Analysis of inherent limitations in localizing step-like singularities in a continuous signal

The Cramer-Rao lower bound (CRLB) on the estimation error of the time of arrival of a continuous waveform with step-like singularities cannot be evaluated directly. Other performance bounds result in expressions which ignore the effect of finite processing band on the achievable performance. This paper presents a close-form expression for a Cramer-Rao type bound which describes the achievable performance of a processor of finite bandwidth in localizing a continuous signal with a step-like singularity in noise. The bound is put in terms of a wavelet expansion of the signal. Employing results from the theory of the wavelet transform, this expression is used to study inherent limitation, of the estimation problem. The validity of the analysis is verified by comparing it to the performance of the optimal processor, using Monte-Carlo simulations.