Unknown Noise Variance Research Articles

A Super-Resolution Direction of Arrival Estimation Algorithm for Coprime Array via Sparse Bayesian Learning Inference

In this paper, we address the problem of direction of arrival (DOA) estimation with coprime array in the context of sparse signal reconstruction to fully exploit the enhanced degrees of freedom (DOF) offered by the difference coarray. The proposed method is based on the framework of sparse Bayesian learning and can jointly refine the unknown DOAs and the sparse signals in a gradual and interweaved manner. Specifically, the proposed approach is constructed by iteratively decreasing a surrogate function majorizing a given objective function, which results in accelerating the speed to converge to the global minimum. Furthermore, for facilitating a noise-free sparse representation, a customized linear transformation is judiciously incorporated in our sparsity-inducing DOA estimator to eliminate the unknown noise variance, and in the mean time, the sample covariance matrix perturbation can be normalized to an identity matrix as a by-product. Extensive simulation experiments under different conditions finally demonstrate the superiority of our suggested algorithm in terms of mean-squared DOA estimation error, DOF and resolution ability over state-of-the-art techniques.

Identification of transmissivity fields using a Bayesian strategy and perturbative approach

The paper deals with the crucial problem of the groundwater parameter estimation that is the basis for efficient modeling and reclamation activities. A hierarchical Bayesian approach is developed: it uses the Akaike's Bayesian Information Criteria in order to estimate the hyperparameters (related to the covariance model chosen) and to quantify the unknown noise variance. The transmissivity identification proceeds in two steps: the first, called empirical Bayesian interpolation, uses Y* (Y = lnT) observations to interpolate Y values on a specified grid; the second, called empirical Bayesian update, improve the previous Y estimate through the addition of hydraulic head observations. The relationship between the head and the lnT has been linearized through a perturbative solution of the flow equation. In order to test the proposed approach, synthetic aquifers from literature have been considered. The aquifers in question contain a variety of boundary conditions (both Dirichelet and Neuman type) and scales of heterogeneities (σY2=1.0 and σY2=5.3). The estimated transmissivity fields were compared to the true one. The joint use of Y* and head measurements improves the estimation of Y considering both degrees of heterogeneity. Even if the variance of the strong transmissivity field can be considered high for the application of the perturbative approach, the results show the same order of approximation of the non-linear methods proposed in literature. The procedure allows to compute the posterior probability distribution of the target quantities and to quantify the uncertainty in the model prediction. Bayesian updating has advantages related both to the Monte-Carlo (MC) and non-MC approaches. In fact, as the MC methods, Bayesian updating allows computing the direct posterior probability distribution of the target quantities and as non-MC methods it has computational times in the order of seconds.

Recursive Least Squares for Censored Regression

Censored observations are encountered naturally in many engineering tasks. Conventional estimation algorithms may suffer from significant performance degradation when the observations are undesirably censored. This work focuses on adaptively estimating the regression parameter in a censored regression (CR) model. We first consider that the noise variance and censored thresholds are known a priori, and solve the online CR problem by computing the maximum-likelihood estimate in an expectation-maximization framework. This strategy yields a recursive least-squares algorithm for the CR (CR-RLS), and we prove its convergence and present analytical results for the steady-state error. Next, we extend the CR-RLS to the case of unknown noise variance and censored thresholds. Theoretical analysis and numerical simulation indicate that the CR-RLS performs significantly better than other competing algorithms in terms of both the estimation accuracy and convergence rate. Especially, for different censored thresholds, the CR-RLS can always achieve good performance, and its steady-state solution is almost as accurate as that of the RLS algorithm with the uncensored (complete) observations.

Distributed incremental bias-compensated RLS estimation over multi-agent networks

In this paper, we study the problem of distributed bias-compensated recursive least-squares (BC-RLS) estimation over multi-agent networks, where the agents collaborate to estimate a common parameter of interest. We consider the situation where both input and output of each agent are corrupted by unknown additive noise. Under this condition, traditional recursive least-squares (RLS) estimator is biased, and the bias is induced by the input noise variance. When the input noise variance is available, the effect of the noise-induced bias can be removed at the expense of an increase in estimation variance. Fortunately, it has been illustrated that distributed collaboration between agents can effectively reduce the variance and can improve the stability of the estimator. Therefore, a distributed incremental BC-RLS algorithm and its simplified version are proposed in this paper. The proposed algorithms can collaboratively obtain the estimates of the unknown input noise variance and remove the effect of the noise-induced bias. Then consistent estimation of the unknown parameter can be achieved in an incremental fashion. Simulation results show that the incremental BC-RLS solutions outperform existing solutions in some enlightening ways.

