Cramer-Rao Bound for Finite Streams of an Arbitrary Number of Pulses
Abstract
Sampling a finite stream of filtered pulses violates the bandlimited assumption of the Nyquist-Shannon sampling theory. However, recent low rate sampling schemes have shown that these sparse signals can be sampled with perfect reconstruction at their rate of innovation which is smaller than the Nyquist's rate. To reach this goal in the presence of noise, an estimation procedure is needed to estimate the time-delay and the amplitudes of each pulse. To assess the quality of any estimator, it is standard to use the Cramer-Rao Bound (CRB) which provides a lower bound on the Mean Squared Error (MSE) of any estimator. In this work, analytic expressions of the Cramer-Rao Bound are proposed for an arbitrary number of filtered pulses. Using orthogonality properties on the filtering kernels, an approximate compact expression of the CRB is provided. The choice of kernel is discussed from the point of view of estimation accuracy.