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This paper considers the problem of distilling a secret key in a Gaussian multiple-input multiple-output (MIMO) scenario with two legitimate nodes and an eavesdropper. Focusing on the realistic case without perfect knowledge of the eavesdropper channel, and following a conservative practical approach based on the maximization of the worst case secret key capacity (SKC), the problem of designing the optimal transmit covariance matrix is reformulated as a convex optimization problem. In the limiting case in which the eavesdropper channel can not be estimated, or when the estimate is highly unreliable, the optimal covariance matrix can be obtained by means of waterfilling or matched filtering like algorithms. Additionally, we illustrate the benefits of allowing time sharing between transmissions of the two legitimate nodes, and provide an efficient algorithm for obtaining the optimal transmit covariance matrices and time-sharing factor. ; This work was supported by the Spanish Government, Ministerio de Ciencia e Innovación (MICINN), under projects COSIMA (TEC2010-19545-C04-03) and COMONSENS (CSD2008-00010, CONSOLIDER INGENIO 2010 Program).

This paper revisits the linear model with noisy inputs, in which the performance of the total least squares (TLS) method is far from acceptable. Under the assumption of Gaussian noises, the maximum likelihood (ML) estimation of the system response is reformulated as a general balanced least squares (BLS) problem. Unlike TLS, which minimizes the trace of the product between the empirical and inverse theoretical covariance matrices, BLS promotes solutions with similar values of both the empirical and theoretical error covariance matrices. The general BLS problem is reformulated as a semidefinite program with a rank constraint, which can be relaxed in order to obtain polynomial time algorithms. Moreover, we provide new theoretical results regarding the scenarios in which the relaxation is tight, as well as additional insights on the performance and interpretation of BLS. Finally, some simulation results illustrate the satisfactory performance of the proposed method. ; This work has been supported by the Spanish Government, Ministerio de Ciencia e Innovación, under project RACHEL (TEC2013-47141-C4-3-R)

This paper focuses on a linear model with noisy inputs in which the performance of the conventional Total Least Squares (TLS) approach is (maybe surprisingly) far from satisfactory. Under the typical Gaussian assumption, we obtain the maximum likelihood (ML) estimator of the system response. This estimator promotes a reasonable balance between the empirical and theoretical variances of the residual errors, which suggests the name of Balanced Least Squares (BLS). The solution of the associated optimization problem is based on its reformulation as a rank constrained semidefinite program (SDP), for which we show that the relaxation is tight with probability one. Both TLS and BLS can be seen as regularized LS estimators, but the (possibly negative) regularization in BLS is softer than its TLS counterpart, which avoids the inconsistency of TLS in our particular model. ; This work has been supported by the Spanish Government, Ministerio de Ciencia e Innovación, under project RACHEL (TEC2013-47141-C4-3-R).

Discriminatory channel estimation emerges as a promising method of not only increasing the secrecy rates in conventional wiretap channels, but also providing a valuable tool for solving the authentication problem. In this paper, we revisit the discriminatory channel estimation method by Chang et al. and propose a generalization to the challenging scenario in which the number of antennas at the legitimate receiver equals or exceeds those of the transmitter. The proposed method is based on the simple idea of dividing the receiver antennas into smaller groups. However, the direct application of previous approaches would result into security problems due to the multiple observations of the eavesdropper, and therefore the transmission system needs to be designed taking this fact into account. The performance of the proposed technique is illustrated by means of some numerical examples, which clearly show the feasibility of discriminatory channel estimation even in the case of systems with more antennas at the receiver side. ; This work has been supported by the Spanish Government, Ministerio de Ciencia e Innovación, under project RACHEL (TEC2013-47141-C4-3-R).

In this paper, we describe an efficient iterative algorithm for finding sparse solutions to a linear system. Apart from the well-known L1 norm regularization, we introduce an additional cost term promoting solutions without too-close activations. This additional term, which is expressed as a sum of cross-products of absolute values, makes the problem nonconvex and difficult to solve. However, the application of the successive convex approximations approach allows us to obtain an efficient algorithm consisting in the solution of a sequence of iteratively reweighted LASSO problems. Numerical simulations on randomly generated waveforms and ECG signals show the good performance of the proposed method. ; This work has been partly funded by the Spanish government through the KERMES excellence network (ref. TEC2016-81900-REDT).

In this work we consider a two-channel passive detection problem, in which there is a surveillance array where the presence/absence of a target signal is to be detected, and a reference array that provides a noise-contaminated version of the target signal. We assume that the transmitted signal is an unknown rank-one signal, and that the noises are uncorrelated between the two channels, but each one having an unknown and arbitrary spatial covariance matrix. We show that the generalized likelihood ratio test (GLRT) for this problem rejects the null hypothesis when the largest canonical correlation of the sample coherence matrix between the surveillance and the reference channels exceeds a threshold. Further, based on recent results from random matrix theory, we provide an approximation for the null distribution of the test statistic. ; The work of I. Santamaría was supported by the Spanish Government through grants PRX14/0028 (Estancias de Movilidad de Profesores, Ministerio de Educación) and by project RACHEL (TEC2013-47141-C4-3-R) funded by the Ministerio de Economía y Competitividad (MINECO). The work of L. Scharf and D. Cochran was supported in part by a sub-contract with Matrix Research for research sponsored by the Air Force Research Laboratory under contract FA8650-14-D-1722.

