Unknown Noise Variance Research Articles

Maximum-Likelihood, Magnitude-Based, Amplitude and Noise Variance Estimation

Maximum likelihood (ML) amplitude and noise variance estimation without having to jointly estimate the frequency and the phase and based only on information from the noisy received signal magnitude, is studied for a single sinusoid in complex additive white Gaussian noise. This estimation problem is equivalent to the classic problem of parameter estimation for the Rician distribution. While solving the likelihood equation is impossible in general, we propose a new approach based on a large argument approximation. For the case with known noise variance, a closed-form ML amplitude estimator is obtained, which outperforms the conventional root-mean-square estimator. For the case with unknown noise variance, the closed-form joint amplitude and noise variance estimators obtained do not require prior knowledge of one another.

The ORLS-Based DoA Estimation for Unknown Mixtures of Uncorrelated and Coherent Signals Under Unknown Number of Sources

In this letter, the order recursive least squares (ORLS) is applied to the azimuth-only ULA to probe the DoA estimation under the unknown number of external sources. In the ORLS-based method, the iterative measured matrix equations are feasibly constructed by combining the two spatial modified Yule-Walker (MYW) systems of linear equations and are required to remain Hankel-block-matrix structure of augmented matrices after arrangement order of the unknowns and the maximum number of detectable sources are preset. Under the stationary assumption on source signals and noises, the remarkable advantage of the proposed method lies in that it can theoretically provide the zero LS error as iteration times is equal to the number of sources. The zero LS error is an evident mark to judge the number of sources, especially for the scenario of the unknown noise variances. The proposed method is free of uncorrelated and coherent signals and the corresponding various mixtures and remains computationally efficient in estimation owing to no eigenvalue decomposition (EVD) and matrix inversion operation. The effectiveness of the determination of the number and azimuth angles of sources versus the different SNRs and numbers of snapshots is numerically confirmed.

An improved time support estimation method for overlapping automatic dependent surveillance‐broadcast signals in low signal‐to‐noise ratio region

SummaryIn the satellite‐based application, due to the low signal‐to‐noise ratio (SNR) of received signals, it is highly difficult to estimate the time support of overlapping automatic dependent surveillance‐broadcast (ADS‐B) signals, resulting in the failure of many separation algorithms. Besides, it is troublesome to determine the detection threshold for unknown noise variance. In this paper, we first present three estimation methods of noise variance based on eigenvalue decomposition, which are suitable for constant and slowly varying noise variance, respectively. The proposed time support estimation is therefore of constant false alarm rate property. Moreover, according to the characteristic of ADS‐B signals, we design a matched filter and propose two estimation schemes of direct form and indirect form, which significantly improves the estimation accuracy and detection probability. Finally, in combination of the well‐known projection algorithm (PA), it is concluded that with the proposed estimation method, PA exhibits a satisfactory separation performance at low SNRs.

Super-resolving multiple scatterers detection in synthetic aperture radar tomography assisted by correlation information

This paper proposes a method for detecting multiple scatterers (targets) in the elevation direction for synthetic aperture radar (SAR) tomography. The proposed method can resolve closely spaced targets through a twostep procedure. In the first step, coarse detection is performed with a successive cancellation scheme in which possible locations of targets are marked. Then, in the second step, by searching in the reduced search space which is finely 10 gridded, the accurate location of the targets is found. For estimating the actual number of targets, a model order selection scheme is used in two cases of known and unknown noise variance. Also, by analytical investigation of the probability of detection for the proposed method, the effect of the influential parameters on the detection ability is explicitly demonstrated. Compared to the super-resolution methods based on compressed sensing (CS), the proposed method has a lower computational cost and higher estimation accuracy, especially at low signal-to-noise ratio regime. 15 Simulation results show the superiority of the proposed method in terms of both 3D scatterer reconstruction and detection ability

Joint Scattering Center Enumeration and Parameter Estimation in GTD Model

In this article, a novel multistage CLEAN-based method based on generalized likelihood ratio test (GLRT) is proposed for scattering centers’ enumeration and their parameter estimation. To analyze the performance of the proposed scattering center enumeration algorithm, the probability density function (pdf) of the GLRT is derived for both the known noise variance case and the unknown noise variance case. In addition, we derived a closed form for the probability of correct enumeration of scattering centers, obtained in the case of the known model parameter. A method to reduce the computational complexity of the proposed algorithm has also been introduced. Despite the conventional methods which are based on the damped exponential models, the proposed method uses the geometrical theory of diffraction (GTD) model whose performance does not degrade as the bandwidth increases. The simulation results show an appropriate performance of this method at low signal-to-noise ratio (SNR) situations.

