Prediction of signal-to-noise ratio gain for passive higher-order correlation detection of energy transients

Abstract

In general, higher-order correlation detectors perform well in passive detection for signals of high third- and fourth-order moments. Previous studies by the authors have shown that the normalized third- and fourth-order signal moments are reliable indicators of higher-order correlation detector performance [Pflug et al. (1992b)]. For a deterministic energy transient of known moments through fourth order, it is possible to predict theoretically the amount of gain over an ordinary cross-correlation detector for a bicorrelation or tricorrelation detector applied in a noise environment of known variance. In this paper, formulas that predict detector performance for passive detection at the minimum detectable level are derived. The noise is assumed to be stationary and zero mean with Gaussian correlation central ordinate probability density functions. To test the formulas, SNR detection and gain curves are generated using hypothesis testing and Monte Carlo simulations on a set of test signals. The test signals are created by varying the time width of a pulse-like signal in a sampling window of fixed time duration, resulting in a set of test signals with varying signal moments. Good agreement is found between the simulated and theoretical results. The effects of observation time (length of detection window) and sampling interval on detector performance are also discussed and illustrated with computer simulations. The prediction formulas indicate that decreasing the observation time or the sampling interval (assuming the signal is sufficiently sampled and the detection window contains the entire signal) improves detection performance. However, the rate of improvement is different for the three detectors. The SNR required to achieve the minimum detectable level of detection performance at a given probability of false alarm (Pfa) decreases with the fourth root of the observation time and sampling interval for the cross-correlation detector, the sixth root for the bicorrelation detector, and the eighth root for the tricorrelation detector. Relative detector performance also varies with Pfa. The probability of detection (Pd) for higher-order detectors degrades less rapidly with decreasing Pfa than the Pd for ordinary correlations. Thus higher-order correlators can be especially appropriate when a very low Pfa is required.