Lower bounds on worst case probability of large error for two channel time delay estimation

A lower bound on the worst case probability of large error for two channel time delay estimation using random signals is developed. The bound is based on the minimum probability of error M-ary hypothesis test using a maximum likelihood estimator (MLE). The bound can only be evaluated approximately; however, it can be determined arbitrarily well via simulation if the MLE can be instrumented. The bound is evaluated in detail for uniform, lowpass signal and noise spectra and large bandwidth time product. It is then compared with another lower bound based on a binary hypothesis test, with computer simulation results for an approximate MLE instrumentation, and with an upper bound established via computer simulation for a correlator. For a uniform, lowpass spectrum the M-ary bound is shown to be very nearly a greatest lower bound over the full range of input signal to noise ratio. Further, the correlator is shown to be very nearly an optimal instrumentation in the sense of reaching the minimum probability of large error.