The problem of measuring conditional dependence between two random phenomena arises when a third one (a confounder) has a potential influence on the amount of information between them. A typical issue in this challenging problem is the inversion of ill-conditioned autocorrelation matrices. This letter presents a novel measure of conditional dependence based on the use ofincompleteunbiased statistics of degree two, which allows to re-interpret independence as uncorrelatedness on a finite-dimensional feature space. This formulation enables to prune data according to observations of the confounder itself, thus avoiding matrix inversions altogether. The proposed approach is articulated as an extension of the Hilbert-Schmidt independence criterion, which becomes expressible through kernels that operate on 4-tuples of data.

The Majorization-Minimization (MM) framework is widely used to derive efficient algorithms for specific problems that require the optimization of a cost function (which can be convex or not). It is based on a sequential optimization of a surrogate function over closed convex sets. A natural extension of this framework incorporates ideas of Block Coordinate Descent (BCD) algorithms into the MM framework, also known as block MM. The rationale behind the block extension is to partition the optimization variables into several independent blocks, to obtain a surrogate for each block, and to optimize the surrogate of each block cyclically. However, known convergence proofs of the block MM are only valid under the assumption that the constraint sets are closed and convex. Hence, the global convergence of the block MM is not ensured for non-convex sets by classical proofs, which is needed in iterative schemes that naturally emerge in a wide range of subspace-based signal processing applications. For this purpose, the aim of this letter is to review the convergence proof of the block MM and extend it for blocks constrained in the Grassmann manifold.

With the goal of enabling ultrareliable and low-latency wireless communications for industrial Internet of Things (IIoT), this paper studies the use of energy-based modulations in noncoherent massive single-input multiple-output (SIMO) systems. We consider a one-shot communication over a channel with correlated Rayleigh fading and colored Gaussian noise, in which the receiver has statistical channel state information (CSI). We first provide a theoretical analysis on the limitations of unipolar pulse-amplitude modulation (PAM) in systems of this kind, based on maximum likelihood detection. The existence of a fundamental error floor at high signal-to-noise ratio (SNR) regimes is proved for constellations with more than two energy levels, when no (statistical) CSI is available at the transmitter. In the main body of the paper, we present a design framework for quadratic detectors that generalizes the widely-used energy detector, to better exploit the statistical knowledge of the channel. This allows us to design receivers optimized according to information-theoretic criteria that exhibit lower error rates at moderate and high SNR. We subsequently derive an analytic approximation for the error probability of a general class of quadratic detectors in the large array regime. Finally, we numerically validate it and discuss the outage probability of the system.

This paper explores the problem of estimating a temporal series measured from multiple independent sensors with unequal and stationary measurement errors with unknown variances. By formulating the data fusion problem as a joint Maximum Likelihood estimation of sensor covariances and a fusion rule, a batch data driven method is derived involving a residual covariance determinant minimization of a diagonal matrix. It is shown that yielding useful learning from data with good generalization properties in the joint regression and fusion approach requires the assumption of some structure on the sensor noises and/or on the temporal series to be estimated. An efficient data driven algorithm is proposed to obtain the best linear sensor combiner, whose performance is numerically analyzed and compared with the Cramer-Rao Lower Bound of the estimated parameters.

It is shown that R´enyi's entropy of a Gaussian mixture with entropic index α ∊ (1,∞] is upper-bounded by the cluster with minimum variance. This basic idea leads to a clean worst-case formulation of the minimum error entropy principle in the context of linear multi-sensor fusion by using a largely contaminated Gaussian distribution to model sensor errors with outliers. The obtained entropic best linear unbiased estimator leads to an operational interpretation in terms of a precision/reliability trade-off, it resonates closely with model order selection methods, and it provides a possible information theoretic root to sparsity-promoting regularization.

This article explores a plug-in estimator of second-order Tsallis entropy based on Kernel Density Estimation KDE and its implicit regularization process. First, it is shown that the expected value of the estimator corresponds to the entropy of an (AWGN) model. Then, we prove various relevant properties of the Tsallis entropy: it is monotonically non-decreasing under random variables addition, its derivative with respect to the Gaussian noise power is monotonically non-increasing, and it is concave in the additive noise power. From these, we derive an information metric that provides an alternative to the strategy of regularization.

