Pseudo-covariance Matrices Research Articles

Off‐grid direction‐of‐arrival estimation for wideband noncircular sources

AbstractResearchers have recently shown an increased interest in estimating the direction‐of‐arrival (DOA) of wideband noncircular sources, but existing studies have been restricted to subspace‐based methods. An off‐grid sparse recovery‐based algorithm is proposed in this paper to improve the accuracy of existing algorithms in low signal‐to‐noise ratio situations. The covariance and pseudo covariance matrices can be jointly represented subject to block sparsity constraints by taking advantage of the joint sparsity between signal components and bias. Furthermore, the estimation problem is transformed into a single measurement vector problem utilizing the focused operation, resulting in a significant reduction in computational complexity. The proposed algorithm's error threshold and the Cramer–Rao bound for wideband noncircular DOA estimation are deduced in detail. The proposed algorithm's effectiveness and feasibility are demonstrated by simulation results.

Waveform covariance matrix design using Fourier series coefficients

Multiple-input multiple-output (MIMO) radars may outperform other radar systems such as phased array radars, in terms of higher resolution, better detection probability in the presence of interferences, better parameter identifiability and more flexibility in beampattern design. Waveform covariance matrix design, because of its role in the beampattern synthesis process, is one of the most important problems in MIMO radar systems. In this study, the authors have proposed a closed-form solution based on Fourier series coefficients to design a covariance matrix. The resulting covariance matrix fulfils the practical constraints, i.e. positive semi-definiteness and the uniform elemental power constraint. It also provides performance similar to that of iterative methods, while requires lower computation time and provides better mean square error with respect to other existing closed-form methods. Eigenvalue decomposition is also utilised to convert the possible resulted pseudo-covariance matrices (pseudo-CM), which are not guaranteed to be positive semidefinite, into a covariance matrix. Simulation results show the performance of the proposed method.

Multi-Agent Distributed Beamforming With Improper Gaussian Signaling for MIMO Interference Broadcast Channels

For rate optimization in interference limited networks, improper Gaussian signaling has shown its ability to outperform conventional proper Gaussian signaling. In this paper, we study a weighted sum-rate maximization problem with improper Gaussian signaling for the multiple-input multiple-output interference broadcast channel. To solve this nonconvex and NP-hard problem, we propose an effective separate covariance and pseudo-covariance matrices optimization algorithm. In the covariance optimization, a weighted minimum mean square error algorithm is adopted, and in the pseudo-covariance optimization, an alternating optimization (AO) algorithm is proposed, which guarantees convergence to a stationary solution and ensures a sum-rate improvement over proper Gaussian signaling. An alternating direction method of multipliers-based multi-agent distributed algorithm is proposed to solve an AO subproblem with the globally optimal solution in a parallel and scalable fashion. The proposed scheme exhibits favorable convergence, optimality, and complexity properties for future large-scale networks. The simulation results demonstrate the superior sum-rate performance of the proposed algorithm as compared with the existing schemes with proper as well as improper Gaussian signaling under various network configurations.

A perspective on CLMS as a deficient length augmented CLMS: Dealing with second order noncircularity

Mean and mean square performance analyses of the strictly linear complex least mean square (CLMS) algorithm are addressed for widely linear estimation (WLE) of second order noncircular (improper) Gaussian inputs, for which both the covariance and pseudo-covariance matrices contain nonzero off-diagonal elements. A detailed performance analysis of standard CLMS in this ‘suboptimal’ context is not trivial but is important for practical applications. To this end, we here consider the strictly linear CLMS as a ‘deficient length’ version of the widely linear augmented CLMS (ACLMS) algorithm, which is second order optimal for improper inputs. Rigorous performance analysis is provided, which also statistically quantifies the suboptimality of CLMS in both the transient and steady state stages. The recently introduced approximate uncorrelating transform (AUT) is employed to derive closed-form expressions for the mean square stability and the steady-state performance of CLMS. In addition, since WLE with second order noncircular inputs is a general linear estimation problem in the complex domain C, these results provide a generalised framework from which current statistical descriptions of CLMS for strictly linear estimation (SLE) can be deduced as special cases. Simulations in system identification settings validate the findings.

Correlation Assisted Strong Uncorrelating Transform Complex Common Spatial Patterns for Spatially Distant Channel Data.

The Strong Uncorrelating Transform Complex Common Spatial Patterns (SUTCCSP) algorithm, designed for multichannel data analysis, has a limitation on keeping the correlation information among channels during the simultaneous diagonalization process of the covariance and pseudocovariance matrices. This paper focuses on the importance of preserving the correlation information among multichannel data and proposes the correlation assisted SUTCCSP (CASUT) algorithm to address this issue. The performance of the proposed algorithm was demonstrated by classifying the motor imagery electroencephalogram (EEG) dataset. The features were first extracted using CSP algorithms including the proposed method, and then the random forest classifier was utilized for the classification. Experiments using CASUT yielded an average classification accuracy of 78.10 (%), which significantly outperformed those of original CSP, Complex Common Spatial Patterns (CCSP), and SUTCCSP with p-values less than 0.01, tested by the Wilcoxon signed rank test.

