Minimum Mean Square Error Criterion Research Articles

Closed-Form Distance Estimators under Kalman Filtering Framework with Application to Object Tracking

In this paper, the minimum mean square error (MMSE) estimation problem for calculation of distances between two signals via the Kalman filtering framework is considered. The developed algorithm includes two stages: the Kalman estimate of a state vector computed at the first stage is nonlinearly transformed at the second stage based on a distance function and the MMSE criterion. In general, the most challenging aspect of application of the distance estimator is calculation of the multivariate Gaussian integral. However, it can be successfully overcome for the specific metrics between two points in line, between point and line, between point and plane, and others. In these cases, the MMSE estimator is defined by an analytical closed-form expression. We derive the exact closed-form bilinear and quadratic MMSE estimators that can be effectively applied for calculation of an inner product, squared norm, and Euclidean distance. A novel low-complexity suboptimal estimator for special composite functions of linear, bilinear, and quadratic forms is proposed. Radar range-angle responses are described by the functions. The proposed estimators are validated through a series of experiments using real models and metrics. Experimental results show that the MMSE estimators outperform existing estimators that calculate distance and angle in nonoptimal manner.

Minimum kernel risk sensitive mean p-power loss algorithms and their performance analysis

The least mean square (LMS) algorithm is optimal for combating Gaussian noises owing to the used minimum mean square error (MSE) criterion in its loss function. However, the MSE criterion is not efficient for non-Gaussian noises. To this end, a new robust minimum kernel risk sensitive mean p-power loss (MKRSP) algorithm is proposed to provide robustness against impulsive noises and improve filtering accuracy, simultaneously. The MKRSP algorithm can generalize the minimum kernel risk sensitive loss (MKRSL) algorithm by setting p=2, and improve filtering accuracy by setting a proper p. Further, the random-Fourier-features MKRSP (RMKRSP) algorithm is developed for improving robustness and filtering accuracy of MKRSP. And the steady-state excess mean square errors (SEMSEs) of MKRSP and RMKRSP are calculated for theoretical analysis. The correctness of the obtained SEMSEs are verified by simulations, and the accuracy advantages of MKRSP and RMKRSP are also confirmed in different noise environments.

Robust Power System State Estimation With Minimum Error Entropy Unscented Kalman Filter

The unscented Kalman filter (UKF) provides a powerful tool for power system forecasting-aided state estimation (FASE). However, when the power systems are affected by the abnormal operating situations, i.e., the non-Gaussian communication noises, sudden loads or state changes, and instrument failures, the original UKF based on the minimum mean square error (MMSE) criterion may suffer from performance degradation. In contrast to the MMSE criterion, the minimum error entropy (MEE) exhibits the robustness with respect to complex non-Gaussian disturbances. In this article, we develop a new unscented Kalman-type filter based on the MEE criterion, termed MEE-UKF. To derive the MEE-UKF, a statistical linearization approach is adopted in the augmented model such that the state and measurement errors are combined in the MEE cost function simultaneously. Then, a fixed-point iteration algorithm is used to recursively update the posterior estimates and covariance matrix. Apart from the impulsive noises, the MEE-UKF can deal with complex multimodal distribution noises in both process and measurement. The high accuracy and strong robustness of MEE-UKF are confirmed by the simulation results on IEEE 14, 30, and 57 bus test systems under different non-Gaussian disturbances.

Linear Precoding and Stream Combining for Rate Splitting in Multiuser MIMO Systems

This letter develops practical linear precoding and stream combining techniques for rate-splitting (RS) in multiuser multiple-input multiple-output (MIMO) systems. We propose linear combining techniques based on the Min-Max, the maximum ratio and the minimum mean-square error criteria along with Regularized Block Diagonalization (RBD) precoders for RS-based multiuser multiple-antenna systems. An analysis of the sum rate performance is carried out, leading to closed-form expressions. Simulations show that the proposed combining schemes offer a significant sum rate performance gain over conventional linear precoding schemes.

Recursive Polynomial Minimum Mean-Square Error Estimation with Applications to Orbit Determination

This paper presents a systematic generalization of the linear update structure associated with the extended Kalman filter for high-order polynomial estimation of nonlinear dynamical systems. The minimum mean-square error criterion is used as the cost function to determine the optimal polynomial update during the estimation process. The high-order series representation is implemented effectively using differential algebra techniques. Numerical examples show that the proposed algorithm, named the high-order differential algebra Kalman filter, provides superior robustness and/or mean-square error performance as compared to linear estimators under the condition considered.