This paper aims to estimate the information between two random phenomena by using consolidated second-order statistics tools. The squared-loss mutual information, a surrogate of the Shannon mutual information, is chosen due to its property of being expressed as a second-order moment. We first review the rationale for i.i.d. discrete sources, which involves mapping the data onto the simplex space, and we highlight the links with other well-known related concepts in the literature based on local approximations of information-theoretic measures. Then, the problem is translated to analog sources by mapping the data onto the characteristic space, focusing on the adaptability between the discrete and the analog case and its limitations. The proposed approach gains interpretability and scalability for its use on large data sets, providing a unified rationale for the free regularization parameters. Moreover, the structure of the proposed mapping allows resorting to Szegö’s theorem to reduce the complexity for high dimensional mappings, exhibiting a strong duality with spectral analysis. The performance of the developed estimators is analyzed using Gaussian mixtures.

This article explores U-Statistics as a tool for testing conditional correlation between two multivariate sources with respect to a potential confounder. The proposed approach is effectively an instance of weighted U-Statistics and does not impose any statistical model on the processed data, in contrast to other well-known techniques that assume Gaussianity. By avoiding determinants and inverses, the method presented displays promising robustness in small-sample regimes. Its performance is evaluated numerically through its MSE and ROC curves.

This paper explores the asymptotic spectral decomposition of periodically Toeplitz matrices with finite summable elements. As an alternative to polyphase decomposition and other approaches based on Gladyshev representation, the proposed route exploits the Toeplitz structure of cyclic autocorrelation matrices, thus leveraging on known asymptotic results and providing a more direct link to the cyclic spectrum and spectral coherence. As a concrete application, the problem of cyclic linear prediction is revisited, concluding with a generalized Kolmogorov-Szegö theorem on the predictability of cyclostationary signals. These results are finally tested experimentally in a prediction setting for an asynchronous mixture of two cyclostationary pulse-amplitude modulation signals.

In this paper, we consider the problem of estimating and tracking an R-dimensional subspace with relevant information embedded in an N-dimensional ambient space, given that N>>R. We focus on a formulation of the signal subspace that interprets the problem as a least squares optimization. The approach we present relies on the geometrical concepts behind the Affine Projection Algorithms (APA) family to obtain the Affine Projection Subspace Tracking (APST) algorithm. This on-line solution possesses various desirable tracking capabilities, in addition to a high degree of configurability, making it suitable for a large range of applications with different convergence speed and computational complexity requirements. The APST provides a unified framework that generalises other well-known techniques, such as Oja’s rule and stochastic gradient based methods for subspace tracking. This algorithm is finally tested in a few synthetic scenarios against other classical adaptive methods.

We study the problem of estimating the overall mutual information in M independent parallel discrete-time memory-less Gaussian channels from N independent data sample pairs per channel (inputs and outputs). We focus on the case where the number of active channels L is sparse in comparison with the total number of channels (L ≪ M), for which the direct application of the maximum likelihood principle is problematic due to overfitting, especially for moderate to small N. For this regime, we show that the bias of the mutual information estimate is reduced by resorting to the minimum description length (MDL) principle. As a result, simple pre-processing based on a per-channel threshold on the empirical squared correlation coefficient is required with a fixed threshold that monotonically decreases with N as 1 - N-1/N, for N ≥ 4. The resulting improvement is shown in terms of the estimated information bias.

Squared-loss mutual information (SMI) is a surrogate of Shannon mutual information that is more advantageous for estimation. On the other hand, the coherence matrix of a pair of random vectors, a power-normalized version of the sample cross-covariance matrix, is a well-known second-order statistic found in the core of fundamental signal processing problems, such as canonical correlation analysis (CCA). This paper shows that SMI can be estimated from a pair of independent and identically distributed (i.i.d.) samples as a squared Frobenius norm of a coherence matrix estimated after mapping the data onto some fixed feature space. Moreover, low computation complexity is achieved through the fast Fourier transform (FFT) by exploiting the Toeplitz structure of the involved autocorrelation matrices in that space. The performance of the method is analyzed via computer simulations using Gaussian mixture models.