Performance analysis of the deficient length augmented CLMS algorithm for second order noncircular complex signals

The augmented complex LMS (ACLMS) algorithm deals with second order noncircular (improper) input signals, based on widely linear modelling and the use of full second order statistical information. In current analyses of ACLMS, it is implicitly or explicitly assumed that the length of the adaptive filter is equal to that of the unknown system’s impulse response (optimal model order). In many applications, however, the length of the adaptive filter is smaller than required, the so called deficient length case, which renders the analysis for a ‘sufficient length’ ACLMS inadequate. To this end, we examine the statistical behaviour of the ACLMS algorithm in undermodelling situations. Exact expressions are developed to completely characterise both the transient and steady-state mean and mean square performances of the deficient length ACLMS for general second order noncircular Gaussian input signals. This is achieved using the recently introduced approximate uncorrelating transform (AUT), in order to jointly diagonalise the covariance and pseudo-covariance matrices with a single singular value decomposition (SVD), which both simplifies the analysis and enables a link between the degree of input noncircularity and the steady state mean square error (MSE) performance of the deficient length ACLMS. Simulations in system identification settings support the analysis.

Structure-Induced Complex Kalman Filter for Decentralized Sequential Bayesian Estimation

The letter considers a multi-sensor state estimation problem configured in a decentralized architecture where local complex statistics are communicated to the central processing unit for fusion instead of the raw observations. Naive adaptation of the augmented complex statistics to develop a decentralized state estimation algorithm results in increased local computations, and introduces extensive communication overhead, making it practically unattractive. The letter proposes a structure-induced complex Kalman filter framework with reduced communication overhead. In order to further reduce the local computations, the letter proposes a non-circularity criterion which allows each node to examine the non-circularity of its local observations. A local sensor node disregards its extra second-order statistical information when the non-circularity coefficient is small. In cases where the local observations are highly non-circular, an intuitively pleasing circularization approach is proposed to avoid computation and communication of the pseudo-covariance matrices. Simulation results indicate that the proposed structured-induced complex Kalman filter (SCKF) provides significant performance improvements over its traditional counterparts.

Improper Complex-Valued Bhattacharyya Distance

Motivated by application of complex-valued signal processing techniques in statistical pattern recognition, classification, and Gaussian mixture (GM) modeling, this paper derives analytical expressions for computing the Bhattacharyya coefficient/distance (BC/BD) between two improper complex-valued Gaussian distributions. The BC/BD is one of the most widely used statistical measures for evaluating class separability in classification problems, feature extraction in pattern recognition, and for GM reduction (GMR) purposes. The BC provides an upper bound on the Bayes error, which is commonly known as the best criterion to evaluate feature sets. Although the computation of the BC/BD between real-valued signals is a well-known result, it has not yet been extended to the case of improper complex-valued Gaussian densities. This paper addresses this gap. We analyze the role of the pseudocovariance matrix, which characterizes the noncircularity of the signal, and show that it carries critical second-order statistical information for computing the BC/BD. We derive upper and lower bounds on the BD in terms of the eigenvalues of the covariance and pseudocovariance matrices of the underlying densities. The theoretical bounds are then used to introduce the concept of β -dominance in the context of statistical distance measures. The BC is pseudometric, since it fails to satisfy the triangle inequality. Using the Matusita distance (a full-metric variant of the BC), we propose an intuitively pleasing indirect distance measure for comparing two general GMs. Finally, we investigate the application of the proposed BC/BD measures for GMR purposes and develop two BC-based GMR algorithms.

Optimized Transmission with Improper Gaussian Signaling in the K-User MISO Interference Channel

This paper studies the achievable rate region of the K-user Gaussian multiple-input single-output interference channel (MISO-IC) with the interference treated as noise, when improper or circularly asymmetric complex Gaussian signaling is applied. The transmit optimization with improper Gaussian signaling involves not only the signal covariance matrix as in the conventional proper or circularly symmetric Gaussian signaling, but also the signal pseudo-covariance matrix, which is conventionally set to zero in proper Gaussian signaling. By exploiting the separable rate expression with improper Gaussian signaling, we propose a separate transmit covariance and pseudo-covariance optimization algorithm, which is guaranteed to improve the users' achievable rates over the conventional proper Gaussian signaling. In particular, for the pseudo-covariance optimization, we establish the optimality of rank-1 pseudo-covariance matrices, given the optimal rank-1 transmit covariance matrices for achieving the Pareto boundary of the rate region. Based on this result, we are able to greatly reduce the number of variables in the pseudo-covariance optimization problem and thereby develop an efficient solution by applying the celebrated semidefinite relaxation (SDR) technique. Finally, we extend the result to the Gaussian MISO broadcast channel (MISO-BC) with improper Gaussian signaling or so-called widely linear transmit precoding.