Enhanced Detection Algorithms Based on Eigenvalues and Energy in Random Matrix Theory Paradigm

This paper considers the problem of spectrum sensing in multi-antenna cognitive radio networks. Energy detection (ED) method for spectrum sensing does not require any information of the source signal and channel, as well as it is suitable for detecting independent identically distributed signals. Since covariance matrix catches the signal correlations well, the maximum eigenvalue detection (MED) method is more competitive than the ED method for correlated signals. Under the framework of random matrix theory, this paper firstly proposes two enhanced detection algorithms based on the maximum eigenvalue and energy of the signal to achieve performance improvement while preserving the advantages of the two algorithms. The proposed algorithms are a generalization of the ED and MED methods. To render the proposed algorithms more practical, we propose two other new blind spectrum sensing algorithms based on the maximum likelihood estimate of unknown noise variance. Using random matrix theory, the theoretical analysis on detection probability, false alarm probability and threshold are given. Finally, simulation results show the effectiveness and robustness of the proposed algorithms.

Asymptotic Performances of a Signal-To-Noise Ratio Moment-Based Estimator for Real Sinusoids in Additive Noise

We considered the problem of the estimation of signal-to-noise ratio (SNR) with a real deterministic sinusoid with unknown frequency, phase and amplitude in additive Gaussian noise of unknown variance. A blind SNR estimator that does not require the knowledge of the instantaneous frequency of the sinusoid, through separate estimation of signal and noise power, was derived using the method of moments, a general method to derive estimators based on high-order moments. Statistical performances of the proposed estimators were studied theoretically through derivation of Cramer–Rao lower bounds (CRLBs) and asymptotic variances. Furthermore, results from Monte-Carlo simulations that confirm the validity of the theoretical analysis are presented along with some comments on the use of proposed estimators in practical applications.

Fusion identification and estimation of multisensor multichannel AR signals with missing measurements and sensor biases

This paper studies the problem of fusion identification and estimation for multisensor multichannel autoregressive (AR) signals with unknown missing measurement rates, model parameters (MPs), sensor biases and measurement noise variances. Optimal local filters, cross-covariance matrices (CCMs), and a distributed fusion filter for AR signal are presented when the system model is accurately known, respectively. A self-tuning (ST) fusion filter is presented when MPs of AR signal, sensor biases, missing measurement rates and measurement noise variances in the system model are unknown. First, MPs of AR signal and sensor biases are identified by using a multi-dimensional recursive extended least squares (MRELS) algorithm. Further, distributed fusion estimates of MPs of AR signal are obtained by using a matrix-weighted optimal fusion estimation algorithm in the sense of linear unbiased minimum variance (LUMV). Then, receiving measurement rates and measurement noise variances are identified by correlation functions. Finally, substituting the identified parameters into the proposed optimal filtering algorithms, a matrix-weighted self-tuning (ST) fusion filtering algorithm is obtained. An example verifies the effectiveness of algorithms.

Self-Tuning Distributed Fusion Filter for Multi-Sensor Networked Systems with Unknown Packet Receiving Rates, Noise Variances, and Model Parameters.

In this study, we researched the problem of self-tuning (ST) distributed fusion state estimation for multi-sensor networked stochastic linear discrete-time systems with unknown packet receiving rates, noise variances (NVs), and model parameters (MPs). Packet dropouts may occur when sensor data are sent to a local processor. A Bernoulli distributed stochastic variable is adopted to depict phenomena of packet dropouts. By model transformation, the identification problem of packet receiving rates is transformed into that of unknown MPs for a new augmented system. The recursive extended least squares (RELS) algorithm is used to simultaneously identify packet receiving rates and MPs in the original system. Then, a correlation function method is used to identify unknown NVs. Further, a ST distributed fusion state filter is achieved by applying identified packet receiving rates, NVs, and MPs to the corresponding optimal estimation algorithms. It is strictly proven that ST algorithms converge to optimal algorithms under the condition that the identifiers for parameters are consistent. Two examples verify the effectiveness of the proposed algorithms.

A state of charge estimation method for lithium-ion batteries based on fractional order adaptive extended kalman filter

This paper focuses on the state of charge (SOC) estimation of a lithium-ion battery based on a fractional-order adaptive extended Kalman filter (FO-AEKF). First, a fractional order model (FOM) is introduced to describe the physical behavior of the battery which is superior than the integral-order model (IOM), because there are diffused and decentralized characteristics in battery inner parameters. Then, the parameters of the FOM are identified by a genetic algorithm which can realize optimal parameter identification. After that, the FO-AEKF algorithm is developed, which combines the advantages of the FOM and the adaptive strategy. Consequently, the FO-AEKF can quickly track the unknown and time-invariant (or slow time-varying) noise variance and then improve the accuracy of SOC estimation. Finally, two types of lithium-ion batteries and two dynamic operation conditions are given to show the efficiency of the FO-AEKF by comparing with the extended Kalman filter (EKF) for FOM and the adaptive extended Kalman filter (AEKF) for IOM.