Minimum Error Entropy Kalman Filter

To date, most linear and nonlinear Kalman filters (KFs) have been developed under the Gaussian assumption and the well-known minimum mean square error (MMSE) criterion. In order to improve the robustness with respect to impulsive (or heavy-tailed) non-Gaussian noises, the maximum correntropy criterion (MCC) has recently been used to replace the MMSE criterion in developing several robust Kalman-type filters. To deal with more complicated non-Gaussian noises such as noises from multimodal distributions, in this article, we develop a new Kalman-type filter, called minimum error entropy KF (MEE-KF), by using the minimum error entropy (MEE) criterion instead of the MMSE or MCC. Similar to the MCC-based KFs, the proposed filter is also an online algorithm with the recursive process, in which the propagation equations are used to give prior estimates of the state and covariance matrix, and a fixed-point algorithm is used to update the posterior estimates. In addition, the MEE extended KF (MEE-EKF) is also developed for performance improvement in the nonlinear situations. The high accuracy and strong robustness of MEE-KF and MEE-EKF are confirmed by experimental results.

Using Generalized Gaussian Distributions to Improve Regression Error Modeling for Deep Learning-Based Speech Enhancement

From a statistical perspective, the conventional minimum mean squared error (MMSE) criterion can be considered as the maximum likelihood (ML) solution under an assumed homoscedastic Gaussian error model. However, in this paper, a statistical analysis reveals the super-Gaussian and heteroscedastic properties of the prediction errors in nonlinear regression deep neural network (DNN)-based speech enhancement when estimating clean log-power spectral (LPS) components at DNN outputs with noisy LPS features in DNN input vectors. Accordingly, we propose treating all dimensions of the prediction error vector as statistically independent random variables and model them with generalized Gaussian distributions (GGDs). Then, the objective function with the GGD error model is derived according to the ML criterion. Experiments on the TIMIT corpus corrupted by simulated additive noises show consistent improvements of our proposed DNN framework over the conventional DNN framework in terms of various objective quality measures under 14 unseen noise types evaluated and at various signal-to-noise ratio levels. Furthermore, the ML optimization objective with GGD outperforms the conventional MMSE criterion, achieving improved generalization and robustness.

Kernel Kalman Filtering With Conditional Embedding and Maximum Correntropy Criterion

The Hilbert space embedding provides a powerful and flexible tool for dealing with the nonlinearity and high-order statistics of random variables in a dynamical system. The kernel Kalman filtering based on the conditional embedding operator (KKF-CEO) shows significant performance improvements over the traditional Kalman filters in the noisy nonlinear time-series prediction. However, KKF-CEO based on the minimum mean-square-error (MMSE) criterion is sensitive to the outliers or heavy-tailed noises. In contrast to the MMSE criterion, the maximum correntropy criterion (MCC) can achieve more robust performance in the presence of outliers. In this paper, we develop a novel kernel Kalman-type filter based on MCC, referred to kernel Kalman filtering with conditional embedding operator and maximum correntropy criterion (KKF-CEO-MCC). The proposed KKF-CEO-MCC can capture higher order statistics of errors and is robust to outliers. In addition, two simplified versions of KKF-CEO-MCC are developed, namely, KKF-CEO-MCC-O and KKF-CEO-MCC-NA. The former is an online approach and the latter is based on Nystrom approximation. Simulations on noisy nonlinear time-series prediction confirm the desirable accuracy and robustness of the new filters.

Joint Iterative Channel Estimation and Frequency-Domain Turbo Equalization for Single-Carrier Spatial Modulation

Single-carrier frequency-domain turbo equalization (SC-FDTE) has gained widespread adoption in the emerging broadband spatial modulation (SM) systems operating in frequency-selective channels, where the channel model considered is a quasi-static Rayleigh fading channel. In this paper, a new class of robust FDTE designs based on the minimum mean-square error (MMSE) criterion is conceived for broadband single-carrier SM (SC-SM) systems relying on realistic imperfect channel knowledge. First, a robust time-domain soft-decision feedback (TDSDF)-aided FDTE is proposed to cope with channel estimation errors at the receiver. Furthermore, its robust frequency-domain soft-decision feedback (FDSDF)-aided counterpart is derived to offer a low-complexity approximate solution. Finally, by exploiting the carefully selected reliable soft-decision output of the channel decoder as pilots, we refine the resultant decision-directed channel estimation. As a benefit, the performance of the two robust FDTEs can be further improved. Both our simulation results and our extrinsic information transfer (EXIT) chart analysis demonstrate that the proposed robust FDTEs achieve significant performance improvements over the conventional FDTEs.

