Differential delay-Doppler estimation using second- and higher-order ambiguity functions

Estimation of differential delay-Doppler parameters in passive sonar and radar applications is addressed. Without assuming Gaussianity it is shown that the conventional ambiguity function is asymptotically chi-squared and yields consistent delay-Doppler estimates for narrowband signals observed in uncorrelated sensor noises. Related asymptotic covariance expressions and their computable forms are also derived. It is further shown that the performance of the ambiguity function degrades severely when the sensor noises are correlated. To deal with such noises a novel third-order ambiguity function is proposed and is shown to be consistent and theoretically immune to Gaussian and symmetrically distributed disturbances. Further, to take into account the errors due to variances of sample statistics, alternative delay-Doppler estimators are defined to minimize a cumulant matching criterion and related issues are discussed. The case of wideband signals is also addressed and, finally, extensions to kth-order ambiguity functions are proposed. Computer simulations are performed to confirm the theory. Throughout the paper the analysis treats both deterministic and stochastic signals on a common framework. Connections with active sonar and radar problems are also shown as a special case.