Estimating The Period Of A Pulse Train From A Noisy Measurements

Abstract

The problem of estimation of the period of a pulse train from a set of sparse, noisy measurements of the times-ofarrival, and the association of pulse indices with the measurements, is considered for a pair of statistical models of the measurement process. We find that the estimation and association problem can be formulated as a simultaneous Diophantine approximation problem. We propose an algorithm for obtaining estimates and associations based on the LLL algorithm [SI. We also show a relationship with the maximisation of a certain trigonometric sum, which can be regarded as the periodogram of the measurements. We present some numerical results which indicate that the algorithm is able to make correct associations of pulse indices, and therefore accurate estimates of the period, with very high probability even for very sparse, short and noisy records. This is demonstrated in examples in which 99.9% of pulses are missiig and as few as 9 pulses are recorded with average errors of up t o 1% of the period on each measurement and yet the experimental frequency of correct association was more than 99%.