The problem of blind sparse analysis of electrogram (EGM) signals under atrial fibrillation (AF) conditions is considered in this paper. A mathematical model for the observed signals that takes into account the multiple foci typically appearing inside the heart during AF is firstly introduced. Then, a reconstruction model based on a fixed dictionary is developed and several alternatives for choosing the dictionary are discussed. In order to obtain a sparse solution, which takes into account the biological restrictions of the problem at the same time, the paper proposes using a Least Absolute Shrinkage and Selection Operator (LASSO) regularization followed by a post-processing stage that removes low amplitude coefficients violating the refractory period characteristic of cardiac cells. Finally, spectral analysis is performed on the clean activation sequence obtained from the sparse learning stage in order to estimate the number of latent foci and their frequencies. Simulations on synthetic signals and applications on real data are provided to validate the proposed approach. ; This work has been partly financed by the Spanish government through the CONSOLIDER-INGENIO 2010 program (COMONSENS project, ref. CSD2008-00010), as well as projects COSIMA (TEC2010-19545-C04-03), ALCIT (TEC2012 38800- C03-01), COMPREHENSION (TEC2012-38883-C02-01) and DISSECT (TEC2012-38058-C03-01).

The spectrum sensing accuracy has been improved by the introduction of cooperative spectrum sensing (CSS) strategies where the spatial diversity is exploited among nonlegacy users. However, these CSS strategies also bring new impairments, such as the interference from other sources that severely degrade the sensing performance. In this paper, we evaluate experimentally our recent proposal for CSS based on kernel canonical correlation analysis (KCCA), where the effect of an interferer is also modeled. The experiments are conducted on a cognitive radio platform composed of several Universal Radio Peripheral (USRP) nodes, and the measurements show that our scheme is able of implicitly learning the surrounding environment by only exploiting the non-linear correlation among the receiver signals of each SU. Eventually, we provide comparative results where a considerable gain over a conventional energy detector is obtained in spite of the impairments provoked by external interferers. ; The research leading to these results has received funding from the Spanish Government (MICINN) under projects TEC2010-19545-C04-03 (COSIMA) and CONSOLIDERINGENIO 2010 CSD2008-00010 (COMONSENS).

In this paper, a new technique for the blind estimation of frequency and/or time-selective multiple-input multiple-output (MIMO) channels under space-time block coding (STBC) transmissions is presented. The proposed method relies on a basis expansion model (BEM) of the MIMO channel, which reduces the number of parameters to be estimated, and includes many practical STBC-based transmission scenarios, such as STBC-orthogonal frequency division multiplexing (OFDM), space-frequency block coding (SFBC), time-reversal STBC, and time-varying STBC encoded systems. Inspired by the unconstrained blind maximum likelihood (UML) decoder, the proposed criterion is a subspace method that efficiently exploits all the information provided by the STBC structure, as well as by the reduced-rank representation of the MIMO channel. The method, which is independent of the specific signal constellation, is able to blindly recover the MIMO channel within a small number of available blocks at the receiver side. In fact, for some particular cases of interest such as orthogonal STBC-OFDM schemes, the proposed technique blindly identifies the channel using just one data block. The complexity of the proposed approach reduces to the solution of a generalized eigenvalue (GEV) problem and its computational cost is linear in the number of sub-channels. An identifiability analysis and some numerical examples illustrating the performance of the proposed algorithm are also provided ; This work was supported by the Spanish Government under projects TEC2007-68020-C04-02/TCM (MultiMIMO) and CONSOLIDER-INGENIO 2010 CSD2008-00010 (COMONSENS).

