Ignacio Santamaría

Generalized correlation function: definition, properties, and application to blind equalization

With an abundance of tools based on kernel methods and information theoretic learning, a void still exists in incorporating both the time structure and the statistical distribution of the time series in the same functional measure. In this paper, a new generalized correlation measure is developed that includes the information of both the distribution and that of the time structure of a stochastic process. It is shown how this measure can be interpreted from a kernel method as well as from an information theoretic learning points of view, demonstrating some relevant properties. To underscore the effectiveness of the new measure, a simple blind equalization problem is considered using a coded signal.

Maximum Sum-Rate Interference Alignment Algorithms for MIMO Channels

Alternating minimization algorithms are typically used to find interference alignment (IA) solutions for multiple-input multiple-output (MIMO) interference channels with more than K=3 users. For these scenarios many IA solutions exit, and the initial point determines which one is obtained upon convergence. In this paper, we propose a new iterative algorithm that aims at finding the IA solution that maximizes the average sum-rate. At each step of the alternating minimization algorithm, either the precoders or the decoders are moved along the direction given by the gradient of the sum-rate. Since IA solutions are defined by a set of subspaces, the gradient optimization is performed on the Grassmann manifold. The step size of the gradient ascent algorithm is annealed to zero over the iterations in such a way that during the last iterations only the interference leakage is being minimized and a perfect alignment solution is finally reached. Simulation examples are provided showing that the proposed algorithm obtains IA solutions with significant higher throughputs than the conventional IA algorithms.

Detection of Rank-

Spectrum sensing is a key component of the cognitive radio paradigm. Primary signals are typically detected with uncalibrated receivers at signal-to-noise ratios (SNRs) well below decodability levels. Multiantenna detectors exploit spatial independence of receiver thermal noise to boost detection performance and robustness. We study the problem of detecting a Gaussian signal with rank-P unknown spatial covariance matrix in spatially uncorrelated Gaussian noise with unknown covariance using multiple antennas. The generalized likelihood ratio test (GLRT) is derived for two scenarios. In the first one, the noises at all antennas are assumed to have the same (unknown) variance, whereas in the second, a generic diagonal noise covariance matrix is allowed in order to accommodate calibration uncertainties in the different antenna frontends. In the latter case, the GLRT statistic must be obtained numerically, for which an efficient method is presented. Furthermore, for asymptotically low SNR, it is shown that the GLRT does admit a closed form, and the resulting detector performs well in practice. Extensions are presented in order to account for unknown temporal correlation in both signal and noise, as well as frequency-selective channels.

P

Gaussian processes (GPs) are versatile tools that have been successfully employed to solve nonlinear estimation problems in machine learning but are rarely used in signal processing. In this tutorial, we present GPs for regression as a natural nonlinear extension to optimal Wiener filtering. After establishing their basic formulation, we discuss several important aspects and extensions, including recursive and adaptive algorithms for dealing with nonstationarity, low-complexity solutions, non-Gaussian noise models, and classification scenarios. Furthermore, we provide a selection of relevant applications to wireless digital communications.

Signals in Cognitive Radio Networks With Uncalibrated Multiple Antennas

In this paper, the second-order circularity of quaternion random vectors is analyzed. Unlike the case of complex vectors, there exist three different kinds of quaternion properness, which are based on the vanishing of three different complementary covariance matrices. The different kinds of properness have direct implications on the Cayley-Dickson representation of the quaternion vector, and also on several well-known multivariate statistical analysis methods. In particular, the quaternion extensions of the partial least squares (PLS), multiple linear regression (MLR) and canonical correlation analysis (CCA) techniques are analyzed, showing that, in general, the optimal linear processing is full-widely linear. However, in the case of jointly Q-proper or Cη-proper vectors, the optimal processing reduces, respectively, to the conventional or semi-widely linear processing. Finally, a measure for the degree of improperness of a quaternion random vector is proposed, which is based on the Kullback-Leibler divergence between two zero-mean Gaussian distributions, one of them with the actual augmented covariance matrix, and the other with its closest proper version. This measure quantifies the entropy loss due to the improperness of the quaternion vector, and it admits an intuitive geometrical interpretation based on Kullback-Leibler projections onto sets of proper augmented covariance matrices.