Self‐tuning full‐order WMF Kalman filter for multisensor descriptor systems

For the multisensor descriptor systems with correlated measurement noises and unknown noise variances, a self-tuning full-order weighted measurement fusion (WMF) Kalman filter is presented. First, based on the controlled autoregressive moving average innovation model of the multisensor descriptor systems, the consistent estimates of the unknown noise variances are obtained applying correlation function method. A compressed measurement equation for the multisensor descriptor systems is obtained by the WMF method, and an optimal full-order WMF descriptor Kalman filter is given. Based on the optimal full-order WMF Kalman filter and the estimates of noise variances, a self-tuning full-order WMF Kalman filter with the self-tuning descriptor Riccati equation is presented. By the dynamic variance error system analysis method, it is proven that the solution of the self-tuning descriptor Riccati equation converges to the solution of the optimal descriptor Riccati equation. Then, the convergence of the presented self-tuning full-order WMF Kalman filter is proven. A simulation example of a six-sensor descriptor system verifies the effectiveness and convergence of the presented algorithms.

Spectrum Sensing for Cognitive Radios With Unknown Noise Variance and Time-variant Fading Channels

The unknown noise variance and time-variant fading channels make the spectrum sensing design a challenging task for cognitive radios. Most existing sensing methods suffer from the information uncertainty and can hardly acquire promising performances in the adverse situations. To address this challenge, in this paper, we first formulate a dynamic state-space model for spectrum sensing, in which the unknown noise variance and time-variant flat fading channels are all taken into considerations. The dynamic behaviors of both primary user states and fading channels are characterized by two discrete state Markov chains. Based on this model, a novel spectrum sensing scheme is designed to recursively estimate the occupancy state of primary users, by estimating the time-variant fading channel gain and noise parameters jointly. The joint estimation is primarily premised on a maximum a posteriori probability criterion and the marginal particle filtering schemes. Simulation results are provided to demonstrate the advantages of our proposed method, which can significantly improve the sensing performance over time-variant flat fading channels, even with unknown noise variance.

An Auxiliary Variable-Aided Hybrid Message Passing Approach to Joint Channel Estimation and Decoding for MIMO-OFDM

This letter deals with message passing receiver design for joint channel estimation and decoding in MIMO-OFDM with unknown noise variance. The conventional factor graph representation for the system involves observation factors, which are functions of a number of variables in the form of multiplication and summation. In this work, by introducing some auxiliary variables, we further break each of the observation factors into several factors, which enables the use of hybrid mean field (MF), belief propagation (BP), and expectation propagation (EP) message passing to tackle the observation factors. It turns out that our approach is much more efficient than the existing approaches, leading to remarkable performance improvement as shown by simulation results.

Probability hypothesis density filter with adaptive parameter estimation for tracking multiple maneuvering targets

The probability hypothesis density (PHD) filter has been recognized as a promising technique for tracking an unknown number of targets. The performance of the PHD filter, however, is sensitive to the available knowledge on model parameters such as the measurement noise variance and those associated with the changes in the maneuvering target trajectories. If these parameters are unknown in advance, the tracking performance may degrade greatly. To address this aspect, this paper proposes to incorporate the adaptive parameter estimation (APE) method in the PHD filter so that the model parameters, which may be static and/or time-varying, can be estimated jointly with target states. The resulting APE-PHD algorithm is implemented using the particle filter (PF), which leads to the PF-APE-PHD filter. Simulations show that the newly proposed algorithm can correctly identify the unknown measurement noise variances, and it is capable of tracking multiple maneuvering targets with abrupt changing parameters in a more robust manner, compared to the multi-model approaches.

