = s(t + r0) + n2 (t), and one desires to estimate the unknown value of T0 ■ s(t), nx(t), and n2(l) are continuous, Gaussian, stationary processes with continuous, monotonoid covariance functions and with Es(t)ni(t') = Es(t)n-2(t') = 0. ih(t) and n2(t) are stationarily cross-correlated with a cross-correlation function that is in general an asymmetric function of time delay. As a criterion for an optimum estimate, To , of t0 , we use the one—commonly used in statistical literature, that t'0 should have minimum variance about the true value t0 ; that is, E(t'0 — t0)2 be a minimum. It is shown that for small values of o>„t0 (where co„ is the highest frequency in the tracking pass band) the normalized error variance E[(t'0 — t0)/t0]2 is identical to the reciprocal of the output signal-to-noise ratio—the criterion of common use in engineering literature on correlation methods. Using the Cramer-Rao inequality, an explicit expression for the minimum variance attainable by any estimate whatsoever is established. Moreover we arrive at the interesting conclusion that even with the use of optimum pre-detection filtering in a correlation system, the variance of the correlator estimate of r0 is always greater than the minimum variance given by the Cramer-Rao inequality. Gains in output signal-to-noise ratio obtained by an optimum system over a correlator vary in accordance with the statistical properties of signal and noise backgrounds. In general the gains are greater with asymmetrical cross-correlation between ni(t) and n2(t) than otherwise. From a practical viewpoint gains obtainable over correlation methods become important for large values of the time-bandwidth product. This is due to the fact that under this condition the maximum likelihood estimate of t0 has an error variance that is approximately equal to the minimum variance obtainable by any estimate. For small values of the time-bandwidth product no theory exists about the variance of the maximum likelihood estimate, since no efficient estimate of t0 exists. For this case, a worthwhile experimental study would be to compare the variance of. the maximum likelihood estimate with the minimum variance calculated from the Cramer-Rao inequality. Finally it is shown how the maximum likelihood estimate is constructed and noted that the construction involves a considerable number of operations performed simultaneously during the time interval T. In order to simplify the system, an approximate maximum likelihood estimate is obtained that is optimum only "on target", and its properties are discussed.