This paper derives an asymptotic generalized likelihood ratio test (GLRT) and an asymptotic locally most powerful invariant test (LMPIT) for two hypothesis testing problems: 1) Is a vector-valued random process cyclostationary (CS) or is it wide-sense stationary (WSS)? 2) Is a vector-valued random process CS or is it nonstationary? Our approach uses the relationship between a scalar-valued CS time series and a vector-valued WSS time series for which the knowledge of the cycle period is required. This relationship allows us to formulate the problem as a test for the covariance structure of the observations. The covariance matrix of the observations has a block-Toeplitz structure for CS and WSS processes. By considering the asymptotic case where the covariance matrix becomes block-circulant we are able to derive its maximum likelihood (ML) estimate and thus an asymptotic GLRT. Moreover, using Wijsman's theorem, we also obtain an asymptotic LMPIT. These detectors may be expressed in terms of the Loe`ve spectrum, the cyclic spectrum, and the power spectral density, establishing how to fuse the information in these spectra for an asymptotic GLRT and LMPIT. This goes beyond the state-of-the-art, where it is common practice to build detectors of cyclostationarity from ad-hoc functions of these spectra. ; The work of P. Schreier was supported by the Alfried Krupp von Bohlen und Halbach Foundation, under its program "Return of German scientists from abroad". The work of I. Santamaría and J. Vía was supported by the Spanish Government, Ministerio de Ciencia e Innovación (MICINN), under project RACHEL (TEC2013-47141-C4-3-R). The work of L. Scharf was supported by the Airforce Office of Scientific Research under contract FA9550-10-1-0241.

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. ; The work of P. Schreier was supported by the Alfried Krupp von Bohlen und Halbach Foundation, under its program "Return of German scientists from abroad". The work of I. Santamaría and J. Vía was supported by the Spanish Government, Ministerio de Ciencia e Innovación (MICINN), under project RACHEL (TEC2013-47141-C4-3-R). The work of L. Scharf was supported by the Airforce Office of Scientific Research under contract FA9550-10-1-0241.

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 vectorvalued wide-sense stationary (WSS) 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. ; The work of L. Scharf was supported by the Airforce Office of Scientific Research under contract FA9550-10-1-0241. The work of I. Santamaría and J. Vía was supported by the Spanish Government, Ministerio de Ciencia e Innovación (MICINN), under project COSIMA (TEC2010-19545-C04-03) and project COMONSENS (CSD2008-00010, CONSOLIDER-INGENIO 2010 Program). The work of P. Schreier was supported by the Alfried Krupp von Bohlen und Halbach Foundation, under its program "Return of German scientists from abroad.

We derive an estimator of the cycle period of a univariate cyclostationary process based on an information-theoretic criterion. Transforming the univariate cyclostationary process into a vector-valued wide-sense stationary process allows us to obtain the structure of the covariance matrix, which is block-Toeplitz, and its block size depends on the unknown cycle period. Therefore, we sweep the block size and obtain the ML estimate of the covariance matrix, required for the information-theoretic criterion. Since there are no closed-form ML estimates of block-Toeplitz matrices, we asymptotically approximate them as block-circulant. Finally, some numerical examples show the good performance of the proposed estimator. ; The work of P. Schreier was supported by the Alfried Krupp von Bohlen und Halbach Foundation, under its program "Return of German scientists from abroad". The work of I. Santamaría and J. Vía was supported by the Spanish Government, Ministerio de Ciencia e Innovación (MICINN), under project RACHEL (TEC2013-47141-C4-3-R). The work of L. Scharf was supported by the Airforce Office of Scientific Research under contract FA9550-10-1-0241.

Much of the recent work on multiantenna spectrum sensing in cognitive radio (CR) networks has been based on generalized likelihood ratio test (GLRT) detectors, which lack the ability to learn from past decisions and to adapt to the continuously changing environment. To overcome this limitation, in this paper we propose a Bayesian detector capable of learning in an efficient way the posterior distributions under both hypotheses. These posteriors summarize, in a compact way, all information seen so far by the cognitive secondary user. Our Bayesian model places priors directly on the spatial covariance matrices under both hypothesis, as well as on the probability of channel occupancy. Specifically, we use inverse-gamma and complex inverse-Wishart distributions as conjugate priors for the null and alternative hypothesis, respectively; and a binomial distribution as the prior for channel occupancy. At each sensing period, Bayesian inference is applied and the posterior for the channel occupancy is thresholded for detection. After a suitable approximation, the posteriors are employed as priors for the next sensing frame, which forms the basis of the proposed Bayesian learning procedure. We also include a forgetting mechanism that allows to operate satisfactorily on time-varying scenarios. The performance of the Bayesian detector is evaluated by simulations and also by means of CR testbed composed of universal radio peripheral (USRP) nodes. Both the simulations and our experimental measurements show that the Bayesian detector outperforms the GLRT in a variety of scenarios. ; The research leading to these results has received funding from the Spanish Government (MIC INN) under Projects TEC2010-19545-C04-03 (COSIMA) and CONSOLIDER-INGENIO 2010 CSD2008-00010 (COMONSENS). It also has been supported by FPI Grant BES-2011-047647.

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 2.4GHz. 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 3dB loss in performance. The hardware MIMO platform is based on high-performance signal acquisition and generation boards, each one equipped with a 1GB 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. ; This work has been supported by Ministerio de Educación y Ciencia of Spain, Xunta de Galicia and FEDER funds of the European Union under grant numbers TEC2004-06451-C05-02, TEC2004-06451-C05-01, PGIDT05PXIC10502PN, and FPU grants AP2004-5127 and AP2006-2965.