This paper shows that de Bruijn identity, which relates entropy with Fisher information, can be obtained as a particular case of an immediate generalization of Price's theorem, which is a tool used in the analysis of nonlinear memoryless systems with Gaussian inputs. It is shown that, while the general Price's theorem follows since the density of the perturbation satisfies the heat equation, the particular case of de Bruijn identity follows since the score function is zero-mean, which is the well-known condition that provides the insightful Cramér- Rao bound expression based on the negative second derivative of the log-likelihood function. The unified framework uses the characteristic function as a main tool and becomes a more intuitive alternative to the classical technical proof obtained by integrating by parts. Second-order Tsallis entropy is also briefly explored under this general framework.

A generic software tool to evaluate the impact of ionospheric disturbances is presented, including low and high latitude physically-based models, and low and high frequency fluctuations. This tool has been developed specifically to assess the performance of navigation receivers, but it is ready to simulate other frequencies and even receivers in dynamic conditions, which allows it to be used in other applications such as communications, GNSS-R, radar altimetry or SAR.

This paper revisits the model order selection problem in the context of second-order spectrum sensing in cognitive radio. Taking advantage of the recent interest on the generalized likelihood ratio (GLR), the asymptotic performance of the minimum description length (MDL) rule under unknown noise variance is addressed. In particular, by exploiting the asymptotically Chi-squared distribution of the GLR, a complete characterization of the error probability is reported, instead of approximating only the missed-detection probability as done in the literature.

The frequency diversity exhibited by cyclostationary signals is exploited in this paper. A novel rank-1 frequency-domain representation of a digital waveform is proposed to address the generalized likelihood ratio (GLR) detection of a cyclo-stationary signal with unknown white noise. With the aim of avoiding the well-known sensitivity of cyclostationary-based detectors to frequency-selective fading channels, a parametric channel model based on the coherence bandwidth is adopted and incorporated in the GLR test. The proposed detector outperforms the classical spectral correlation magnitude detectors by exploiting the rank-1 structure of small spectral co-variance matrices.

This study addresses the optimal spectrum sensing detection based on the complete or partial side‐information on the signal and noise statistics. The use of the generalised‐likelihood ratio test (GLRT) involves maximum‐likelihood (ML) estimation of the nuisances. ML estimation of the unknowns is especially challenging for wideband cognitive radio because closed‐form solutions are often not available. Based on the equivalence between the wideband regime and the low‐signal‐to‐noise ratio regime, this study provides a general kernel framework for GLRT spectrum sensing. It is shown that any GLRT detector exclusively depends on the projection of the sample covariance matrix of the data onto a given underlying kernel that reflects the available side‐information in the problem. The kernels in several scenarios of interest are derived, including the widespread single and multi‐frequency channelisation cases. Theoretical interpretations and numerical results show the trade‐off between detection performance and the degree of side‐information on the most informative statistics for detection, that is, the modulation format and spectrum distribution of the primary users.

This paper addresses the problem of blind detection of a wide-sense stationary (WSS) signal over fading channels. We propose a test statistic which is optimal from a correlation-matching perspective that shows invariance with respect to the noise power and the channel gain. In the blind scenario, we derive the quadratic sphericity test (QST) which exploits the structure of the fading channel as a squared mean to arithmetic mean ratio of the eigenvalues of the autocorrelation matrix of the observations. We provide numerical results to assess the performance of the QST in several fading scenarios, as well as the benchmarking to other blind and non-blind detectors.

In this work, we show the existence of sampling walls in signal detection of nonuniformly sampling wideband signals in the presence of noise uncertainty. A sampling wall is the sampling rate below which the target error probabilities (namely the missed detection and false alarm probabilities) cannot be achieved at a given signal to noise ratio (SNR) regardless the number of acquired samples. Contrary to existing works, we address the signal detection as a binary hypotheses testing problem without having to reconstruct neither the signal nor the spectrum from the observations. Specifically, we adopt a Bernoulli distributed sampler as it exhibits good tradeoff properties between complexity and performance. We show that Bernoulli nonuniform sampling suffers from noise enhancement, which translates into a whitening effect in the correlation of the legacy signal. Therefore, in the presence of noise uncertainty, we derive explicit expressions for sampling walls as a function of the legacy signal occupation, the SNR and the noise uncertainty level. Finally, numerical results are further provided to assess the behavior of the sampling walls and signal detection performance.

