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

Abstract

Abstract-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.

Analysis of Multicomponent Polynomial Phase Signals

“…2). For the PHAF, the set of lags are (16,14,13,11), in which the first lag is optimized in the HAF sense. Note that for a second order mc-PPS, we need only one lag for each set.…”

“…Hence, the element of Fisher's information matrix that is due to the interaction between and is (11) Using the same notation, it follows from (8) and (11) that (12) From the likelihood function (3), it can be shown that (13) The Cramér-Rao bound on the covariance of the estimate is found by taking the inverse of Fisher's information matrix .…”

“…Other issues relating to monocomponent polynomial phase signal (mono-PPS) analysis include aliasing [2], nonuniform sampling schemes [13] and bootstrap model selection [32]. However, in a number of practical situations such as analysis of a nonstationary signal in the presence of another nonstationary jamming signal, the multicomponent model is more relevant.…”

Auditory motivated level-crossing approach to instantaneous frequency estimation

“…For a detailed discussion on performance bounds for general nonuniform and random sampling schemes, see [31]. Consider the phase signal in zero-mean, complex additive Gaussian noise of variance to yield the noisy signal .…”

Cramer-Rao bound for a sparse complex model

“…The resulting expression of F p was already obtained in [8] for the irregular sampling case when M = 1, but the authors consider a more general polynomial phase complex-valued signal and they adopt a different parameterization by considering the estimation of the modulus and phase of c 1 instead of its real and imaginary parts. (ii) When a sum of distinct cisoids is considered as previously, a regular sampling is performed by choosing (∀q ∈ {1, .…”