In this work we consider a multistatic passive detection problem, which is motivated by a multiple-input multiple-output (MIMO) passive radar application. Specifically, we consider a single illuminator of opportunity (IO) and several surveillance and reference arrays for the detection of a Gaussian cyclostationary signal in temporally colored and spatially correlated noise. Concretely, 1) we derive the generalized likelihood ratio test (GLRT) for this problem and 2) provide a stochastic representation of the test statistic under the null hypothesis, which allows us to set the threshold for a constant probability of false alarm. Monte Carlo simulations are carried out to investigate the performance of the proposed GLRT.

This paper addresses the passive detection of a common signal in two multi-sensor arrays. For this problem, we derive a detector based on likelihood theory for the case of one-antenna transmitters, independent Gaussian noises with arbitrary spatial structure, Gaussian signals, and known channel subspaces. The detector uses a likelihood ratio where all but one of the unknown parameters are replaced by their maximum likelihood (ML) estimates. The ML estimation of the remaining parameter requires a numerical search, and it is therefore estimated using a sample-based estimator. The performance of the proposed detector is illustrated by means of Monte Carlo simulations and compared with that of the detector for unknown channels, showing the advantage of this knowledge.

Cooperative spectrum sensing has proved to be an effective method to improve the detection performance in cognitive radio systems. This work focuses on centralized cooperative schemes based on the soft fusion of the energy measurements at the cognitive radios (CRs). In these systems, the likelihood ratio test (LRT) is the optimal detection rule, but the sufficient statistic depends on the local signal-to-noise ratio (SNR) at the CRs, which are unknown in most practical cases. Therefore, the detection problem becomes a composite hypothesis test. The generalized LRT is the most popular approach in those cases. Unfortunately, in mobile environments, its performance is well below the LRT because the local energies are measured under varying SNRs. In this work, we present a new algorithm that jointly estimates the instantaneous SNRs and detects the presence of primary signals. Due to its adaptive nature, the algorithm is well suited for mobile scenarios where the local SNRs are time-varying. Simulation results show that its detection performance is close to the LRT in realistic conditions.

This paper addresses the problem of change-point detection in sequences of high-dimensional and heterogeneous observations, which also possess a periodic temporal structure . Due to the dimensionality problem, when the time between change points is of the order of the dimension of the model parameters, drifts in the underlying distribution can be misidentified as changes. To overcome this limitation, we assume that the observations lie in a lower-dimensional manifold that admits a latent variable representation. In particular, we propose a hierarchical model that is computationally feasible, widely applicable to heterogeneous data and robust to missing instances. Additionally, the observations’ periodic dependencies are captured by non-stationary periodic covariance functions . The proposed technique is particularly well suited to (and motivated by) the problem of detecting changes in human behavior using smartphones and its application to relapse detection in psychiatric patients. Finally, we validate the technique on synthetic examples and we demonstrate its utility in the detection of behavioral changes using real data acquired by smartphones.

Change-point detection (CPD) aims to locate abrupt transitions in the generative model of a sequence of observations. When Bayesian methods are considered, the standard practice is to infer the posterior distribution of the change-point locations. However, for complex models (high-dimensional or heterogeneous), it is not possible to perform reliable detection. To circumvent this problem, we propose to use a hierarchical model, which yields observations that belong to a lower-dimensional manifold. Concretely, we consider a latent-class model with an unbounded number of categories, which is based on the chinese-restaurant process (CRP). For this model we derive a continual learning mechanism that is based on the sequential construction of the CRP and the expectation-maximization (EM) algorithm with a stochastic maximization step. Our numerical results show that the proposed method is able to recursively infer the number of underlying latent classes and perform CPD in a reliable manner.

This paper addresses the problem of testing whether a covariance matrix can be expressed by an unknown linear combination of a set of known matrices or by another unknown linear combination of a set of different, but known, matrices. This problem is of interest in a wide range of real-world applications, such as radar, sonar, and spectrum sensing. We study the problem under the Gaussian assumption and derive the generalized likelihood ratio test (GLRT). Since there is no general closed-form solution for the maximum likelihood (ML) estimates of the covariance matrices, which are required for the GLRT, we resort to a powerful inverse iteration algorithm. Finally, an example, along with numerical results, is given to illustrate the methodology.

