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.

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.

The problem of estimating the period of a point process from observations that are both sparse and noisy is considered. By sparse it is meant that only a potentially small unknown subset of the process is observed. By noisy it is meant that the subset that is observed, is observed with error, or noise. Existing accurate algorithms for estimating the period require O(N2) operations where is the number of observations. By quantizing the observations we produce an estimator that requires only O(N log N) operations by use of the chirp z-transform or the fast Fourier transform. The quantization has the adverse effect of decreasing the accuracy of the estimator. This is investigated by Monte-Carlo simulation. The simulations indicate that significant computational savings are possible with negligible loss in statistical accuracy.

We show that for those lattices of Voronoi's first kind with known obtuse superbasis, a closest lattice point can be computed in $O(n^4)$ 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 $O(n^3)$ operations using an algorithm to compute a minimum cut in an undirected flow network.

The problem of synchronizing weak user signals in multiuser code-division multiple access systems is addressed. We integrate timing synchronization into an iterative multiuser detection (MUD) framework for the improved synchronization performance of weak users. First, the effect of correctly detected users on the remaining undetected users is suppressed via soft parallel interference cancellation. Then, the time offset (TO) of each undetected user is estimated. To speed up the MUD, interference cancellation is performed after only a few decoder iterations of the detected users, and the TO estimator is designed taking into account the residual multiple access interference. A low-complexity non-data-aided TO estimator that estimates the TO of each undetected user separately is proposed. The proposed technique significantly improves the synchronization performance of weak users under high near-far conditions compared to the full interference case. Assuming Gaussian distributed log-likelihood ratios at the decoder outputs, we directly relate the missed detection probability of a desired user to the root-mean-squared error of the detected user's data sequences. The extrinsic information transfer chart of the decoder is used to analyze the acquisition performance of the desired user with the number of decoder iterations of the detected users. An upper bound for the missed detection probability of the proposed estimator is derived.

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.

We consider the estimation of signal amplitude, time offset, and the parameters of a polynomial-phase signal from a continuous-phase modulated (CPM) signal that contains both unknown data symbols and known pilot symbols. Transformation of the polynomial-phase signal into an orthogonal basis allows us to derive the modified Cramér-Rao bounds (MCRB) for the problem of vector-parameter estimation in closed form. Numerical results demonstrate that the achievable estimation accuracy significantly depends on the burst structure and highlight the need for properly designed pilot sequences. Since our bounds are easy to evaluate, they can aid the design of pilot and burst configuration for CPM systems.

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.

This paper focuses on the problem of code acquisition in multiuser CDMA systems in a near-far scenario. A new time offset estimator which iteratively cancels interference from correctly acquired users is proposed. Interference cancellation is accomplished by projecting the received signal into the null space of already acquired users. The proposed technique performs well under low SNR conditions in the presence of high multiple access interference. Analytical expressions for the missed detection probability of the proposed estimator are derived and these agree with the results of Monte-Carlo simulations.

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.

We consider the root lattice$A_{n}$and derive explicit recursive formulas for the moments of its Voronoi cell. These formulas enable accurate prediction of the error probability of lattice codes constructed from$A_{n}$.

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.

The Coxeter lattices are a family of lattices containing many of the important lattices in low dimensions. This includesAn,E7,E8and their dualsAn*,E7*, andE8*. We consider the problem of finding a nearest point in a Coxeter lattice. We describe two new algorithms, one with worst case arithmetic complexityO(nlogn) and the other with worst case complexityO(n) wherenis the dimension of the lattice. We show that for the particular latticesAnandAn* the algorithms are equivalent to nearest point algorithms that already exist in the literature.

We propose a new algorithm for noncoherent sequence detection of M-ary phase-shift-keying (M-PSK) symbols transmitted over a block fading channel. The algorithm is of complexity O(T), where T is the sequence length, and is therefore computationally superior to existing maximum-likelihood (ML) detectors of complexity O(T logT). Our detector is based on a new approximation we propose to the noncoherent ML function. We show that by using this close approximation, the detection problem reduces to a nearest lattice point problem for the lattice An*, from which we derive our O(T) approach. Simulation results are provided that show the difference in bit error rate is negligibly small for a wide range of signal-to-noise ratios.

It has been well known for at least twenty years that computing the maximizer of the periodogram, in order to estimate the unknown frequency in a noisy sinusoid, is problematic. In particular, because the periodogram is highly nonlinear, a grid size of order o (T-1) is needed to find the maximizer reliably, where T is the sample size, and that Newton's method may fail to find the zero of the first derivative of the periodogram closest to the maximizer of the periodogram calculated, for example, using the FFT. In this paper, we show that Newton's method does, in fact, work if it is applied to an appropriately chosen monotonic function of the periodogram.

The problem of estimating the period of a periodic point process is considered when the observations are sparse and noisy. There is a class of estimators that operate by maximizing an objective function over an interval of possible periods, notably the periodogram estimator of Fogel & Gavish and the line-search algorithms of Sidiropoulos et al. and Clarkson. For numerical calculation, the interval is sampled. However, it is not known how fine the sampling must be in order to ensure statistically accurate results. In this paper, a new estimator is proposed which eliminates the need for sampling. For the proposed statistical model, it calculates a maximum- likelihood estimate. It is shown that the expected arithmetic complexity of the algorithm is O(n3log n) where n is the number of observations. Numerical simulations demonstrate the superior statistical performance of the new estimator.

The latticeAn*is an important lattice because of its covering properties in low dimensions. Clarkson described an algorithm to compute the nearest lattice point inAn*that requiresO(nlogn) arithmetic operations. In this correspondence, we describe a new algorithm. While the complexity is stillO(nlogn), it is significantly simpler to describe and verify. In practice, we find that the new algorithm also runs faster.