The performance in signal detection is evaluated by the error (false-alarm and missed-detection) probabilities. However, calculating these probabilities is a difficult task in practice. This paper studies the asymptotic behavior of the energy-detector and the estimator-correlator by means of the Stein's lemma. The Stein's lemma is an information-theory result that provides the best achievable error exponent in the error probabilities when the number of observations goes to infinity. The derived closed-form expressions explain how detection performance is driven by the detector parameters and the second-order statistics of the problem. More specifically, it is shown that the error exponents depend on the signal-to-noise ratio (SNR) and the observation size. The prime focus is to establish a link between the required observation size for a fixed error probability as a function of the SNR. Numerical results show the tightness of the lemma.

The spectrum analysis problem based on the periodogram and Capon estimates from a nonuniformly sampled signal is addressed. The nonuniform sampling process is cast as a linear projection matrix which encompasses several recent sampling strategies such as compressive sampling. A rank-1 correlation- matching approach is proposed to derive a general filter-bank framework that allows the formulation of the nonuniform periodogram and Capon estimates as particular cases. The performance of the novel periodogram and Capon estimates is analyzed in a cognitive radio scenario where the primary system is a multiband signal with sparse spectrum. Both theoretical and numerical results show that the denoising process required by correlation-matching plays an important role in spectral analysis based on nonuniform sampling.

In this paper, we consider a novel compressed correlation-matching (CCM) approach for spectrum sensing of wideband sparse signals. We derive a general closed-form estimate of the wideband sparse signal level from compressed observations, while providing physical interpretation of the problem. The formulation allows straightforward application to signal processing problems of interest, such as generalized likelihood ratio test (GLRT) spectrum sensing for wideband cognitive radio. Simulation results are reported to assess the behavior of the CCM method.

The problem of spectrum sensing in multi-frequency cognitive radio systems is addressed. We show that as the sensed bandwidth increases, the primary user detection is governed by a low signal-to-noise ratio (low-SNR) regime. By means of low-SNR approximations, we show that the optimal generalized likelihood ratio test (GLRT) only depends on the second order statistics of the observations and on a shaping kernel that highlights the relevant parameters required for detection. Furthermore, the ML estimates of the unknown model parameters are derived for multi-frequency systems, which allow closed-form expressions for the GLRT statistic. The detection performance and the kernel interpretation are supported with simulation results.

Mobile location using time of arrival (TOA), time difference of arrival (TDOA) or angle of arrival (AOA) measurements has received considerable attention over the last years. Several closed-form algorithms have been presented for the TOA and TDOA case based on approximations of the maximum-likelihood (ML) estimator. In the case of AOA measurements, only ad-hoc estimators have been presented in order to avoid the classical linearization solution that needs an initial guess. This paper presents an approximation of the ML position estimator based on AOA measurements applying the divide-and-conquer approach dividing the ML estimation in smaller problems each one with a closed-form solution. Numerical simulations show that the proposed algorithm outperforms the previous contributions and presents a generic way to combine AOA and TDOA measurements

Geolocation in non line of sight (NLOS) environments is an important issue in wireless communication networks. In several recent publications related with the nature and magnitude of the NLOS phenomenon it is concluded that this is the major source of error in position estimators based on time measurements. This article presents a new approach to ameliorate the effect of the NLOS exploiting the redundant time measurements in scenarios with more than the minimum number of base stations (BS). This redundant data allows to formulate the problem as a test of hypothesis performing a hard decision to discard the BS considered to be in a NLOS scenario. Numerical simulations show that the proposed algorithm can discard the NLOS errors in certain scenarios. The algorithm is also compared with some other existing methods to show the advantage of the new approach.

Geolocation of mobile terminals has become in the last decades an important issue in mobile networks. In the literature, there have been presented several closed-form position estimators based on time-difference-of-arrival (TDOA) measurements. Only Fang's estimator can be considered optimum in the maximum likelihood (ML) sense. Unfortunately, it can only be applied to the particular case of two TDOA measurements for the two dimensional (2D) location case. This paper presents an extension of this closed-form estimator to be applied to an arbitrary number of TDOA measurements by means of a transformation in the maximum likelihood function. This allows the ML function minimization to be split in several partial ML minimizations which only consider a subset of the available measurements, where the original Fang's estimator can be applied. Numerical simulations show that the proposed algorithm, that can be considered asymptotically the ML-estimator, attains the theoretical limits for all range of reasonable SNR values and has a low implementation complexity.

