Threshold Effects in Time-Delay Estimation via Cross-Correlation

The estimate of the difference in time of arrival of a common signal received at two sensors, each of which also receives uncorrelated noise, is shown to exhibit a thresholding effect similar to that in pulse position modulation. Thus, as the post detection signal-to-noise ratio decreases, the probability of an anomalous estimate (an estimate that is outside ±T_C/2, where T_C is the correlation time of the process) increases suddenly and precipitously. Theoretical results for the probability of an anomalous estimate are presented and verified experimentally. Threshold Is shown to depend most strongly on the two quantities M and A. M=2T_O/T_C, where ±T_O is the extent of the cross-correlation function searched for a peak; M is thus the effective number of independent correlation bins. A = (B_ST)^1/2{(S/N)/[(S/N)²+(1+S/N)²]^1/2} where B_S is the statistical bandwidth of the process, T is the total observation time, and S/N is the signal-to-noise ratio at the sensors. Depending on M, threshold is shown to occur in the region 3<; A<; 5.