Known source detection predictions for higher order correlators

Abstract

The problem addressed in this paper is whether higher order correlation detectors can perform better in white noise than the cross correlation detector for the detection of a known transient source signal, if additional receiver information is included in the higher order correlations. While the cross correlation is the optimal linear detector for white noise, additional receiver information in the higher order correlations makes them nonlinear. In this paper, formulas that predict the performance of higher order correlation detectors of energy signals are derived for a known source signal. Given the first through fourth order signal moments and the noise variance, the formulas predict the SNR for which the detectors achieve a probability of detection of 0.5 for any level of false alarm, when noise at each receiver is independent and identically distributed. Results show that the performance of the cross correlation, bicorrelation, and tricorrelation detectors are proportional to the second, fourth, and sixth roots of the sampling interval, respectively, but do not depend on the observation time. Also, the SNR gains of the higher order correlation detectors relative to the cross correlation detector improve with decreasing probability of false alarm. The source signal may be repeated in higher order correlations, and gain formulas are derived for these cases as well. Computer simulations with several test signals are compared to the performance predictions of the formulas. The breakdown of the assumptions for signals with too few sample points is discussed, as are limitations on the design of signals for improved higher order gain. Results indicate that in white noise it is difficult for the higher order correlation detectors in a straightforward application to achieve better performance than the cross correlation. ͓S0001-4966͑98͒01805-0͔