Near Optimum Sampling for Single Tone Frequency Estimation

Work on single tone frequency estimation has focused on uniformly sampled data. However, it has been shown that, for a given number of samples, more information on the frequency of a signal can be gained by non-uniform sampling schemes. Unfortunately, an optimum sampling pattern (that, for example, minimizes the Cramér-Rao bound) does not automatically have a fast and simple algorithm for frequency estimation associated with it. For application in an interferometric measurement system, an algorithm is needed that gathers as much information as possible from a low number of samples, while at the same time keeping the computational effort sufficiently low to process millions of time series in a few seconds. One simple approximation to the optimum pattern can be obtained by 1) uniformly sampling blocks of data, 2) estimating phase and frequency in each of these and 3) exploiting these intermediate results in the final estimation. An approach to do so is investigated in detail. Results are compared to the Cramér-Rao bound (CRB), and it is shown that the algorithm almost reaches this limit on the variance of unbiased estimators, at a computational complexity lower than that of a typical FFT-based approach. For M=32 samples and a signal-to-noise ratio of 10, the standard deviation of the frequency estimate is lower by more than 50% compared to uniform sampling. In addition, the algorithm can easily be applied to poorly characterized systems, e.g. systems for which the noise is not known exactly.