Cramér-Rao Bound for finite streams of pulses

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. 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 unbiased estimator, it is standard to use the Cramér-Rao Bound (CRB) which provides a lower bound on the Mean Squared Error (MSE) of any unbiased estimator. In this work, analytic expressions of the Cramér-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 the kernel is discussed from the point of view of the estimation accuracy.