Fisher’s information concentration
independent time delay and Doppler shift estimations
This paper presents an approach to obtain an invariant statistics for estimation interested signal parameters independently from unwanted parameters in two dimensional parameter problems. Considered parameters are the delay of signals of known shape and the value the Doppler shift of the spectrum.The proposed algorithm is based on the exclusion of Fisher’s information about the unwanted parameters and maintaining information about the parameter of interest. It uses the analysis of matrices composed of derivatives of signals on estimated parameters. These matrices together with the correlation matrix of noise are characterized by the Fisher information on the specific parameters of the signal and the information matrices are called partial.Simultaneous reduction the partial Fisher information matrices to diagonal forms provides the key steps for separation of the signal space into two orthogonal subspaces, containing Fisher’s information about different parameters. The proposed approach requires knowledge of the statistical distributions of signals of interest. Mathematical tool to perform this step is a singular value decomposition of matrices composed of derivatives of the signals on the estimated parameters. We present illustrative examples of the use of chirp signals and phase-shift keyed signals in accordance with the Gold code.The estimated parameters are the signal time delay and Doppler shift. They confirm the possibility of vector invariant statistics, preserving much of the Fisher information about the estimated parameter, which is the key to maintaining high accuracy of the estimate.