Multisnapshot Sparse Bayesian Learning for DOA

The directions of arrival (DOA) of plane waves are estimated from multi-snapshot sensor array data using Sparse Bayesian Learning (SBL). The prior source amplitudes is assumed independent zero-mean complex Gaussian distributed with hyperparameters the unknown variances (i.e. the source powers). For a complex Gaussian likelihood with hyperparameter the unknown noise variance, the corresponding Gaussian posterior distribution is derived. For a given number of DOAs, the hyperparameters are automatically selected by maximizing the evidence and promote sparse DOA estimates. The SBL scheme for DOA estimation is discussed and evaluated competitively against LASSO ($\ell_1$-regularization), conventional beamforming, and MUSIC

Variational Bayesian labeled multi-Bernoulli filter with unknown sensor noise statistics

It is difficult to build accurate model for measurement noise covariance in complex backgrounds. For the scenarios of unknown sensor noise variances, an adaptive multi-target tracking algorithm based on labeled random finite set and variational Bayesian (VB) approximation is proposed. The variational approximation technique is introduced to the labeled multi-Bernoulli (LMB) filter to jointly estimate the states of targets and sensor noise variances. Simulation results show that the proposed method can give unbiased estimation of cardinality and has better performance than the VB probability hypothesis density (VB-PHD) filter and the VB cardinality balanced multi-target multi-Bernoulli (VB-CBMeMBer) filter in harsh situations. The simulations also confirm the robustness of the proposed method against the time-varying noise variances. The computational complexity of proposed method is higher than the VB-PHD and VB-CBMeMBer in extreme cases, while the mean execution times of the three methods are close when targets are well separated.

High SNR Consistent Linear Model Order Selection and Subset Selection

High SNR consistency of model order selection criteria in linear regression models has attracted a lot of attention in the signal processing community recently. It is now known that Exponentially Embedded Family, versions of Minimum Description Length etc. are high SNR consistent. However, a general framework for high SNR consistency in linear regression is still missing. This paper fills this gap by deriving necessary and sufficient conditions (NSCs) for model order selection criteria in both known and unknown noise variance situations to be high SNR consistent. Many popular model order selection criteria are proved to be high SNR consistent using these NSCs. We also provide a convergence rate analysis to discriminate between various high SNR consistent model order selection criteria. A direct application of model order selection techniques to the very important subset selection problem is computationally infeasible. This paper establish the high SNR consistency of an existing t-statistics based index ordering scheme that allows the conversion of a subset selection problem into a model order selection problem. Combining this index ordering with high SNR consistent model order selection criteria leads to a novel subset selection procedure (SSP) that is numerically shown to outperform existing high SNR consistent SSPs.

LPWS Algorithm for Wideband Spectrum Sensing

Wideband or multi-band spectrum sensing is an essential functionality for cognitive radio (CR) networks. It enables a secondary user to identify spectral holes dynamically and transmit opportunistically so as not to interfere with cohabiting primary users over the same bands. This paper presents an effective wideband spectrum sensing method based on linear programming (named as LPWS algorithm) for a CR user equipped with a single receiving antenna. Firstly, the proposed method utilizes the temporal smoothing technique to form a virtual multi-antenna structure. Secondly, the wideband spectrum sensing problem is reformulated as a sparse reconstruction problem by exploiting a sparse representation of the virtual multi-antenna array covariance vector. Finally, making use of $$l_{\infty }$$lź-norm, the sparse reconstruction problem is modelled as a linear programming (LP) problem and hence can be solved efficiently. The presented method offers a number of advantages over other recently proposed methods. For examples, (1) it can reduce system overhead since single antenna is used instead of multiple antennas or sensing nodes. (2) The unknown noise variances can be eliminated effectively by a linear transformation. (3) It is computationally simpler since it is efficiently formulated in terms of the LP problem based on real-valued computation, etc. Simulation results are presented to verify the efficiency of the proposed method.

Estimation of low-rank covariance function

We consider the problem of estimating a low rank covariance function K(t,u) of a Gaussian process S(t),t∈[0,1] based on n i.i.d. copies of S observed in a white noise. We suggest a new estimation procedure adapting simultaneously to the low rank structure and the smoothness of the covariance function. The new procedure is based on nuclear norm penalization and exhibits superior performances as compared to the sample covariance function by a polynomial factor in the sample size n. Other results include a minimax lower bound for estimation of low-rank covariance functions showing that our procedure is optimal as well as a scheme to estimate the unknown noise variance of the Gaussian process.