Gaussian Processes for Nonlinear Signal Processing: An Overview of Recent Advances

In this paper, we introduce a kernel recursive least-squares (KRLS) algorithm that is able to track nonlinear, time-varying relationships in data. To this purpose, we first derive the standard KRLS equations from a Bayesian perspective (including a sensible approach to pruning) and then take advantage of this framework to incorporate forgetting in a consistent way, thus enabling the algorithm to perform tracking in nonstationary scenarios. The resulting method is the first kernel adaptive filtering algorithm that includes a forgetting factor in a principled and numerically stable manner. In addition to its tracking ability, it has a number of appealing properties. It is online, requires a fixed amount of memory and computation per time step, incorporates regularization in a natural manner and provides confidence intervals along with each prediction. We include experimental results that support the theory as well as illustrate the efficiency of the proposed algorithm.

Properness and Widely Linear Processing of Quaternion Random Vectors

This work addresses the problem of deciding whether a set of realizations of a vector-valued time series with unknown temporal correlation are spatially correlated or not. For wide sense stationary (WSS) Gaussian processes, this is a problem of deciding between two different power spectral density matrices, one of them diagonal. Specifically, we show that for arbitrary Gaussian processes (not necessarily WSS) the generalized likelihood ratio test (GLRT) is given by the quotient between the determinant of the sample space-time covariance matrix and the determinant of its block-diagonal version. Furthermore, for WSS processes, we present an asymptotic frequency-domain approximation of the GLRT which is given by a function of the Hadamard ratio (quotient between the determinant of a matrix and the product of the elements of the main diagonal) of the estimated power spectral density matrix. The Hadamard ratio is known to be the GLRT detector for vector-valued random variables and, therefore, what this paper shows is how frequency-dependent Hadamard ratios must be merged into a single test statistic when the vector-valued random variable is replaced by a vector-valued time series with temporal correlation. For bivariate time series, the derived frequency domain detector can be rewritten as a function of the well-known magnitude squared coherence (MSC) spectrum, which suggests a straightforward extension of the MSC spectrum to the general case of multivariate time series. Finally, the performance of the proposed method is illustrated by means of simulations.

Kernel Recursive Least-Squares Tracker for Time-Varying Regression

Canonical correlation analysis (CCA) is a classical tool in statistical analysis to find the projections that maximize the correlation between two data sets. In this work we propose a generalization of CCA to several data sets, which is shown to be equivalent to the classical maximum variance (MAXVAR) generalization proposed by Kettenring. The reformulation of this generalization as a set of coupled least squares regression problems is exploited to develop a neural structure for CCA. In particular, the proposed CCA model is a two layer feedforward neural network with lateral connections in the output layer to achieve the simultaneous extraction of all the CCA eigenvectors through deflation. The CCA neural model is trained using a recursive least squares (RLS) algorithm. Finally, the convergence of the proposed learning rule is proved by means of stochastic approximation techniques and their performance is analyzed through simulations.

Detection of Spatially Correlated Gaussian Time Series

In this letter we derive a tight analytical approximation for the outage capacity of orthogonal space-time block codes (STBCs). The proposed expression is a simple closed-form function of the power covariance matrix of the channel. In the case of uncorrelated channels, the expression only depends on the variances of the channel power gains that can be expressed analytically for the most common fading distributions: Rayleigh, Rice, Nakagami, Weibull, etc. Furthermore, the approximation encompasses different fading distributions and gains between different pairs of transmit and receive antennas, which can occur in distributed STBC networks.

A learning algorithm for adaptive canonical correlation analysis of several data sets

This paper investigates the application of error-entropy minimization algorithms to digital communications channel equalization. The pdf of the error between the training sequence and the output of the equalizer is estimated using the Parzen windowing method with a Gaussian kernel, and then, the Renyis quadratic entropy is minimized using a gradient descent algorithm. By estimating Renyis entropy over a short sliding window, an online training algorithm is also introduced. Moreover, for a linear equalizer, an orthogonality condition for the minimum entropy solution that leads to an alternative fixed-point iterative minimization method is derived. The performance of linear and nonlinear equalizers trained with entropy and mean square error (MSE) is compared. As expected, the results of training a linear equalizer are very similar for both criteria since, even if the input noise is non-Gaussian, the output filtered noise tends to be Gaussian. On the other hand, for nonlinear channels and using a multilayer perceptron (MLP) as the equalizer, differences between both criteria appear. Specifically, it is shown that the additional information used by the entropy criterion yields a faster convergence in comparison with the MSE.