Time delay estimation in a sensor array

The estimation of time delays between elements in an array of M point sensors located in a discrete multipath propagation medium is considered. It is shown that, in fact, time delays are not estimated directly. Rather, a location vector
\underline{\Theta}
which parameterizes the relative source-receiving array geometry is estimated. In general, the relative sensor-to-sensor time delays are nonlinear functions of the location vector elements. The maximum-likelihood (ML) estimation process for
\underline{\Theta}
is established and shown to be realized by an extension of the so-called focused beam-former concept wherein wavefront shape matching time delays are generated. The form of the Cramér-Rao bound for the covariance matrix of the minimum mean-square error unbiased estimate of
\underline{\Theta}
is given.