An efficient off-grid DOA estimation approach for nested array signal processing by using sparse Bayesian learning strategies

In this paper, we address the problem of sparsity-inducing direction-of-arrival (DOA) estimation in the context of array covariance vectors when a recently proposed nested array structure is utilized. The proposed approach is based on a Sparse Bayesian learning (SBL) perspective while the off-grid effects inherent in traditional sparse recovery algorithms are taken into account simultaneously. Specifically, the notable advantages of the developed approach can be attributed to two aspects. First, through a linear transformation, the unknown noise variance can be eliminated effectively and the estimation error between the actual and sample covariance matrix can be normalized to an identity matrix, thereby facilitating a noise-free sparse representation. Second, a computationally tractable polynomial rooting procedure is introduced to calculate the discretized sampling grid error one by one based on the reconstructed spatial power spectrum, thus this novel algorithm can maintain high estimation accuracy even under a coarse sampling grid. Extensive simulation results finally demonstrate the superiority of the proposed approach in terms of DOA estimation precision and computational efficiency over off-the-shelves techniques.

An EM Approach for Cooperative Spectrum Sensing in Multiantenna CR Networks

In this paper, a cooperative wideband spectrum sensing scheme based on the expectation–maximization (EM) algorithm is proposed for the detection of a primary user (PU) system in multiantenna cognitive radio (CR) networks. Given noisy signal observations from $N$ secondary users (SUs) over multiple subbands at a fusion center (FC), prior works on cooperative spectrum sensing often use the set of received subband energy as decision statistics over the sensing interval. However, to achieve satisfactory performance, knowledge of the channel state information (CSI) and the noise variances at all the SUs is required by these algorithms. To overcome this limitation, our proposed method, which is referred to as joint detection and estimation (JDE), adopts the EM algorithm to jointly detect the PU signal and estimate the unknown channel frequency responses and noise variances over multiple subbands in an iterative manner. Various aspects of this proposed EM-JDE scheme are investigated, including a reliable initialization strategy to ensure convergence under practical conditions and a distributed implementation to reduce communication overhead. Under the assumption of perfect estimation for the channel frequency responses and noise variances, we further show that the proposed EM-JDE converges to the maximum-likelihood (ML) solution, which serves as an upper bound on its performance. Monte Carlo simulations over Rayleigh fading channels show that the proposed scheme significantly improves the performance of spectrum detection by exploiting the diversity of the spatially distributed SUs with multiple antennas.

A Novel Spectrum Sensing for Cognitive Radio Networks with Noise Uncertainty

This correspondence investigates a joint spectrum sensing scheme in cognitive radio (CR) networks with unknown and dynamic noise variance. A novel Bayesian solution is proposed to recover the dynamic noise variance and detect the occupancy of primary frequency band simultaneously. The states of primary users are detected based on particle filtering technology, and then the noise parameters are tracked by using finite-dimensional statistics for each particle based on marginalized adaptive particle filtering. Simulation results are provided to validate that the proposed method can improve the sensing performance significantly and target the dynamic noise variance accurately.

GLRT Based Spectrum Sensing for MIMO SC-FDMA Cognitive Radio Networks

This work presents generalized likelihood ratio test (GLRT) based spectrum sensing schemes for multiple-input multiple-output (MIMO) single carrier frequency division multiple access (SC-FDMA) cognitive radio networks. The finite record GLRT is initially derived for the MIMO SC-FDMA cognitive radio scenario. Subsequently, the GLRT is extended to a scenario with unknown noise variance. Closed form expressions are derived for the probabilities of false alarm and detection to characterize the performance of the GLRT for finite sample as well as asymptotic scenarios. Further, the computational complexity of the proposed GLRT-based detectors is analyzed. Simulation results are presented to illustrate the spectrum sensing performance of the proposed tests in comparison with existing detectors in literature such as the conventional energy detector, maximum eigenvalue-based detector and also to verify the analytical expressions derived for the detection performance.

Adjusted least squares fitting of algebraic hypersurfaces

We consider the problem of fitting a set of points in Euclidean space by an algebraic hypersurface. We assume that points on a true hypersurface, described by a polynomial equation, are corrupted by zero mean independent Gaussian noise, and we estimate the coefficients of the true polynomial equation. The adjusted least squares estimator accounts for the bias present in the ordinary least squares estimator. The adjusted least squares estimator is based on constructing a quasi-Hankel matrix, which is a bias-corrected matrix of moments. For the case of unknown noise variance, the estimator is defined as a solution of a polynomial eigenvalue problem. In this paper, we present new results on invariance properties of the adjusted least squares estimator and an improved algorithm for computing the estimator for an arbitrary set of monomials in the polynomial equation.

