On the Cramer Rao bound and maximum likelihood in passive time delay estimation for complex signals

This paper is devoted to time delay estimation for wide sense stationary complex circular or noncircular Gaussian signals. Using a theorem by Whittle that we have extended to complex data, closed-form expressions of the Cramer Rao bound (CRB) are given for the time delay alone in presence of nuisance parameters. In particular, we prove that the CRB for the time delay is weakly reduced for noncircular signals w.r.t. circular signals, except for very low signal to noise ratios (SNR), for which the CRB for rectilinear signals is half of the CRB for circular signals. Then, the maximum likelihood (ML) estimate that extends the generalized cross correlation (GCC) estimate is derived.