Search

We develop an estimator of time offset (or time-of-arrival) of a transmitted communications signal that contains both pilot symbols, known to the receiver, and data symbols, unknown to the receiver. We focus on signalling constellations that have symbols lying on the complex unit circle, such as M-ary phase shift keying (M-PSK). We describe an algorithm for computing the estimator that requires O(L log L) operations in the worst case, where L is the number of transmitted symbols. Our estimator integrates information from the pilot symbols more effectively than popular estimators from the literature that usually split estimation into two subproblems called symbol timing and frame synchronisation. Our estimator combines these subproblems into a single operation, that of estimating time offset. We hypothesise that our estimator will be statistically more accurate. Monte-Carlo simulations support this hypothesis. 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 show that simple linear classification of pairwise products of convolutional features achieves near state-of-the-art performance on some standard labelled image databases. Specifically, we found test classification error rates on the MNIST handwritten digits image database of under 0.5%, and achieved under 19% and under 44% error rates on the CIFAR-10 and CIFAR-100 RGB image databases. Since the number of weights in such a classifier grows with the square of the number of features, we discuss how implementation of such a pair-wise products classifier can be achieved in an SLFN architecture where the hidden unit function is the simple quadratic nonlinearity: we can this a Quadratic Neural Network (QNN). We compare this method to setting the input weights in a QNN randomly, and find optimal performance can be achieved provided the hidden layer is sufficiently large. This analysis provides insight on why 'extreme-learning machines' can achieve classification performance equal to or better than the use of backpropagation training. 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

We consider the problem of distance, or range, estimation by measuring the phase of a sinusoidal signal transmitted between two locations. The distance can only be unambiguously measured if it is contained in an interval of length less than the wavelength of the signal. To increase the length of this interval, multiple phase measurements at different wavelengths can be used. The identifiable range is then extended to an interval of length equal to the least common multiple of the wavelengths. Phase measurements are noisy in practice and the problem of estimating range from multiple noisy phase measurements is non trivial. Existing solutions are based on least squares, the method of excess fractions, and on noise resilient versions of the Chinese remainder theorem (CRT). These estimators all, either explicitly, or implicitly, make an estimate of so called wrapping variables related to the whole number of wavelengths that occur over the range. In this paper, we discover an upper bound such that if all absolute phase measurement errors are less than this bound, then the least squares range estimator is guaranteed to correctly estimate the wrapping variables. We compare this with a similar bound discovered for estimators based on the CRT. The bound for the least squares estimator is often larger. This corroborates with existing empirical evidence suggesting that the least squares estimator is often more accurate. 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

The Coxeter lattices, which we denote $A_{n/m}$, are a family of lattices containing many of the important lattices in low dimensions. This includes $A_n$, $E_7$, $E_8$ and their duals $A_n^*$, $E_7^*$ and $E_8^*$. We consider the problem of finding a nearest point in a Coxeter lattice. We describe two new algorithms, one with worst case arithmetic complexity $O(n\log{n})$ and the other with worst case complexity O(n) where $n$ is the dimension of the lattice. We show that for the particular lattices $A_n$ and $A_n^*$ the algorithms reduce to simple nearest point algorithms that already exist in the literature., Comment: submitted to IEEE Transactions on Information Theory

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

We show that for those lattices of Voronoi’s first kind with known obtuse superbasis,a closest lattice point can be computed in O(n4) operations, where n is the dimension of the lattice.To achieve this a series of relevant lattice vectors that converges to a closest lattice point is found.We show that the series converges after at most n terms. Each vector in the series can be efficiently computed in O(n3) operations using an algorithm to compute a minimum cut in an undirected flow network. Refereed/Peer-reviewed

We examine the problem of estimating the periods of interleaved periodic point processes. We are particularly interested in the case where times of arrival (TOAs) are either measured with noise or not measured at all. This can arise in communications surveillance, where communications signals of different bauds may lie within the same surveillance bandwidth, and likewise in Electronic Surveillance (ES), where pulses or scans from different radars are observed together. In [1], the authors developed a general asymptotic theory for the Bartlett point-process periodogram estimator of the period of a single periodic process. In this paper, we extend the model to multiple periodic processes, each with a distinct period. The TOAs are observed unlabelled and in time order, i.e., they are interleaved. The largest local maximizers of the periodogram are shown to be good estimators of the unknown periods, asymptotically, and central limit theorems are proved. Simulations highlight a number of practical problems, and some problems with outstanding solutions are suggested.