On the Noise Uncertainty for the Energy Detection of OFDM Signals

Orthogonal frequency-division multiplexing (OFDM) is a promising technology for communication systems. This paper investigates the impacts of unknown noise variance on the popular spectrum sensing scheme, i.e., energy detection, for OFDM cognitive radios over multipath fading channels. To study the effects of unknown parameters on the energy detector, a new maximum-likelihood estimation of noise and signal powers employing the cyclic prefix of OFDM is presented in this paper. The mean values and Cram $\acute{\text{e}}$ r–Rao lower bounds of the estimation are obtained. Furthermore, the performances of the energy detector for both hypotheses, i.e., false-alarm rate and detection probability, under the influence of unknown noise variance are validated by both simulation and analytical results. The assessment on the required number of samples for the proposed energy detector is conducted, which indicates that an amount of 40–50% samples can be saved compared to the conventional energy detector.

WMF self-tuning Kalman estimators for multisensor singular system

For the multisensor linear stochastic singular system with unknown noise variances, the weighted measurement fusion (WMF) self-tuning Kalman estimation problem is solved in this paper. The consistent estimates of these unknown noise variances are obtained based on the correlation method. Applying the WMF method and the singular value decomposition (SVD) method yields the WMF reduced-order subsystems. Based on these consistent estimates of unknown noise variances and the new non-singular systems, the WMF self-tuning Kalman estimators of the state components and white noise deconvolution estimators are presented. Then the WMF self-tuning Kalman estimators of the original state are presented, and their convergence has been proved by dynamic error system analysis (DESA) method and dynamic variance error system analysis (DVESA) method. A simulation example of 3-sensors circuits systems verifies the effectiveness, the accuracy relationship and the convergence.

Sparse signal detection without reconstruction based on compressive sensing

This paper considers the problem of compressive detection of sparse signals without signal reconstruction. In the compressed detection model, we present the design of measurement matrix by using the sparsity of a signal, and prove that the variances of the compressive measurements follow two different Gaussian distributions under the binary hypothesis testing. A single measurement vector (SMV) detector is proposed when the knowledge of noise variance is available. Nevertheless, the theoretical analysis shows that the SMV detector has an inferior perfornance in the low signal-noise-ratio regime. To address this situation, the SMV model is extended to the multiple measurement vectors (MMV) model. In the scenario of unknown noise (UN) variance, we propose the UN-MMV detector which requires less prior knowledge. Moreover, the closed-form expressions for the probabilities of false alarm and detection of each detector are derived, respectively. Numerical simulations are conducted to illustrate the performance of the proposed detectors.

Spectrum Sensing Using a Uniform Uncalibrated Linear Antenna Array for Cognitive Radios

In this paper, we study the spectrum sensing problem when the sensing device is equipped with an uncalibrated multiantenna array. We consider an unknown Toeplitz structure for the covariance matrix of the signal of interest and unknown uncorrelated noise variances. We propose two new invariant constant false-alarm rate detectors with less computational cost than the generalized likelihood ratio test: 1) the Rao test and 2) a separating function estimation test (SFET). We calculate approximate expressions for the false-alarm and detection probabilities of the proposed Rao test-based detector. In addition, we derive the exact false-alarm probability of the proposed SFET for the case of two antennas. Moreover, we obtain an accurate approximation for the false-alarm probability of the proposed SFET for the general case using the method of moments. The accuracy of these expressions is validated by Monte-Carlo experiments. The simulation results show that the proposed detectors outperform state-of-the-art detectors.

Distributed adaptive Kalman filter based on variational Bayesian technique

In this paper, distributed Kalman filter design is studied for linear dynamics with unknown measurement noise variance, which modeled by Wishart distribution. To solve the problem in a multi-agent ...

Modulation-Constrained Clustering Approach to Blind Modulation Classification for MIMO Systems

Blind modulation classification is a fundamental step before signal detection in cognitive radio networks where the knowledge of modulation scheme is not completely known. In this paper, a modulation-constrained (MC) clustering classifier is proposed for recognizing the modulation scheme with unknown channel matrix and noise variance for MIMO systems. By recognizing the fact that the received signals within an observation interval form into clusters and exploiting the intrinsic relationships between different digital modulation schemes, the modulation classification is transformed into a number of clustering problems, one for each modulation scheme, with the final classification decision based on the maximum likelihood criterion. To improve the learning efficiency, centroid reconstruction is proposed to recover all cluster centroids through a limited number of parameters by exploiting the structural relationships in constellation diagrams. Furthermore, a method to initialize the cluster centroids is also proposed. The proposed MC classifier together with centroid reconstruction and initialization methods not only reduce the number of parameters to be estimated, but also help to initialize the clustering algorithm for the enhanced convergence performance. Simulation results show that our algorithm can perform excellently even at low SNR and with very short observation interval length.

