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We consider least squares estimators of carrier phase and amplitude from a noisy communications signal that contains both pilot signals, known to the receiver, and data signals, unknown to the receiver. We focus on signaling constellations that have symbols evenly distributed on the complex unit circle, i.e., M-ary phase shift keying. We show, under reasonably mild conditions on the distribution of the noise, that the least squares estimator of carrier phase is strongly consistent and asymptotically normally distributed. However, the amplitude estimator is asymptotically biased and converges to a positive real number that is a function of the true carrier amplitude, the noise distribution and the size of the constellation. This appears to be the first time that the statistical properties of a non-data-aided estimator for carrier amplitude have been analyzed theoretically. The results of Monte Carlo simulations are provided and these agree with the theoretical results. Refereed/Peer-reviewed

Polynomial-phase signals have applications including radar, sonar, biology, and radio communication. Of practical importance is the estimation of the parameters of a polynomial phase signal from a sequence of noisy observations. Assuming that the noise is additive and Gaussian, the direct evaluation of the Cramér–Rao lower bound for this estimation problem involves evaluating the inverse of a matrix. Computing this inverse is numerically difficult for polynomial phase signals of large order. By making use of a family of orthogonal polynomials, we derive formulae for the Cramér–Rao bounds that are numerically stable and easy to compute. Refereed/Peer-reviewed

We consider the problem of estimating the distance, or range, between two locations by measuring the phase of multiple sinusoidal signals transmitted between the locations. Traditional estimators developed for optical interferometry include the beat wavelength and excess fractions methods. More recently, estimators based on the Chinese remainder theorem (CRT) and least squares have appeared. Recent research suggests the least squares estimator to be the most accurate in many cases. The accuracy of all of these range estimators depends upon the wavelengths chosen. This leads to the problem of selecting wavelengths that maximize accuracy. Procedures for selecting wavelengths for the beat wavelength and excess fractions methods have previously been described, but procedures for the CRT and least squares estimators are yet to be developed. In this paper, we develop an algorithm to automatically select wavelengths for use with the least square range estimator. The algorithm minimizes an optimization criterion connected with the mean square error. Interesting properties of a particular class of lattices simplify the criterion allowing minimization by depth first search. Monte-Carlo simulations indicate that wavelengths that minimize the criterion can result in considerably more accurate range estimates than wavelengths selected by ad hoc means. Refereed/Peer-reviewed

Circular statistics has found substantial application in science and engineering. One of the fundamental problems in circular statistics is that of estimating the mean direction of a circular random variable from a number of observations. The standard approach in the literature is called the sample circular mean and its asymptotic properties are well known. It can also be computed efficiently in a number of arithmetic operations that is linear in the number of observations. In this paper we consider an alternative estimator called the sample intrinsic mean that is based on minimizing squared arc length. We show how this estimator can be computed efficiently in a linear number of operations using an algorithm from algebraic number theory and we derive its asymptotic properties. In some scenarios the sample circular mean and the sample intrinsic mean are estimators of the same quantity and can therefore be compared. We show both theoretically and by simulation that in some of these scenarios the sample intrinsic mean is statistically more accurate than the sample circular mean. As such the results in this paper potentially have implications for the wide variety of fields in science, engineering and statistics that currently use the sample circular mean. Refereed/Peer-reviewed

Polynomial-phase signals have numerous applications including radar, sonar, geophysics, and radio communication. Many techniques for estimating the parameters of polynomial-phase signals have been described in the literature. Despite the significant interest, aliasing of polynomial-phase parameters has not been fully clarified. We address the problem of identifiability and aliasing in polynomial-phase signals. We fully describe the region in which aliasing does not occur for polynomial-phase signals of any order. We call this the identifiable region. We find that this region is the Voronoi region of a lattice generated by the coefficients of a set of polynomials known as the integer-valued polynomials. We show how aliasing can be resolved by solving the nearest lattice point problem. We discuss some of the consequences of these results on a popular estimator for polynomial-phase signals that is based on the discrete polynomial phase transform (DPPT). It is shown that the range of parameters suitable for the DPPT estimator is very small compared to the identifiable region.