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.

We consider least squares estimators of carrier phase and amplitude from a noisy communications signal. 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 not consistent, but converges to a positive real number that is a function of the true carrier amplitude, the noise distribution and the size of the constellation. The results of Monte Carlo simulations are provided and these corroborate the theoretical results. Refereed/Peer-reviewed

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

We show that for those lattices of Voronoi's first kind with known obtuse superbasis, a vector of shortest nonzero Euclidean length can computed in polynomial time by computing a minimum cut in a graph. Refereed/Peer-reviewed

Estimating the coefficients of a noisy polynomial phase signal is important in many fields including radar, biology and radio communications. One approach to estimation 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 therefore must be unwrapped before the regression can be performed. A recent approach suggested by the authors is to perform the unwrapping in a least squares manner. It was shown by Monte Carlo simulation that this produces a remarkably accurate estimator. In this paper we describe the asymptotic properties of this estimator, showing that it is strongly consistent and deriving its central limit theorem. We hypothesise that the estimator produces very near maximum likelihood performance. Refereed/Peer-reviewed

We consider the root lattice $A_n$ and derive explicit formulae for the moments of its Voronoi cell. We then show that these formulae enable accurate prediction of the error probability of lattice codes constructed from $A_n$., Comment: submitted to IEEE Transactions on Information Theory

We consider block noncoherent detection of hexagonal quadrature amplitude modulation (QAM). We focus on hexagonal constellations generated from a Voronoi code. We find that these constellations are particularly well suited to noncoherent detection because they avoid most of the identifiability problems that occur with more traditional constellations. We describe a fast, approximate, noncoherent detection algorithm with statistical performance that is almost indistinguishable from an optimal, brute force approach, but with significantly smaller computational requirements. The performance is close to that of coherent detection when the block length is sufficiently large. 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.

The lattice An* is an important lattice because of its covering properties in low dimensions. Two algorithms exist in the literature that compute the nearest point in the lattice AAn* in O(n log n) arithmetic operations. In this paper we describe a new algorithm that requires only O(n) operations. The new algorithm makes use of an approximate sorting procedure called a bucket sort. This is the fastest known nearest point algorithm for this lattice.

It has been well known for at least twenty years[1] that computing the maximizer of the periodogram, in orderto estimate the unknown frequency in a noisy sinusoid, isproblematic. In particular, because the periodogram is highlynonlinear, a grid size of order o T−1 is needed to find themaximizer reliably, where T is the sample size, and that Newton'smethod may fail to find the zero of the first derivative ofthe periodogram closest to the maximizer of the periodogramcalculated, for example, using the FFT. In this paper, we showthat Newton's method does, in fact, work if it is applied to anappropriately chosen monotonic function of the periodogram.

Ryan et al. [1] recently described two polynomial time algorithms for noncoherent detection of square QAM in block fading channels with additive white Gaussian noise (AWGN). The first algorithm is optimal with respect to the generalized likelihood ratio test (GLRT) and requires O(T 3) arithmetic computations, where T is the block length of the noncoherent receiver. The second algorithm requires only O(T 2 log T ) arithmetic computations but is statistically suboptimal. This paper derives a new algorithm that is optimal yet requires only O(T 2 log T ) arithmetic computations. The new algorithm has the geometric interpretation of finding the nearest codeword to a plane (2 dimensional subspace). The nearest codeword is found by testing codewords that are near a finite number of lines formed by the intersection of the plane and the nearest neighbour boundaries of the codewords.