Transmit Optimization With Improper Gaussian Signaling for Interference Channels

This paper studies the achievable rates of Gaussian interference channels with additive white Gaussian noise (AWGN), when improper or circularly asymmetric complex Gaussian signaling is applied. For the Gaussian multiple-input multiple-output interference channel (MIMO-IC) with the interference treated as Gaussian noise, we show that the user's achievable rate can be expressed as a summation of the rate achievable by the conventional proper or circularly symmetric complex Gaussian signaling in terms of the users' transmit covariance matrices, and an additional term, which is a function of both the users' transmit covariance and pseudo-covariance matrices. The additional degrees of freedom in the pseudo-covariance matrix, which is conventionally set to be zero for the case of proper Gaussian signaling, provide an opportunity to further improve the achievable rates of Gaussian MIMO-ICs by employing improper Gaussian signaling. To this end, this paper proposes widely linear precoding, which efficiently maps proper information-bearing signals to improper transmitted signals at each transmitter for any given pair of transmit covariance and pseudo-covariance matrices. In particular, for the case of two-user Gaussian single-input single-output interference channel (SISO-IC), we propose a joint covariance and pseudo-covariance optimization algorithm with improper Gaussian signaling to achieve the Pareto-optimal rates. By utilizing the separable structure of the achievable rate expression, an alternative algorithm with separate covariance and pseudo-covariance optimization is also proposed, which guarantees the rate improvement over conventional proper Gaussian signaling.

Complex-valued independent vector analysis: Application to multivariate Gaussian model

We consider the problem of joint blind source separation of multiple datasets and introduce a solution to the problem for complex-valued sources. We pose the problem in an independent vector analysis (IVA) framework and provide a new general IVA implementation using Wirtinger calculus and a decoupled nonunitary optimization algorithm to facilitate Newton-based optimization. Utilizing the noncircular multivariate Gaussian distribution as a source prior enables the full utilization of the complete second-order statistics available in the covariance and pseudo-covariance matrices. The algorithm provides a principled approach for achieving multiset canonical correlation analysis.

Blind separation of linear instantaneous mixtures of non-stationary signals in the frequency domain

Blind source separation (BSS) methods aim at restoring source signals from their mixtures. For linear instantaneous mixtures of stationary random sources, a natural and widely used approach consists in using some statistics associated to the temporal representation of the signals. On the contrary, we here consider non-stationary real sources and we show that they have interesting frequency-domain properties which motivate the introduction of two new frequency-domain BSS methods. The first method works by diagonalizing a zero-lag, second-order statistics matrix, created using both covariance and pseudo-covariance matrices of Fourier transforms of real-valued observations. In practice, this method is specially suitable for separating cyclo-stationary sources. The second method is particularly important because it allows the existing time-domain algorithms developed for stationary, temporally correlated sources (like AMUSE or SOBI) to be extended to non-stationary, temporally uncorrelated sources just by mapping the mixtures into the frequency domain. Both methods set no constraint on the piecewise stationarity of the sources, unlike most previously reported BSS methods exploiting source non-stationarity. The experimental results using artificial and real-world sources confirm the good performance of the proposed methods for non-stationary sources.

Complex Noise Analysis of DMT

In this paper, we consider discrete multitone (DMT) or baseband orthogonal frequency-division multiplexing (OFDM) modulation and perform a detailed noise analysis which takes into account dependencies and power (variance) differences of real and imaginary part after the complex-valued discrete Fourier transform (DFT). The derivation is based on the so-called pseudocovariance matrix of a complex random vector, which was introduced by Neeser and Massey (1993). We show that the relevant pseudocovariance matrix is not the zero matrix in general, in contrast to pass-band OFDM, for which it can be proven (under certain assumptions) that all occurring pseudocovariance matrices are vanishing. We show that for colored noise rotated rectangular symbol constellations are more appropriate than the common quadratic quadrature amplitude modulation (QAM) symbol constellations with respect to capacity and symbol error probability, and we derive formulas for the rotation angles and constellation sizes/densities. Finally, we extend the results to a multitransceiver [multiple-input-multiple-output (MIMO)] scenario, for which we assume a very general noise model at the inputs of the receivers, allowing correlations between the noise signals of different receivers. This requires the introduction of pseudocross-covariance matrices of complex random vectors, which are the important objects (together with cross-covariance matrices) in the MIMO situation.

Fast Direction Finding Using Modified Pseudocovariance Matrix

A modified pseudocovariance matrix is described which can improve the performance of Kim 's (Fast DOA estimation algorithm using pseudocovariance IEEE Trans. Antennas Propagat., vol. 53, no. 4, pp. 1346-1351, Apr. 2005) algorithm to realize the fast direction finding with single snapshot in coherent or uncorrelated case. By adding the backwards formulation of pseudocovariance matrix, the rank of pseudocovariance matrix is enhanced. Then a set of pseudocovariance matrices are constructed by circularly shifting the matrix. By weighted averaging, a modified pseudocovariance matrix is reconstructed whose noise is reduced. The proposed matrix has less array element loss than that of the original matrix. Moreover, this algorithm can be applied in the lower signal-noise-ratio (SNR) through noise preprocessing. The new matrix has expanded the application of high-resolution direction finding to nonuniform linear array (NLA), since traditional methods usually assume the array is uniform linear when handling coherent sources. Simulations are conducted to compare the performance of the modified method and the original method. The results illustrate that the high performance of DOA estimate could be obtained rapidly