Minimum Mean Square Error Sense Research Articles

Approximate Kalman filtering by both M-robustified dynamic stochastic approximation and statistical linearization methods

The problem of designing a robustified Kalman filtering technique, insensitive to spiky observations, or outliers, contaminating the Gaussian observations has been presented in the paper. Firstly, a class of M-robustified dynamic stochastic approximation algorithms is derived by minimizing at each stage a specific time-varying M-robust performance index, that is, general for a family of algorithms to be considered. The gain matrix of a particular algorithm is calculated at each stage by minimizing an additional criterion of the approximate minimum variance type, with the aid of the statistical linearization method. By combining the proposed M-robust estimator with the one-stage optimal prediction, in the minimum mean-square error sense, a new statistically linearized M-robustified Kalman filtering technique has been derived. Two simple practical versions of the proposed M-robustified state estimator are derived by approximating the mean-square optimal statistical linearization coefficient with the fixed and the time-varying factors. The feasibility of the approaches has been analysed by the simulations, using a manoeuvring target radar tracking example, and the real data, related to an object video tracking using short-wave infrared camera.

Low-Resolution MMSE Equalization for Massive MU-MIMO Systems

Finite-alphabet equalization that represents a spatial equalization matrix with low-resolution coefficients is a promising technique to reduce the power consumption, processing delay, and circuit area of baseband processing in the context of all-digital massive multiuser multiple-input multiple-output uplink systems. However, to minimize the performance loss caused by a coarse-resolution spatial equalization matrix, its coefficients must be carefully designed in the minimum mean-square error sense to achieve the desired bit error rate (BER) performance, which unfortunately constitutes an NP-hard optimization problem. To tackle this problem, we first reformulate the finite-alphabet equalization design problem as an unconstrained optimization on a smooth Riemannian manifold. Then we propose an algorithm based on Riemannian manifold optimization (RMO) to solve the reformulated problem. Based on simulation results, the proposed 2-bit RMO-assisted equalizer outperforms its state-of-the-art counterparts while maintaining the same asymptotic complexity. In addition, the proposed 2-bit RMOassisted equalizer exhibits a loss of only 1.47 dB at a BER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-4</sup> compared to an unquantized linear minimum mean squared error equalizer.

Distributed Kalman Filtering for Interconnected Dynamic Systems.

This article is concerned with the distributed Kalman filtering problem for interconnected dynamic systems, where the local estimator of each subsystem is designed only by its own information and neighboring information. A decoupling strategy is developed to minimize the impact of interconnected terms on the estimation performance, and then the recursive and distributed Kalman filter is derived in the minimum mean-squared error sense. Moreover, by using Lyapunov criterion for linear time-varying systems, stability conditions are presented such that the designed estimator is bounded. Finally, a heavy duty vehicle platoon system is employed to show the effectiveness and advantages of the proposed methods.

Recursive state estimation for a class of nonlinear uncertain coupled complex networks subject to random link failures and packet disorders

This paper is concerned with the recursive state estimation (RSE) problem under minimum mean-square error sense for a class of nonlinear complex networks (CNs) with uncertain inner coupling, random link failures and packet disorders. Firstly, a set of random variables obeying the Bernoulli distribution is adopted to characterize whether there are connections between different network units, i.e., there is no random link failure when the random variable is equal to 1, otherwise the random link failure occurs. In addition, the inner coupling strength is assumed to be varying within a given interval and the phenomenon of packet disorders caused by the random transmission delay (RTD) is also taken into account. In our study, the nonlinearity satisfies the continuously differentiable condition, which can be linearized by resorting to the Taylor expansion. The focus of the addressed RSE problem is on the design of an RSE approach in the mean-square error sense such that, for all uncertain inner coupling, random link failures and packet disorders, a suboptimal upper bound of the state estimation error covariance is obtained and minimized by parameterizing the state estimator gain with explicit expression form. Furthermore, a sufficient condition with respect to the uniform boundedness of state estimation error in mean-square sense is elaborated. Finally, a numerical experiment is introduced to demonstrate the validity of the presented RSE approach.