Non-linear state recovery in power system under bad data and cyber attacks

The problems of recovering the state of power systems and detecting the instances of bad data have been widely studied in literature. Nevertheless, these two operations have been designed and optimized for the most part in isolation. Specifically, state estimators are optimized based on the minimum mean-square error criteria, which is only optimal when the source of distortions in the data is Gaussian random noise. Hence, the state estimators fail to perform optimality when the data is further contaminated by bad data, which cannot necessarily be modeled by additive Gaussian terms. The problem of power state estimation has been studied extensively. But the fundamental performance limits and the attendant decision rules are unknown when the data is potentially compromised by random bad data (due to sensor failures) or structured bad data (due to cyber attacks, which are also referred to false data injection attacks). This paper provides a general framework that formalizes the underlying connection between state estimation and bad data detection routines. We aim to carry out the combined tasks of detecting the presence of random and structured bad data, and form accurate estimations for the state of power grid. This paper characterizes the optimal detectors and estimators. Furthermore, the gains with respect to the existing state estimators and bad data detectors are established through numerical evaluations.

Two‐stage BFGS‐based hybrid precoding for mmWave multiuser MIMO systems

Millimetre wave (mmWave) communications are the most promising candidate for the future fifth-generation mobile broadband networks, which offer greater available spectrum than current cellular. However, the huge path loss and rain attenuation due to the characteristic of the mmWave channel make it difficult to realise. Thanks to the small wavelength of mmWave signals, massive multi-input-multi-output (MIMO) systems can be leveraged to overcome the path loss. Unfortunately, in contrast to conventional MIMO systems, high-power consumption and hardware cost make fully digital precoding impractical. In this study, the authors study the hybrid precoding structure for multiuser mmWave systems, which compromises the cost and performance. First, an extension algorithm from single user systems, which directly decomposes the optimal fully digital precoder into a baseband precoder and an analogue radio-frequency (RF) precoder, is introduced according to the recent work. Then, the drawback of this kind of algorithm is analysed at high signal-to-noise ratios. Subsequently, a two-stage hybrid precoding algorithm, which is based on the minimum mean-squared error criterion is proposed, where the modified Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm is presented to reduce the reconstruction error of computing the analogue RF precoder. Simulation results show that the proposed precoding algorithm can significantly improve the performance of the system sum-rate.

The Online Random Fourier Features Conjugate Gradient Algorithm

Kernel conjugate gradient (KCG) algorithms have been proposed to improve the convergence rate and filtering accuracy of kernel adaptive filters (KAFs). However, sparsification is still required in the KCG algorithms to curb the growth of network structure for online applications. To address this issue, a novel online random Fourier features conjugate gradient (RFFCG) algorithm based on the minimum mean-square error criterion is proposed by approximating the kernel with the random Fourier features. With no requirement of sparsification, the proposed RFFCG significantly reduces the computational and storage complexities in KAFs. Meanwhile, the RFFCG efficiently uses only one error in the loss function to approach the filtering accuracy of the KAF with sparsification based on all the errors in the loss function. Monte Carlo simulations on short-term chaotic time-series prediction validate the superiorities of the proposed RFFCG algorithm.

