Analytic inversion of Fisher´s information matrix for delay, delay rate, and higher derivatives

Fisher´s information matrix for delay (which corresponds to range), Doppler (range rate), and higher derivatives was recently presented by Schultheiss and Weinstein [1], [2]. Its inverse, which provides a lower bound on the error covariance matrix, was obtained numerically by them. An analytic inversion is presented here. It applies to the case of one reference signal (single-frequency) and one received signal, delayed and contaminated by white noise. The analytic inversion provides an insight on the increases in the variances of delay and Doppler estimations in the presence of higher derivatives. Asymptotically, the delay variance increases linearly with the highest derivative order, and the Doppler variance increases cubically.