FDA-MIMO radar covariance matrix estimation via shrinkage processing

Frequency diverse array multiple-input multiple-output (FDA-MIMO) produces an angle-range-dependent and time-varying transmit beampattern due to the small frequency increment across its array elements, which provides potential applications in new radar techniques. Due to this particularity, multiple signal classification (MUSIC) algorithm can be used to simultaneously estimate the angle and range of the target through a two-dimensional search. The MUSIC need to estimate the covariance matrix. In this paper, we introduce a shrinkage processing for estimating large covariance matrices when the number of samples is substantially fewer than the number of variables which is more common in FDA-MIMO than phase array (PA). Firstly, we propose a receiving model of multi-channel mixing multi-matched filter to eliminate time-varying effect. Secondly, the taper processing, Toeplitz rectification processing and shrinkage-to-identity processing are analyzed. Then, we propose two algorithms, namely, shrinkage-to-taper and shrinkage-to-Toeplitz-rectification on the basis of these algorithms. The proposed processing method improves upon both shrinkage and Toeplitz rectification or tapering processing by shrinking the sample covariance matrix (SCM) to its Toeplitz rectification or tapered version. Thirdly, a closed form expression of shrinkage coefficient for the shrinkage-to-Toeplitz-rectification, and shrinkage-to-taper are used. Finally, simulations results for FDA-MIMO radar show that the proposed algorithm is superior to the sample covariance matrix, shrinkage LW algorithm proposed by Ledoit-Wolf and shrinkage RBLW algorithm which is improved shrinkage LW according to the Rao-Blackwell theorem, in the sense of minimum mean-squared error (MMSE).

Tobit Kalman filter with channel fading and dead-zone-like censoring

In complex control systems, the channel fading and the measurement censoring are often unavoidable due to sudden environmental changes, intermittent transmission congestions, random failures and repairs of components. To this end, the Tobit Kalman filtering problem for discrete-time systems with the dead-zone-like censoring and channel fading is studied where the censoring and fading phenomena are, respectively, described by the Tobit measurement model and the -order Rice fading channel model. The design of the filter is carried out in two stages: (1) the state augmentation technique is firstly adopted to remove the time-delay effect caused by the fading channel; and (2) the statistical information of censored measurements is used to determine the optimal filter structure in the minimum mean-square error sense, where the orthogonal projection principle is utilised to avoid the deep correlations among the multiple random variables induced by measurement censoring and channel fading. Besides, the desired filter is also devised based on the approach of the conditional expectation. For the convenience of calculation, the filter with the dead-zone-like censoring is converted into an equivalent one with one-side censoring. Finally, a numerical example of estimating the ballistic ball rates is given to verify the effectiveness of the proposed filters.

Dual Pilot Scheme (DPS) and Its Application in Massive MIMO

The pilot scheme currently used in 5th generation (5G) cellular networks assigns the same set of orthogonal pilot signals to all cells. This results in inter-cell interference, also known as pilot contamination , which can significantly degrade performance, especially in massive multi-input multi-output (MIMO) systems. To mitigate this interference, we propose a novel Dual Pilot Scheme (DPS) that assigns a slightly modified set of nearly-orthogonal pilot signals. DPS is a general scheme that can be implemented in any wireless communication system, including 5G and beyond. We demonstrate the integration of DPS in a massive MIMO system in both microscopic and macroscopic levels and analytically prove that DPS enables more accurate estimates of the channel state information in the minimum mean-squared error sense, under the independent identically distributed (i.i.d.) and the correlated Rayleigh fading wireless communication channel models. We further validate and demonstrate the advantages of DPS over various channel models of massive MIMO 5G technology by extensive simulations.