This paper presents the theoretical accuracy limits of the geolocation algorithms based on TOA measurements exploiting the fact that the mobile is moving in a known or unknown direction. The developed expressions show us the possible improvements in terms of accuracy and/or availability due to the diversity created with the movement. A simple algorithm based on the use of TOA drift estimation is also presented in order to compare its performance with the developed theoretical limits. The proposed estimator attains the theoretical limits under certain conditions.

Estimation of frequency and symbol timing in continuous phase modulated (CPM) signals is investigated. Several well-known statistical approaches, classically applied to the sensor array problem, are used to derive non-data-aided (NDA) algorithms under a unifying general framework (estimation-directed). A new cost function is proposed which is shown to provide a good compromise between additive and pattern noise cancellation, when the additive noise power is unknown.

The study of parameter estimation under additive and multiplicative noise terms constitutes an interesting signal processing research topic with important consequences in the development of high-performance digital communication receivers. In this sense, the paper addresses the analysis of non-assisted digital synchronizers based on up to fourth-order moments and it shows how this alternative outperforms the conventional second-order techniques. The study is performed for MSK (minimum shift keying) as a particular case of the binary CPM modulations. This transmission scheme is adopted because it allows a simple extension to any linear digital modulation and to any multiple access modulation, as well. Simulation results show that the higher-order techniques exhibit a parameter estimation variance closer to the so-called modified Cramer-Rao bound for moderate-to-high SNR if compared to second-order techniques.

The use of (spectrally efficient) CPM modulations may lead to a serious performance degradation of the classical non-data-aided (NDA) frequency and timing estimators due to the presence of self noise. The actual performance of these estimators is usually much worse than that predicted by the classical modified Cramer-Rao bound. We apply some well known results in the field of signal processing to these two important problems of synchronization. In particular we propose and explain the meaning of the unconditional CRB in the synchronization task. Simulation results for MSK and GMSK, along with the performance of some classical and previously proposed synchronizers, show that the proposed bound (along with the MCRB) is useful for a better prediction of the ultimate performance of the NDA estimators.

The conditional maximum likelihood (CML) principle, well known in the context of sensor array processing, is applied to the problem of timing recovery. A new self-noise free CML-based timing error detector is derived. Additionally, a new (conditional) Cramer-Rao bound (CRB) for timing estimation is obtained, which is more accurate than the extensively used modified CRB (MCRB).

The use of spectrally efficient continuous phase modulations for mobile communications may lead to a serious performance degradation of the classical frequency error detectors (FEDs) due to the presence of self-noise. This article presents a new statistically efficient frequency estimation algorithm for staggered modulations. The cancellation of the self-noise is accomplished by the use of the conditional ML principle, well known in the context of array processing, as an alternative to the unconditional ML, typically applied in the communications field. The paper also provides a new Cramer Rao bound (CRB) which is more accurate than the so-called modified CRB (MCRB) extensively applied to synchronization problems.

A new algorithm for signal selective tracking of the directions-of-arrival (DOAs) of multiple moving targets with an array of passive sensors is presented. A new method based on the principles of maximum likelihood estimation and cyclostationarity is used to generate initial angle estimates which, in turn, are refined by a Kalman filter. Source angle dynamics are used to achieve correct data association. High performance is obtained with relatively low computational complexity.

We present a new method for recursively estimating the frequency and timing parameters of a second order cyclostationary signal with known cyclic autocorrelation matrix (CAM). This problem appears in the context of radar as well as asynchronous communications in highly non-stationary environments (e.g. telemetry) due to the Doppler effect. The parameters evolution is modelled by a zero-order random walk. The estimates are obtained from the instantaneous CAM (ICAM) of the signal. While nonlinear Kalman filter theory is a possible approach to this problem, Bayes risk theory is used instead. In this way the Riccati equation and gain matrices become independent of the estimates, thus allowing a look-up table solution. The folded normal density (FND) is assumed as parameters' prior. The prior is recursively updated in mean (estimates) and variance. By comparing the obtained equations with linear Kalman filter equations we show that with few modifications a first-order random walk model can be easily incorporated to cope with highly non-stationary frequency evolution.>