A Sparse-Based Off-Grid DOA Estimation Method for Coprime Arrays

Recently, many sparse-based direction-of-arrival (DOA) estimation methods for coprime arrays have become popular for their excellent detection performance. However, these methods often suffer from grid mismatch problem due to the discretization of the potential angle space, which will cause DOA estimation performance degradation when the target is off-grid. To this end, we proposed a sparse-based off-grid DOA estimation method for coprime arrays in this paper, which includes two parts: coarse estimation process and fine estimation process. In the coarse estimation process, the grid points closest to the true DOAs, named coarse DOAs, are derived by solving an optimization problem, which is constructed according to the statistical property of the vectorized covariance matrix estimation error. Meanwhile, we eliminate the unknown noise variance effectively through a linear transformation. Due to finite snapshots effect, some undesirable correlation terms between signal and noise vectors exist in the sample covariance matrix. In the fine estimation process, we therefore remove the undesirable correlation terms from the sample covariance matrix first, and then utilize a two-step iterative method to update the grid biases. Combining the coarse DOAs with the grid biases, the final DOAs can be obtained. In the end, simulation results verify the effectiveness of the proposed method.

Distributed parameter estimation in wireless sensor networks in the presence of fading channels and unknown noise variance

Parameter estimation is one of the most important research areas in wireless sensor networks. In this study, we consider the problem of estimating a deterministic parameter over fading channels with unknown noise variance. Owing to the bandwidth constraints in wireless sensor networks, sensor observations are quantized and subsequently transmitted to the fusion center. Two types of communication channels are considered, namely, parallel-access channels and multiple-access channels. Based on the knowledge of channel statistics, the power of the received signals at the fusion center can be described by the mode of the exponential mixture distribution. The expectation maximization algorithm is used to determine maximum likelihood solutions for this mixture model. A new estimator based on the expectation maximization algorithm is subsequently proposed. Simulation results show that this estimator exhibits superior performance compared to the method of moments estimator in both parallel- and multiple-access schemes. In addition, we determine that the parallel-access scheme outperforms the multiple-access scheme when the noise variance is small and it loses its superiority when the noise variance is large.

Power of the spacing test for least-angle regression

Recent advances in Post-Selection Inference have shown that conditional testing is relevant and tractable in high-dimensions. In the Gaussian linear model, further works have derived unconditional test statistics such as the Kac–Rice Pivot for general penalized problems. In order to test the global null, a prominent offspring of this breakthrough is the Spacing test that accounts the relative separation between the first two knots of the celebrated least-angle regression (LARS) algorithm. However, no results have been shown regarding the distribution of these test statistics under the alternative. For the first time, this paper addresses this important issue for the Spacing test and shows that it is unconditionally unbiased. Furthermore, we provide the first extension of the Spacing test to the frame of unknown noise variance. More precisely, we investigate the power of the Spacing test for LARS and prove that it is unbiased: its power is always greater or equal to the significance level $\alpha$. In particular, we describe the power of this test under various scenarii: we prove that its rejection region is optimal when the predictors are orthogonal; as the level $\alpha$ goes to zero, we show that the probability of getting a true positive is much greater than $\alpha$; and we give a detailed description of its power in the case of two predictors. Moreover, we numerically investigate a comparison between the Spacing test for LARS, the Pearson’s chi-squared test (goodness of fit) and a numerical testing procedure based on the maximal correlation. When the noise variance is unknown, our analysis unleashes a new test statistic that can be computed in cubic time in the population size and which we refer to as the $t$-Spacing test for LARS. The $t$-Spacing test involves the first two knots of the LARS algorithm and we give its distribution under the null hypothesis. Interestingly, numerical experiments witness that the $t$-Spacing test for LARS enjoys the same aforementioned properties as the Spacing test.

Angular Statistical Resolution Limit of Two Closely-Spaced Point Targets: A GLRT-Based Study

The study of statistical resolution limit (SRL) of two closely spaced targets has attracted considerable interests in the last decade. Two definitions for SRL have been proposed: one is based on the Cramer–Rao bound (CRB) and the other is based on a decision-based process. In this paper, we focus on the latter one. Different from the existing study, which assumes that the center parameter of interests (POIs) is known a priori , we use a more general model where all the POIs are unknown. We exploit the first-order Taylor expansion of the signals to get an approximate linear model with respect to the tested parameter, namely, the separation of directions of arrival of two sources. Then, we apply the general likelihood rate test to get a closed-form expression of SRL. We consider both the cases with known and unknown noise variance. Moreover, we analyze the impact of some parameters (including the resolution rate, the false-alarm rate, and the waveforms) on the SRL. For comparison, we also derive the CRB-based SRL, which is essentially different with our decision-based counterpart. Numerical simulation results demonstrate the validity of our theoretical results.