Performance bounds for polynomial phase parameter estimation with nonuniform and random sampling schemes

Estimating the parameters of a cisoid with an unknown amplitude and polynomial phase using uniformly spaced samples can result in ambiguous estimates due to Nyquist sampling limitations. It has been shown previously that nonuniform sampling has the advantage of unambiguous estimates beyond the Nyquist frequency; however, the effect of sampling on the Cramér–Rao bounds is not well known. This paper first derives the maximum likelihood estimators and Cramér–Rao bounds for the parameters with known, arbitrary sampling times. It then outlines two methods for incorporating random sampling times into the lower variance bounds, describing one in detail. It is then shown that for a signal with additive white, Gaussian noise the bounds for the estimation with nonuniform sampling tend toward those of uniform sampling. Thus, nonuniform sampling overcomes the ambiguity problems of uniform sampling without incurring the penalty of an increased variance in parameter estimation.