ЗАСТОСУВАННЯ ЛОКАЛЬНО-АДАПТИВНОЇ МІРІАДНОЇ ФІЛЬТРАЦІЇ ДЛЯ КОМПЛЕКСНОЇ МОДЕЛІ ОДНОМІРНОГО СИГНАЛУ

Local adaptive algorithms of myriad filtering with adaptation of a sample myriad linearity parameter K depending upon local estimates of signal scale and "hard" switching of parameters of sliding window lengths and coefficients that influence on parameter K are proposed. Statistical estimates of filters quality are obtained using criterion of minimum mean-square error for a complex model of one-dimensional signal that includes elementary signals such as a constant signal, step edge, piecewise linear fragments, linearly increasing and decreasing signals, a peak-like maximum, a piecewise function that consists of constant signal and polynomial curve, a parabolic maximum under conditions of additive Gaussian noise with zero mean and different variances and possible spikes presence. It is shown that the proposed myriad locally-adaptive filters can preserve fragments of rapid changing of signal as step edge and other discontinues due to high dynamic properties in nonlinear mode of myriad filtering and small length of the sliding window and can effectively suppress noise while processing fragments of linear signal behavior and polynomial curves by adjusting the parameter K to a linear mode and increasing the window length. Having high efficiency for all fragments of the complex signal, one of the proposed algorithms provides practically optimal noise suppression at the fragment of linear change of the signal, the other one provides higher quality of step-like and constant signals processing. As a result of application of the proposed myriad locally-adaptive filters, improvement of integral and local performance indicators is shown in comparison to the high effective non-linear locally adaptive algorithm that adaptively switches the output signals between median filter with small window length and alpha-trimmed filters with middle and large windows and local-adaptive myriad algorithm with adaptation of the linearity parameter K which are used for the considered test complex signal. Due to the use of multithreading in programing for parallel calculations, all the considered nonlinear algorithms have possibility to be implemented in real time. The more appropriate algorithm for calculating a sample myriad is the algorithm of minimization of myriad cost function based on a numerical Newton technique because of its best performance in step-like signal fragments and best robustness. In order to ensure better spike removal, it is expedient to pre-process the signal by robust myriad filter.

Near Optimum User Selection for Minimum Mean Square Error Based Tomlinson Harashima Precoding

This paper proposes novel user selection techniques for the Tomlinson-Harashima precoding (THP) based on the minimum mean square error (MMSE) criterion. These techniques utilize a part of the intermediate values in the derivation of the THP weight, which makes it possible to implement the techniques with lower computational complexity. For further complexity reduction, we propose successive user selection techniques based on the techniques proposed above where user terminals are selected one by one. It is confirmed by computer simulations that the proposed techniques outperform conventional selection techniques. The computational complexity of the successive selection techniques is about 1/20 as small as that of the original selection techniques when 4 user terminals are selected out of 10 terminals.

Vector Perturbation Precoding Under Imperfect CSI and Inaccurate Power Scaling Factors

We focus on the design of vector perturbation (VP) precoding for multiuser multiple-input-single-output (MU-MISO) broadcast channel systems where the centralized transmitter equipped with multiple antennas and communicates simultaneously to multiple single-antenna receivers. While conventional VP requires the feedback of the channel matrix at the transmitter for precoding and the power scaling factor at the receivers for detection, VP precoding has so far been developed and analyzed under assumptions that the transmitter has perfect channel state information (CSI) or the receivers have perfect knowledge of the channel- and data-dependent power scaling factors. In practical limited feedback scenarios, wireless communication systems suffer from limited time and frequency resource for pilots to feed-forward information and only a quantized version of power scaling factors is available at the receivers; under such limitations, the performance of VP precoding will degrade significantly compared with ideal scenarios and would always encounter an error floor at mid-to-high signal-to-noise ratio (SNR) regions. Motivated by such observations, we propose a robust VP precoder design, which takes the imperfectness of CSI and power scaling factor jointly into account under the criterion of minimum mean-square error (MMSE). The closed-form expressions of the proposed precoder are then derived. As illustrated by the simulation results, the proposed VP precoder is less sensitive to CSI and power scaling factor imperfections compared with the classic VP precoder and other existing MMSE-based VP precoders, as it has a lower error floor when imperfectness is assumed to be fixed, and power scaling factor accuracy is shown to offer a non-linear performance gain compared with that of the linear gain CSI accuracy could offer.

Sliding Window Filtering for Ground Moving Targets with Cross-Correlated Sensor Noises

This paper reports a sliding window filtering approach for ground moving targets with cross-correlated sensor noise and uncertainty. In addition, the effect of uncertain parameters during a tracking error on the model performance is considered. A distributed fusion sliding window filter is also proposed. The distributed fusion filtering algorithm represents the optimal linear combination of local filters under the minimum mean-square error criterion. The derivation of the error cross-covariances between the local sliding window filters is the key to the proposed method. Simulation results of the motion of the ground moving target a demonstrate high accuracy and computational efficiency of the distributed fusion sliding window filter.