Spectrum Sensing for Self-Organizing Network in the Presence of Time-Variant Multipath Flat Fading Channels and Unknown Noise Variance

Cognitive radio, as an important means of implementing spectrum-awareness and dynamic sharing, is of great significance to the future deployment of self-organizing networks. Given its practical application, cognitive radio technology may operate in various adverse self-organizing networks environments. For example, in wireless mobile communication, the multipath propagation with time-varying fading coefficients and unknown noise variance becomes inevitable. Unfortunately, most existing spectrum sensing methods could hardly acquire good performance in this adverse situation. To overcome this difficulty, in this paper we present a novel spectrum sensing algorithm in realistic cognitive radio applications. Firstly, we establish a dynamic state-space model which gives full consideration to the evolution characteristics of primary user's state and time-variant multipath flat fading channel, while the received signal processed by matched filtering is viewed as the observation. On this basis, spectrum sensing is realized by estimating the primary user's state, multipath channel impulse response and noise variance jointly and iteratively. This formulated problem is solved based on maximum a posteriori probability criterion and marginal particle filtering technology. Simulations demonstrate that the sensing performance achieved by proposed algorithm is satisfactory and at the same time, the channel response and noise variance are estimated accurately.

Editorial for “SON and Automatic Configuration/Optimization in LTE-A Networks: Challenges and Practical Solutions”

Editorial: This special issue features six selected papers with high quality on the topic of SON and Automatic Configuration/ Optimization in LTE-A Networks: Challenges and Practical Solutions. The first article, BA Pricing Power Control Scheme with Statistical Delay QoS Provisioning in Uplink of Two-tier OFDMAFemtocell Networks”, co-authored by Shenghua He, Zhaoming Lu, Xiangming Wen, Zhicai Zhang, Jun Zhao and Wenpeng Jing, investigates the interference mitigation problem in LTE-A self-organizing networks (SON). In particular, this work has adopted a price-based power control strategy, where the macrocell base station (MBS) protects itself by pricing the interference from femtocell users (FUs). Based on a mathematical analysis, the authors formulate the problem using game theory and propose an effective power allocation algorithm named PSOPA. The simulation results demonstrate that PSOPA can magnificently improve the average effective capacity of each FU while guaranteeing their statistical delay QoS. The second article titled BA Stackelberg Game Based Intertier Spectrum Sharing Scheme for LTE-A SON^ from Songlin Sun, Liang Gong, Bo Rong, Abdel Mouaki and Amir Basri, develops a spectrum sharing scheme between Micro-eNodeB (MeNB) and Pico-eNodeB (PeNB) in LTE-A HetNets. Their proposed model takes into account a number of LTE-A SON techniques, such as eICIC and CRE, and employs Stackelberg game as the mathematical tool. The authors present a variety of simulation results and conclude that their spectrum sharing scheme can allocate frequency resource efficiently between MeNB and PeNB with respect to the change of user density. The next article investigates spectrum sensing for selforganizing Network. The authors (Mengwei Sun, Shenghong Li, Chenglin Zhao and Bin Li) propose a new spectrum sensing scheme to address the grand challenges engendered by time-varying multipath fading channels and unknown noise variance in realistic cognitive radio (CR) applications. CR is of great significance to the future deployment of self-organizing networks due to the spectrum scarcity problem. To make better use of CR, the authors create a dynamic state-space model, based on which spectrum sensing is realized by estimating the primary user’s state, multipath channel impulse response and noise variance jointly and iteratively. Numerical results show that the proposed scheme provides a promising solution of spectrum sensing in practical adverse communication environments. The fourth article, BData and Control Plane Traffic Modelling for LTE Networks,^ from Dima Dababneh, Marc St-Hilaire and Christian Makaya, develops a traffic model that involves bandwidth, signalling, voice BHSA (BHVSA), data BHSA (BHDSA), and EPSB. The proposed model provides network operators with realistic traffic parameters at the deployment phase of LTE networks. In particular, those parameters cover different aspects of the traffic and consider a variety of practical factors such as signalling, bandwidth, busy hour session attempts and number of simultaneous EPS (Evolved Packet Systems) bearers. The results of this work serve as an important guidance in the future network planning * Min Zhang mzhang@bupt.edu.cn