In array signal processing, the detection of the number of sources is an important step. Most approaches assume the signals to be embedded in white noise. However, this assumption is unrealistic in many scenarios. In this letter, we propose a strategy that can handle colored noise. We model the source detection as a regression problem and apply information-theoretic criteria to determine the model order of the regression. We show simulations of different scenarios, where our approach outperforms traditional techniques.

This paper introduces a novel method to assess the social activity maintained by psychiatric patients using information and communication technologies. In particular, we model the daily usage patterns of phone calls and social and communication apps using point processes. We propose a novel nonhomogeneous Poisson process model with periodic (circadian) intensity function using a truncated Fourier series expansion, which is inferred using a trust-region algorithm. We also extend the model using a mixture of periodic intensity functions to cope with the different daily patterns of a person. The analysis of the usage of phone calls and social and communication apps of a cohort of 259 patients reveals common patterns shared among patients with relatively high homogeneity and differences among patient pathologies.

This paper generalizes the classical joint problem of signal demixing and blind deconvolution to the realm of graphs. We investigate a setup where a single observation formed by the sum of multiple graph signals is available. The main assumption is that each individual signal is generated by an originally sparse input diffused through the graph via the application of a graph filter. In this context, we address the related problems of: 1) separating the individual graph signals, 2) identifying the unknown input supports, and 3) estimating the coefficients of the diffusing graph filters. We first consider the case where each signal - prior to mixing - is diffused in a different graph. We then particularize the results for the more challenging case where all the signals are diffused in the same graph. The corresponding demixing and blind graph-signal deconvolution problems are formulated, convex relaxations are presented, and recovery conditions are discussed. Numerical experiments in both the single and multiple graph cases show the capabilities of demixing in synthetic and biology-inspired graphs.

This work addresses the problem of determining whether two multivariate random time series have the same power spectral density (PSD), which has applications, for instance, in physical-layer security and cognitive radio. Remarkably, existing detectors for this problem do not usually provide any kind of optimality. Thus, we study here the existence under the Gaussian assumption of optimal invariant detectors for this problem, proving that the uniformly most powerful invariant test (UMPIT) does not exist. Thus, focusing on close hypotheses, we show that the locally most powerful invariant test (LMPIT) only exists for univariate time series. In the multivariate case, we prove that the LMPIT does not exist. However, this proof suggests two LMPIT-inspired detectors, one of which outperforms previously proposed approaches, as computer simulations show.

We propose a technique that jointly detects the presence of almost-cyclostationary (ACS) signals in wide-sense stationary noise and provides an estimate of their cycle period. Since the cycle period of an ACS process is not an integer, the approach is based on a combination of a resampling stage and a multiple hypothesis test, which deal separately with the fractional part and the integer part of the cycle period. The approach requires resampling the signal at many different rates, which is computationally expensive. For this reason, we propose a filter bank structure that allows us to efficiently resample a signal at many different rates by identifying common interpolation stages among the set of resampling rates.

In spectrum sensing for cognitive radio, the presence of a primary user can be detected by making use of the cyclostationarity property of digital communication signals. For the general scenario of a cyclostationary signal in temporally colored and spatially correlated noise, it has previously been shown that an asymptotic generalized likelihood ratio test (GLRT) and locally most powerful invariant test (LMPIT) exist. In this paper, we derive detectors for the presence of a cyclostationary signal in various scenarios with structured noise. In particular, we consider noise that is temporally white and/or spatially uncorrelated. Detectors that make use of this additional information about the noise process have enhanced performance. We have previously derived GLRTs for these specific scenarios; here, we examine the existence of LMPITs. We show that these exist only for detecting the presence of a cyclostationary signal in spatially uncorrelated noise. For white noise, an LMPIT does not exist. Instead, we propose tests that approximate the LMPIT, and they are shown to perform well in simulations. Finally, if the noise structure is not known in advance, we also present hypothesis tests using our framework.

