Performance bound for time delay and amplitude estimation from low rate samples of pulse trains

The problem considered is estimating the time delays and amplitudes of a train of pulses of known shape. It was recently shown that in the absence of noise, if the pulses are short relative to the interval separating them, the stream of pulses can be sampled at rates lower than the Nyquist rate without any loss of information. This paper investigates the performance of time delay and amplitude estimation from samples taken at low (sub-Nyquist) rates when the stream of pulses is affected by additive white Gaussian noise. To this end, the Cramér-Rao bound (CRB) is developed. For a particular setup, the CRB for time delay estimation is inverse proportional to the cube of the sampling rate, while the CRB for amplitude estimation is inverse proportional to the sampling rate. Numerical simulations confirm this result.