We consider the problem of estimating the distance, or range, between two locations by measuring the phase of a sinusoidal signal transmitted between the locations. This method is only capable of unambiguously measuring range within an interval of length equal to the wavelength of the signal. To address this problem signals of multiple different wavelengths can be transmitted. The range can then be measured within an interval of length equal to the least common multiple of these wavelengths. Estimation of the range requires solution of a problem from computational number theory called the closest lattice point problem. Algorithms to solve this problem require a basis for this lattice. Constructing a basis is non-trivial and an explicit construction has only been given in the case that the wavelengths can be scaled to pairwise relatively prime integers. In this paper we present an explicit construction of a basis without this assumption on the wavelengths. This is important because the accuracy of the range estimator depends upon the wavelengths. Simulations indicate that significant improvement in accuracy can be achieved by using wavelengths that cannot be scaled to pairwise relatively prime integers., Comment: submitted to IEEE Signal Processing Letters

Estimating the coefficients of a noisy polynomial phase signal is important in fields including radar, biology and radio communications. One approach attempts to perform polynomial regression on the phase of the signal. This is complicated by the fact that the phase is wrapped modulo 2π and must be unwrapped before regression can be performed. In this paper, we consider an estimator that performs phase unwrapping in a least squares manner. We call this the least squares unwrapping (LSU) estimator. The LSU estimator can be computed in a reasonable amount of time for data sets of moderate size using existing general purpose algorithms from algebraic number theory. Under mild conditions on the distribution of the noise we describe the asymptotic properties of this estimator, showing that it is strongly consistent and asymptotically normally distributed. A key feature is that the LSU estimator is accurate over a far wider range of parameters than many popular existing estimators. Monte-Carlo simulations support our theoretical results and demonstrate the excellent statistical performance of the LSU estimator when compared with existing state-of-the-art estimators. Refereed/Peer-reviewed

The problem discussed is that of estimating the period of a sequence of periodic events when the occurrence time measurements are noisy and sparse. The problem arises in signal processing applications such as baud estimation from zero-crossings in telecommunications and in pulse repetition interval estimation in electronic support measures. Estimation techniques have been based on periodogram maximisation [1][2], Euclidean algorithms [3][4][5], least squares line search [6], lattice line search [7], Gaussian maximum likelihood [8] and least squares [9]. Aside from [9], there has been no rigorous statistical analysis. In this paper, we show that the periodogram maximiser has excellent (theoretical) asymptotic statistical properties, illustrating them via simulation.

The problem discussed in this paper is that of estimating the period of a sequence of periodic events when the measurements of the occurrence times are noisy and sparse. The problem is common to many signal processing applications, such as baud estimation from zero-crossings in telecommunications and in pulse repetition interval estimation in electronic support measures. Previous algorithms have been based on periodogram maximisation [1, 2], Euclidean algorithms [3-5], least-squares line search [6], lattice line search [7] and Gaussian maximum likelihood [8]. Until now, very little has been known about the asymptotic statistical properties of any such algorithm. In this paper, a new algorithm is proposed, based on a modified least-squares approach. Under very general properties, the estimators of the system parameters are shown to have excellent (theoretical) asymptotic statistical properties. These properties are illustrated using a number of simulations. Refereed/Peer-reviewed

Single frequency estimation is a long-studied problem with application domains including radar, sonar, telecommunications, astronomy and medicine. One method of estimation, called phase unwrapping, attempts to estimate the frequency by performing linear regression on the phase of the received signal. This procedure is complicated by the fact that the received phase is ‘wrapped’ modulo ???? and therefore must be ‘unwrapped’ before the regression can be performed. In this paper, we propose an estimator that performs phase unwrapping in the least squares sense. The estimator is shown to be strongly consistent and its asymptotic distribution is derived.We then show that the problem of computing the least squares phase unwrapping is related to a problem in algorithmic number theory known as the nearest lattice point problem. We derive a polynomial time algorithm that computes the least squares estimator. The results of various simulations are described for different values of sample size and SNR.