Optimum vs. correlation methods in tracking random signals in background noise

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 three-dimensional problem utilizing three or more receivers. With $N$ receivers the computational difficulties increase as $N^2$. The tracking problem considered here is formulated as follows. During a time interval $0 \le t \le T$ one observes random processes $x_1(t) = s(t) + n_1(t)$ and $x_2(t) = s(t + \tau _0) + n_2 (t)$, and one desires to estimate the unknown value of $\tau _0 \cdot s(t), n_1(t)$, and $n_2(t)$ are continuous, Gaussian, stationary processes with continuous, monotonoid covariance functions and with $Es(t) n_1(t’) = Es(t) n_2(t’) = 0$. $n_1(t)$ and $n_2(t)$ 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, $\tau _0’$, of $\tau _0$, we use the one—commonly used in statistical literature, that $\tau _0’$ should have minimum variance about the true value $\tau _0$; that is, $E(\tau _0’ - \tau _0)^2$ be a minimum. It is shown that for small values of $\omega _u \tau _0$ (where $\omega _u$ is the highest frequency in the tracking pass band) the normalized error variance $E\left [ (\tau _0’ - \tau _0)/\tau _0 \right ]^2$ 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 Cramér-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 ${\tau _0}$ is always greater than the minimum variance given by the Cramér-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 $n_1(t)$ and $n_2(t)$ 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 $\tau _0$ 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 $\tau _0$ 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 Cramér-Rao inequality.