An Adaptive Approach for the Joint Antenna Selection and Beamforming Optimization

Adaptive beamforming techniques are widely known for their capability of leveraging the performance of antenna arrays. The effectiveness of such techniques typically increases as the number of antennas grows. In contrast, computational and hardware costs very often limit the deployment of beamforming in large-scale arrays. To circumvent this problem, antenna selection strategies have been developed aiming to maintain much of the performance gain obtained by using a large array while keeping computational and hardware costs at acceptable levels. In this context, the present paper is dedicated to the development of two new adaptive algorithms for solving the problem of joint antenna selection and beamforming for uplink reception in mobile communication systems. Both algorithms are based on an alternating optimization strategy and are designed to operate with a limited number of radio-frequency chains. The main difference between the proposed algorithms is that the first is formulated by considering the minimum mean-square error (MMSE) criterion, while the second is based on the minimum-variance distortionless-response (MVDR) approach. The numerical simulation results confirm the effectiveness of the proposed algorithms.

Linear precoder design for non-orthogonal AF MIMO relaying systems based on MMSE criterion

Multiple-input multiple-output (MIMO) relaying system has attracted the attention of cooperative network researchers, due to its advantage over the conventional single antenna system, in terms of system capacity and spatial diversity. Precoder design is a processing scheme implemented at a source and relay node to improve system performance. We propose a linear precoder design for non-orthogonal amplify-and-forward MIMO relaying systems based on the minimum mean square error (MMSE) criterion. We analyze an upper bound of MMSE using a convenient expression to determine the structure of precoding matrices using the singular-value decomposition technique. Simulation results demonstrate that the proposed precoded scheme outperforms both unprecoded and existing precoded schemes.

Soft-Feedback Time-Domain Turbo Equalization for Single-Carrier Generalized Spatial Modulation

In this paper, a new class of soft-decision feedback aided time-domain turbo equalizer (TDTE) based on the minimum mean-square error criterion is proposed for broadband single-carrier (SC) generalized spatial modulation (GSM) systems. First, a time-domain soft-decision feedback (TDSDF) aided TDTE is proposed, which operates on a vector-by-vector basis. In order to provide a flexible tradeoff between complexity and performance, time-invariant filter coefficients are derived to provide a low-complexity approximate solution to the proposed TDSDF-TDTE. Furthermore, a soft-feedback interference cancelation (SFIC) aided TDTE is proposed for SC-GSM systems with a lower complexity, where the computation of the second-order moment of matrices is completely avoided. Finally, bi-directional soft-decision feedback aided TDTEs are proposed by using a time-reversed TDSDF-TDTE/SFIC-TDTE in conjunction with a ordinary TDSDF-TDTE/SFIC-TDTE so as to harvest time-reverse diversity and to mitigate the error propagation. An extrinsic information transfer chart based analysis and bit error rate simulation results are presented to demonstrate the superior performance of the proposed soft-decision feedback aided TDTEs as opposed to the original TDTEs, while achieving a lower complexity.

Random Fourier Filters Under Maximum Correntropy Criterion

Random Fourier adaptive filters (RFAFs) project the original data into a high-dimensional random Fourier feature space (RFFS) such that the network structure of filters is fixed while achieving similar performance with kernel adaptive filters. The commonly used error criterion in RFAFs is the well-known minimum mean-square error (MMSE) criterion, which is optimal only under the Gaussian noise assumption. However, the MMSE criterion suffers from instability and performance deterioration in the presence of non-Gaussian noises. To improve the robustness of RFAFs against large outliers, the maximum correntropy criterion (MCC) is applied to RFFS, generating a novel robust random Fourier filter under maximum correntropy (RFFMC). To further improve the filtering accuracy, a random-batch RFFMC (RB-RFFMC) is also presented. In addition, a theoretical analysis on the convergence characteristics and steady-state excess mean-square error of RFFMC and RB-RFFMC is provided to validate their superior performance. Simulation results illustrate that RFFMC and its extension provide desirable filtering performance from the aspects of filtering accuracy and robustness, especially in the presence of impulsive noises.