Best integer equivariant estimation: performance analysis using real data collected by low-cost, single- and dual-frequency, multi-GNSS receivers for short- to long-baseline RTK positioning

The key to precise global navigation satellite system (GNSS) positioning is carrier phase integer ambiguity resolution with a high success rate. On the other hand when the success rate is too low, the user will normally prefer the float solution. The alternative can be to use the best integer equivariant (BIE) estimator, since it is optimal in the minimum mean squared error (MMSE) sense. Low-cost receiver real-time kinematic precise positioning has become possible through the many signals that can be obtained by combining several GNSSs, such as BDS, Galileo, QZSS and GPS. In this contribution, we will use both simulations and such low-cost multi-GNSS data to compare the performance of the BIE and integer least squares (ILS) estimator, based on full ambiguity resolution. The GNSS data are evaluated in Dunedin, New Zealand, with a short- (670 m) and long-baseline (112.9 km) where the relative atmospheric delays can be neglected and need to be estimated, respectively. We compare the BIE and ILS results by using both single-frequency and dual-frequency (DF) low-cost and survey-grade receivers and antennas. We demonstrate, for the first time, the distributional properties of BIE positioning, where it will be shown that a ‘star-like’ pattern reveals itself once the model gets stronger and the ILS success rate increases. It will further be shown that the DF low-cost receivers give a very good positioning performance, but still not yet competitive to the survey-grade counterparts for the long-baseline. We will also demonstrate that the positioning performance of the BIE estimator will always equal or be better than that of the float solutions. It will finally be shown that BIE will always be better in the MMSE sense than the ILS solution when the success rate is at low to medium levels, whereas for high success rates we get a similar performance to ILS.

Target State and Markovian Jump Ionospheric Height Bias Estimation for OTHR Tracking Systems

Ionospheric heights provided by ionosondes are vital for over-the-horizon radar (OTHR) target tracking. However, biases contained in the provided ionospheric heights definitely cause degradation of the tracking performance. This paper is concerned with the joint estimation problem of the target state and the ionospheric height bias for OTHR target tracking subject to multipath and cluttered measurements. A Markovian jump ionospheric height bias model is presented by simultaneously considering the intermittent and abrupt ionosphere changes. Meanwhile, a set of stochastic variables are adopted to depict association uncertainties among measurements, clutters, and propagation modes so that association is embedded in the resultant measurement model with random coefficients. Through such modeling transformation from association uncertainties to parameter randomness, the coupling processing of data association and state estimate is equivalent to pure state estimate subject to stochastic parameters. Furthermore, an optimal linear joint estimator containing causality constraints is developed in the minimum mean-squared error sense, and further extended to the case of nonlinear measurement model via iterative optimization. A target tracking example with four resolvable propagation modes illustrates the effectiveness of the proposed estimation scheme.

An Optimal Filter for Singular Systems With Stochastic Multiplicative Disturbance

This brief proposes an optimal filtering algorithm for linear singular systems with stochastic multiplicative noise by using the second restricted equivalent transformation (SRET) to reduce computation load, avoid curse of dimensionality and improve precision. The presented filter can be guarantee optimum in the sense of linear minimum mean-square error (LMMSE) by employing the projection theorem. A performance analysis and simulation results verify the proposed algorithm.

Recursive state estimation for linear systems with lossy measurements under time-correlated multiplicative noises

This paper is concerned with the recursive state estimation problem for a class of linear discrete-time systems with lossy measurements and time-correlated multiplicative noises (TCMNs). The lossy measurements result from one-step transmission delays and packet dropouts. Different from the traditional white multiplicative noises, TCMNs are included in the measurement model in order to reflect engineering practice. Utilizing the state augmentation approach, the system under investigation is first converted into a stochastic parameter system, and some new recursive terms (including the estimation for the product of state and multiplicative noises) are introduced to handle the difficulties caused by the TCMNs. Then, by the well-known projection theorem, recursive state estimation algorithms are developed in the sense of minimum mean-square error, which facilitate the design of the filter, the multi-step predictor and the smoothers. The proposed algorithms are explicitly dependant on the key system parameters including the covariances of the TCMNs, the occurrence probabilities of the transmission delays and the packet losses. Finally, simulation results illustrate the effectiveness of the presented estimation algorithms.

An Optimal Data Fusion Algorithm in the Presence of Unknown Cross Covariances

This paper presents an optimal data fusion formulation and algorithm in the sense of minimum mean-square error when some or all cross covariances are unknown. This algorithm is generic in that it is capable of processing any number of measurement vectors of any dimension with any pattern of unknown cross covariances. Closed-form solution is provided for the case when all cross covariances are unknown and all measurement covariance matrices are diagonal. Numerical projected subgradient optimal fusion algorithm is provided for the most generic case. The well known covariance intersection method is shown to have a weaker upper bound of this formulation.