This paper investigates the problems of signal reconstruction and blind deconvolution for graph signals that have been generated by an originally sparse input diffused through the network via the application of a graph filter operator. Assuming that the support of the sparse input signal is unknown, and that the diffused signal is observed only at a subset of nodes, we address the related problems of: 1) identifying the input and 2) interpolating the values of the diffused signal at the non-sampled nodes. We first consider the more tractable case where the coefficients of the diffusing graph filter are known and then address the problem of joint input and filter identification. The corresponding blind identification problems are formulated, novel convex relaxations are discussed, and modifications to incorporate a priori information on the sparse inputs are provided.

One approach to spectrum sensing for cognitive radio is the detection of cyclostationarity. We extend an existing multi-antenna detector for cyclostationarity proposed by Ramírez et al. [1], which makes no assumptions about the noise beyond being (temporally) wide-sense stationary. In special cases, the noise could be uncorrelated among antennas, or it could be temporally white. The performance of a general detector can be improved by making use of a priori structural information. We do not, however, require knowledge of the exact values of the temporal or spatial noise covariances. We develop an asymptotic generalized likelihood ratio test and evaluate the performance by simulations.

We propose a new detector of primary users in cognitive radio networks. The main novelty of the proposed detector in comparison to most known detectors is that it is based on sound statistical principles for detecting cyclostationary signals. In particular, the proposed detector is (asymptotically) the locally most powerful invariant test, i.e. the best invariant detector for low signal-to-noise ratios. The derivation is based on two main ideas: the relationship between a scalar-valued cyclostationary signal and a vector-valued wide-sense stationary signal, and Wijsman's theorem. Moreover, using the spectral representation for the cyclostationary time series, the detector has an insightful interpretation, and implementation, as the broadband coherence between frequencies that are separated by multiples of the cycle frequency. Finally, simulations confirm that the proposed detector performs better than previous approaches.

We derive the generalized likelihood ratio test (GLRT) for detecting cyclostationarity in scalar-valued time series. The main idea behind our approach is Gladyshev's relationship, which states that when the scalar-valued cyclostationary signal is blocked at the known cycle period it produces a vector-valued wide-sense stationary process. This result amounts to saying that the covariance matrix of the vector obtained by stacking all observations of the time series is block-Toeplitz if the signal is cyclostationary, and Toeplitz if the signal is wide-sense stationary. The derivation of the GLRT requires the maximum likelihood estimates of Toeplitz and block-Toeplitz matrices. This can be managed asymptotically (for large number of samples) exploiting Szegö's theorem and its generalization for vector-valued processes. Simulation results show the good performance of the proposed GLRT.

Detecting and analyzing directional structures in images is important in many applications since one-dimensional patterns often correspond to important features such as object contours or trajectories. Classifying a structure as directional or nondirectional requires a measure to quantify the degree of directionality and a threshold, which needs to be chosen based on the statistics of the image. In order to do this, we model the image as a random field. So far, little research has been performed on analyzing directionality in random fields. In this paper, we propose a measure to quantify the degree of directionality based on the random monogenic signal, which enables a unique decomposition of a 2-D signal into local amplitude, local orientation, and local phase. We investigate the second-order statistical properties of the monogenic signal for isotropic, anisotropic, and unidirectional random fields. We analyze our measure of directionality for finite-size sample images and determine a threshold to distinguish between unidirectional and nonunidirectional random fields, which allows the automatic classification of images.

This work presents a variation of canonical correlation analysis (CCA), where the correlation coefficient between the instantaneous power of the projections is maximized, rather than between the projections themselves. The resulting optimization problem is not convex, and we have to resort to a sub-optimal approach. Concretely, we propose a two-step solution consisting of the singular value decomposition (SVD) of a “coherence” matrix followed by a rank-one matrix approximation. This technique is applied to blindly recovering signals in a model that is motivated by the study of neuronal dynamics in humans using electroencephalography (EEG) and magnetoencephalography (MEG). A distinctive feature of this model is that it allows recovery of amplitude-amplitude coupling between neuronal processes.

