Maximum likelihood real part of coherence between two noisy sensor outputs with zero-delay signal inputs

We seek, in particular at low signal-to-noise ratio, the coherence between the outputs of two sensors that receive a signal with ten, time-of-arrival difference but whose outputs include additive independent noises. This situation arises when the spectrum of a source is being investigated in a laboratory situation. The coherence measure may be used as a detection statistic for power at any FFT or DFT frequency; it may also be used indirectly as a signal power estimator since it is a function of SNR, and noise power is known. Because of the zero delay difference, ideal complex coherence has a zero imaginary part, and direct estimation of the real part seems appropriate. The maximum likelihood estimator for this and the corresponding Cramer-Rao bound are derived. Extensive simulations verify the derivations. The small-coherence approximation to the maximum likelihood solution gives a significant improvement in efficiency. Both the maximum likelihood solution and its small-coherence approximation give computational advantages over the conventional coherence estimator