Quantized Kalman Filter Tracking in Directional Sensor Networks

Moving target tracking in directional sensor networks has recently attracted attention by right of special directional sensing features. Unlike omnidirectional sensors, the directional sensor senses the target only in the direction of its orientation. It can provide quantized direction that indicates the presence or absence of the target in the sensing field, rather than just the analog measurement of sensing signal with respect to the detected target. A quantized Kalman filter (QKF) based on both quantized directions and analog ranging measurements is derived in the minimum mean-square error (MMSE) sense. Its performances of mean square estimation error (MSE) and complexity are also analyzed. Then, a reduced-complexity QKF of high-accuracy is pursued to facilitate its implementation. It is proved that the QKF yields a smaller MSE than the traditional extended Kalman filter (EKF) merely based on analog measurements. The posterior Cram $\acute{e}$ r-Rao lower bound (PCRLB) is introduced as the performance measure. The performance advantages of the proposed QKF are demonstrated using Monte Carlo simulations in a target tracking application using ultrasonic ranging sensors.

State estimation for networked systems with Markovian communication constraints and multiple packet dropouts

In this paper, the state estimation problem for discrete-time networked systems with communication constraints and random packet dropouts is considered. The communication constraint is that, at each sampling instant, there is at most one of the various transmission nodes in the networked systems is allowed to access a shared communication channel, and then the received data are transmitted to a remote estimator to perform the estimation task. The channel accessing process of those transmission nodes is determined by a finite-state discrete-time Markov chain, and random packet dropouts in remote data transmission are modeled by a Bernoulli distributed white sequence. Using Bayes’ rule and some results developed in this study, two state estimation algorithms are proposed in the sense of minimum mean-square error. The first algorithm is optimal, which can exactly compute the minimum mean-square error estimate of system state. The second algorithm is a suboptimal algorithm obtained under a lot of Gaussian hypotheses. The proposed suboptimal algorithm is recursive and has time-independent complexity. Computer simulations are carried out to illustrate the performance of the proposed algorithms.

Massive MIMO Pilot Decontamination and Channel Interpolation via Wideband Sparse Channel Estimation

We consider a massive MIMO system based on time division duplexing (TDD) and channel reciprocity, where the base stations (BSs) learn the channel vectors of their users via the pilots transmitted by the users in the uplink (UL). It is well-known that, in the limit of very large number of BS antennas, the system performance is limited by pilot contamination, due to the fact that the same set of orthogonal pilots is reused in multiple cells. In the regime of moderately large number of antennas, another source of degradation is channel interpolation because the pilot signal of each user probes only a limited number of orthogonal frequency division multiplexing (OFDM) subcarriers, and the channel must be interpolated over the other subcarriers, where no pilot symbol is transmitted. In this paper, we propose a low-complexity algorithm that uses the received UL wideband pilot snapshots in an observation window comprising several coherence blocks (CBs) to obtain an estimate of the angle-delay power spread function (PSF) of the received signal. This is generally given by the sum of the angle-delay PSF of the desired user and the angle-delay PSFs of the copilot users, i.e., the users re-using the same pilot dimensions in other cells/sectors. We propose supervised and unsupervised clustering algorithms to decompose the estimated PSF and isolate the part corresponding to the desired user only. We use this decomposition to obtain an estimate of the covariance matrix of the user wideband channel vector, which we exploit to decontaminate the desired user channel estimate by applying minimum mean squared error (MMSE) smoothing filter, i.e., the optimal channel interpolator in the MMSE sense. We also propose an effective low-complexity approximation/implementation of this smoothing filter. We use numerical simulations to assess the performance of our proposed method, and compare it with other recently proposed schemes that use the same idea of separability of users in the angle-delay domain.