In this paper, we study the existence of locally most powerful invariant tests (LMPIT) for the problem of testing the covariance structure of a set of Gaussian random vectors. The LMPIT is the optimal test for the case of close hypotheses, among those satisfying the invariances of the problem, and in practical scenarios can provide better performance than the typically used generalized likelihood ratio test (GLRT). The derivation of the LMPIT usually requires one to find the maximal invariant statistic for the detection problem and then derive its distribution under both hypotheses, which in general is a rather involved procedure. As an alternative, Wijsman's theorem provides the ratio of the maximal invariant densities without even finding an explicit expression for the maximal invariant. We first consider the problem of testing whether a set ofN-dimensional Gaussian random vectors are uncorrelated or not, and show that the LMPIT is given by the Frobenius norm of the sample coherence matrix. Second, we study the case in which the vectors under the null hypothesis are uncorrelated and identically distributed, that is, the sphericity test for Gaussian vectors, for which we show that the LMPIT is given by the Frobenius norm of a normalized version of the sample covariance matrix. Finally, some numerical examples illustrate the performance of the proposed tests, which provide better results than their GLRT counterparts.

Spectrum sensing is a key component of the cognitive radio (CR) paradigm. Among CR detectors, multiantenna detectors are gaining popularity since they improve the detection performance and are robust to noise uncertainties. Traditional approaches to multiantenna spectrum sensing are based on the generalized likelihood ratio test (GLRT) or other heuristic detectors, which are not optimal in the Neyman-Pearson sense. In this work, we derive the locally most powerful invariant test (LMPIT), which is the optimal detector, among those preserving the problem invariances, in the low SNR regime. In particular, we apply Wijsman's theorem, which provides us an alternative way to derive the ratio of the distributions of the maximal invariant statistic. Finally, numerical simulations illustrate the performance of the proposed detector.

The monogenic signal allows us to decompose a two-dimensional real signal into a local amplitude, a local orientation, and a local phase. In this paper, we introduce the random monogenic signal and study its second-order statistical properties. The monogenic signal may be represented as a quaternion-valued signal. We show that for homogeneous random fields, we need exactly two quaternion-valued covariance functions for a complete second-order description. We also introduce a stochastic model for unidirectional signals and a measure of unidirectionality.

Spectrum sensing is a challenging key component of the Cognitive Radio paradigm, since primary signals must be detected in the face of noise uncertainty and 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. Here, we study the problem of detecting Gaussian signals with unknown rank-P spatial covariance matrix when the noise at the receiver is independent across the antennas and with unknown power. A generic diagonal noise covariance matrix is allowed to model calibration uncertainties in the different antenna frontends. We derive the generalized likelihood ratio test (GLRT) for this detection problem. Although, in general, the corresponding statistic must be obtained by numerical means, in the low SNR regime the GLRT does admit a closed form. Numerical simulations show that the proposed asymptotic detector offers good performance even for moderate SNR values.

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. Specifically, the spatial correlation is induced by a colored source over a frequency-flat single-input multiple-output (SIMO) channel distorted by independent and identically distributed noises with temporal correlation. The generalized likelihood ratio test (GLRT) for this detection problem does not have a closed-form expression and we have to resort to numerical optimization techniques. In particular, we apply the successive convex approximations approach which relies on solving a series of convex problems that approximate the original (non-convex) one. The proposed solution resembles a power method for obtaining the dominant eigenvector of a matrix, which changes over iterations. Finally, the performance of the proposed detector is illustrated by means of computer simulations showing a great improvement over previously proposed detectors that do not fully exploit the temporal structure of the source.

