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
\pm 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
\pm 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_{s}T)^{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.