State estimation for discrete-time Markov jump linear systems with time-correlated and mode-dependent measurement noise

The state estimation problem for discrete-time Markov jump linear systems corrupted by time-correlated and mode-dependent measurement noise is considered where the time-correlated and mode-dependent measurement noise is described via a discrete-time stochastic system with Markov parameter and Kronecker delta function. By defining the measurement noise in this manner, both time-correlation and periodic step change caused by the change of system environment or structure can be embodied in the measurement noise. A novel “distributed measurement differencing method” is applied to the problem of state estimation under consideration so that two algorithms are obtained using some results presented in this paper. The first algorithm is optimal in the sense of minimum mean-square error, which can exactly compute the minimum mean-square error estimate of system state. The second algorithm is suboptimal and the suboptimality of the algorithm is caused by using some Gaussian hypotheses. The two proposed algorithms are recursive and the proposed suboptimal algorithm has a time-independent complexity. The performance of the proposed suboptimal algorithm is illustrated using computer simulations.

Time-Varying Weighted Optimal Empirical Mode Decomposition Using Multiple Sets of Basis Functions

Empirical mode decomposition (EMD) is a favorite tool for analyzing nonlinear and non-stationary signals. It decomposes any signal into a finite set of oscillation modes consisting of intrinsic mode functions and a residual function. Superimposing all these modes reconstructs the signal without any information loss. In addition to satisfying the perfect reconstruction property, however, there is no implication about the reconstruction optimality of the EMD. The lack of optimality restricts the signal recovery capability of the EMD in the presence of disturbances. Only a few attempts are made to meet this deficiency. In this paper, we propose a new algorithm named as time-varying weighted EMD. By this algorithm, original signal is reconstructed in the minimum mean-square error sense through the EMD followed by time-varying weightings of the oscillation modes. Determining the time-varying weights for the oscillation modes constitutes the backbone of the algorithm. Aiming to determine the time-varying weights of the oscillation modes; we use multiple sets of basis functions. The effectiveness of the proposed algorithm is demonstrated by computer simulations involving real biomedical signals. Simulation results show that the proposed algorithm exhibits better performance than that of its existing counterparts in terms of lower mean-square error and higher signal-to-error ratio.

Optimal signal reconstruction based on time-varying weighted empirical mode decomposition

Empirical mode decomposition (EMD) is a tool developed for analyzing nonlinear and non-stationary signals. It is capable of splitting any signal into a set of oscillation modes known as intrinsic mode functions and a residual function. Although the EMD satisfies the perfect signal reconstruction property by superimposing all the oscillation modes, it is not based on any optimality criterion. The lack of optimality limits the signal recovery performance of the EMD in the presence of disturbances such as noise and interference. In this paper, we propose a new algorithm, termed, time-varying weighted EMD, which gives the best estimate of a given signal in the minimum mean-square error sense. The main idea of the proposed algorithm is to reconstruct the original signal through the EMD followed by time-varying weightings of the oscillation modes. Simulations including two real-life signals are performed to show the superiority of the proposed algorithm.

State estimation for discrete-time Markov jump linear systems with time-correlated measurement noise

In this paper, the state estimation problem for discrete-time Markov jump linear systems affected by time-correlated measurement noise is considered where the time-correlated measurement noise is described by a linear system model with white noise. As a result, two algorithms are proposed to estimate the state of the system under consideration based on a measurement sequence. The first algorithm is optimal in the sense of minimum mean-square error, which is obtained based on the measurement differencing method, Bayes’ rule and some results derived in this paper. The second algorithm is a suboptimal algorithm obtained by using a lot of Gaussian hypotheses. The proposed suboptimal algorithm is finite-dimensionally computable and does not increase computational and storage load with time. Computer simulations are carried out to evaluate the performance of the proposed suboptimal algorithm.

Optimal Filtering for Discrete-Time Linear Systems With Time-Correlated Multiplicative Measurement Noises

In this note, the filtering problem for discrete-time linear systems with time-correlated multiplicative measurement noises is considered where the vector consisting of all the multiplicative measurement noises can be described by a linear system model with white noise. By introducing several new recursive terms, a novel algorithm for optimal filtering of the system under consideration is proposed in the sense of linear minimum mean-square error. The proposed algorithm is recursive and has time-independent complexity. Computer simulations are provided to illustrate the performance of the proposed algorithm.