Cramer-Rao lower bound of Doppler stretch and delay in wideband signal model

Abstract

Narrowband signal model, which approximates Doppler effect as a frequency shift, is inappropriate for some problems of radiolocation, hydroacoustics or passive location. In this case it is necessary to use Doppler stretch (time-scaling) as a signal parameter. In this paper Cramer-Rao lower bound of joint Doppler stretch and delay estimation for wideband signal model is derived. Maximum likelihood estimations in presence of nuisance parameters can be obtained by two ways. First method implies estimation of all unknown signal parameters and subsequent rejecting of nuisance. Second method involves averaging of conditional probability density function by nuisance parameters with considering it's priori probability density. Although Fisher information matrices of signal parameters are different for specified methods, estimation variances of Doppler stretch and delay are equal. In wideband signal model, the variance of time delay estimation depends on choice of reference axis. Varying the reference time leads to modifying of measurement moment with respect to signal. Since time delay is various parameter, its estimation variance is also variant. Appropriate choice of reference time results minimization of estimation variance and diagonalization of FIM. In the narrowband limit proposed consideration results in well-known CRLB of Doppler shift and delay estimation.