One of the key problems in cognitive radio (CR) is the detection of primary activity in order to determine which parts of the spectrum are available for opportunistic access. This detection task is challenging, since the wireless environment often results in very low SNR conditions. Moreover, calibration errors and imperfect analog components at the CR spectral monitor result in uncertainties in the noise spectrum, making the problem more difficult. In this work, we present a new multiantenna detector which is based on the fact that the observation noise processes are spatially uncorrelated, whereas any primary signal present should result in spatial correlation. In particular, we derive the generalized likelihood ratio test (GLRT) for this problem, which is given by the quotient between the determinant of the sample covariance matrix and the determinant of its block-diagonal version. For stationary processes the GLRT tends asymptotically to the integral of the logarithm of the Hadamard ratio of the estimated power spectral density matrix. Additionally, we present an approximation of the frequency domain detector in the low SNR regime, which results in computational savings. The performance of the proposed detectors is evaluated by means of numerical simulations, showing important advantages over existing detectors.

In this paper the two main definitions of quaternion properness (or second order circularity) are reviewed, showing their connection with the structure of the optimal quaternion linear processing. Specifically, we present a rigorous generalization of the most common multivariate statistical analysis techniques to the case of quaternion vectors, and show that the different kinds of quaternion improperness require different kinds of widely linear processing. In general, the optimal linear processing is full-widely linear, which requires the joint processing of the quaternion vector and its involutions over three pure unit quaternions. However, in the case of jointly ℚ-proper and ℂη-proper vectors, the optimal processing reduces, respectively, to the conventional and semi-widely linear processing, with the latter only requiring to operate on the quaternion vector and its involution over the pure unit quaternion η. Finally, a simulation example poses some interesting questions for future research.

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.

The coherence spectrum is a well-known measure of the linear statistical relationship between two time series. In this paper, we extend this concept to several processes and define the generalized magnitude squared coherence (GMSC) spectrum as a function of the largest eigenvalue of a matrix containing all the pairwise complex coherence spectra. The GMSC is bounded between zero and one, and attains its maximum when all the processes are perfectly correlated at a given frequency. Furthermore, three different GMSC spectrum estimators, extending those previously proposed for the MSC of two processes, are presented. Specifically, we compare the Welch method, the minimum variance distortionless response (MVDR) estimator and a new estimator based on canonical correlation analysis (CCA).

Abstract In this paper we employ a 2 × 2 Multiple‐Input Multiple‐Output (MIMO) hardware platform to evaluate, in realistic indoor scenarios, the performance of different space‐time block coded (STBC) transmissions at 2.4 GHz. In particular, we focus on the Alamouti orthogonal scheme considering two types of channel state information (CSI) estimation: a conventional pilot‐aided supervised technique and a recently proposed blind method based on second‐order statistics (SOS). For comparison purposes, we also evaluate the performance of a Differential (non‐coherent) space‐time block coding (DSTBC). DSTBC schemes have the advantage of not requiring CSI estimation but they incur in a 3 dB loss in performance. The hardware MIMO platform is based on high‐performance signal acquisition and generation boards, each one equipped with a 1 GB memory module that allows the transmission of extremely large data frames. Upconversion to RF is performed by two RF vector signal generators whereas downconversion is carried out with two custom circuits designed from commercial components. All the baseband signal processing is implemented off‐line in MATLAB®, making the MIMO testbed very flexible and easily reconfigurable. Using this platform we compare the performance of the described methods in line‐of‐sight (LOS) and non‐line‐of‐sight (NLOS) indoor scenarios. Copyright © 2007 John Wiley & Sons, Ltd.

In this paper, a new second-order statistics (SOS) based method for blind decoding of orthogonal space time block coded (OSTBC) systems with only one receive antenna is proposed. To avoid the inherent ambiguities of this problem, the spatial correlation matrix of the source signals must be non-white and known at the receiver. In practice, this can be achieved by a number of simple linear precoding techniques at the transmitter side. More specifically, it is shown in the paper that if the source correlation matrix has different eigenvalues, then the decoding process can be formulated as the problem of maximizing the sum of a set of weighted variances of the signal estimates. Exploiting the special structure of OSTBCs, this problem can be reduced to a principal component analysis (PCA) problem, which allows us to derive computationally efficient batch and adaptive blind decoding algorithms. The algorithm works for any OSTBC (including the popular Alamouti code) with a single receive antenna. Some simulation results are presented to demonstrate the potential of the proposed procedure