Maximum Correntropy Estimation Is a Smoothed MAP Estimation

The multikernel adaptive filters based on the minimum mean square error (MMSE) criterion have been proposed to improve the performance of the kernel least mean square (KLMS), efficiently. However, these multikernel methods suffer from large computational burden as well as instability in impulsive noises. To address these issues, a novel multikernel method is proposed by replacing the trace operation of a matrix with the inner product of two vectors, thus leading to higher computational efficiency, significantly. Based on the maximum correntropy criterion, the multikernel maximum correntropy (MKMC) algorithm is therefore proposed. To further reduce the complexity, an online vector quantization strategy is presented for MKMC to generate the quantized MKMC (QMKMC) algorithm. Monte Carlo simulations on different nonlinear examples validate the superiorities of the proposed two algorithms.

This paper investigates the kernel entropy based extended Kalman filter (EKF) as the navigation processor for the Global Navigation Satellite Systems (GNSS), such as the Global Positioning System (GPS). The algorithm is effective for dealing with non-Gaussian errors or heavy-tailed (or impulsive) interference errors, such as the multipath. The kernel minimum error entropy (MEE) and maximum correntropy criterion (MCC) based filtering for satellite navigation system is involved for dealing with non-Gaussian errors or heavy-tailed interference errors or outliers of the GPS. The standard EKF method is derived based on minimization of mean square error (MSE) and is optimal only under Gaussian assumption in case the system models are precisely established. The GPS navigation algorithm based on kernel entropy related principles, including the MEE criterion and the MCC will be performed, which is utilized not only for the time-varying adaptation but the outlier type of interference errors. The kernel entropy based design is a new approach using information from higher-order signal statistics. In information theoretic learning (ITL), the entropy principle based measure uses information from higher-order signal statistics and captures more statistical information as compared to MSE. To improve the performance under non-Gaussian environments, the proposed filter which adopts the MEE/MCC as the optimization criterion instead of using the minimum mean square error (MMSE) is utilized for mitigation of the heavy-tailed type of multipath errors. Performance assessment will be carried out to show the effectiveness of the proposed approach for positioning improvement in GPS navigation processing.

A Bayesian estimation approach for enhancing speech signals which have been degraded by statistically independent additive noise is motivated and developed. In particular, minimum mean square error (MMSE) and maximum a posteriori (MAP) signal estimators are developed using hidden Markov models (HMMs) for the clean signal and the noise process. It is shown that the MMSE estimator comprises a weighted sum of conditional mean estimators for the composite states of the noisy signal, where the weights equal the posterior probabilities of the composite states given the noisy signal. The estimation of several spectral functionals of the clean signal such as the sample spectrum and the complex exponential of the phase is also considered. A gain-adapted MAP estimator is developed using the expectation-maximization algorithm. The theoretical performance of the MMSE estimator is discussed, and convergence of the MAP estimator is proved. Both the MMSE and MAP estimators are tested in enhancing speech signals degraded by white Gaussian noise at input signal-to-noise ratios of from 5 to 20 dB. >

In wireless sensor networks (WSN), measurements are always corrupted by outliers or impulsive noise. Cubature information filtering (CIF) is founded based on minimum mean square error (MMSE) criterion, which is not applicable to non-Gaussian noise. Hence, a novel robust CIF (RCIF) is derived based on maximum correntropy criterion (MCC) to enhance the robustness of state estimation in the local node. For the information fusion, weighted average consensus (WAC) based distributed RCIF (DRCIF) is founded to improve the stability of sensor networks and the accuracy of state estimation. The estimation error of DRCIF is proved to be bounded in mean square. Numerical simulations are provided to evaluate the effectiveness of proposed algorithms.

In this paper, we propose a distributed source localization algorithm based on the Maximum A Posteriori (MAP) criterion, where the observations generated by each of the distributed sensors are quantized before being transmitted to a fusion node for localization. If the source signal energy is known, each quantized sensor reading corresponds to a region in which the source can be located. Aggregating the information obtained from multiple sensors corresponds to generating intersections between the regions. In our previous work we developed quantizer design techniques aimed at optimizing localization accuracy for a given aggregate rate. In this paper we develop localization algorithms based on estimating the likelihood of each of the intersection regions. This likelihood can incorporate uncertainty about the source signal energy as well as measurement noise. We show that the computational complexity of the algorithm can be significantly reduced by taking into account the correlation of the received quantized data. We also propose a technique, based on a weighted average of estimators, to address the case when the signal energy is unknown. Our simulation results show that our localization algorithm achieves good performance with reasonable complexity as compared with Minimum Mean Square Error (MMSE) estimation.

This paper presents novel kernel adaptive filters with feedback, namely, kernel recursive maximum correntropy with multiple feedback (KRMC-MF) and its simplified version, a linear recurrent kernel online learning algorithm based on maximum correntropy criterion (LRKOL-MCC). In LRKOL-MCC and KRMC-MF, single output and multiple outputs based on single delay are utilized to construct their feedback structure, respectively. Compared with the minimum mean square error criterion, the maximum correntropy criterion (MCC) adopted by LRKOL-MCC and KRMC-MF captures higher order statistics of errors. The proposed filters are, therefore, robust against outliers. Therefore, the past information can be reused to improve filtering performance in terms of the steady-state mean square error. The convergence characteristics of the filter parameters in LRKOL-MCC and KRMC-MF are also derived. Simulations on chaotic time-series prediction and nonlinear regression illustrate the desirable accuracy and robustness of the proposed filters.

We generalize the multiuser (CDMA spread spectrum) communication systems to the fading environments. We first extend Verdu's (1986) conventional optimum receiver to Rayleigh-fading environments and then evaluate its performance. Having no knowledge of the received power at the receiving end, we therefore need an estimator to efficiently estimate the received signal strength of each user in fading environments. A linear minimum mean square error (LMMSE) unbiased estimator is proposed to attain this goal. By using the minimum mean square error (MMSE) Bayesian estimation, we further propose the LMMSE bit estimator for efficient demodulation. Its performance is close to the optimum multiuser receiver but with a much simpler polynomial complexity. To further reduce the complexity, we extend the LMMSE estimator to the sequential LMMSE estimator. In sequential estimation, we do not need to implement the matched filter banks and to perform the matrix inversion when estimating. In addition, it converges after approximately 2k iterations, where k is the number of users. With this fast convergence property and the simple structure, the sequential LMMSE estimator provides an attractive alternative to the implementation of a multiuser system.

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.

In present days, there has been a race among the communication systems in different schemes and generations. Gradually the transmission rate of data is increasing tremendously and breaking the previous generation's record. Now people are thinking of transmitting data beyond imagination. Using multiple antennas both at transmitter and receiver side, MIMO wireless technology efficiently provides very high spectral efficiency, boost the data transmission rate, enhance link reliability and improve system coverage. While transmitting data over mobile radio channels at high bit rates, the impulse response of the channel can be extended over a lot of symbol periods, leading fading and Inter-symbol interference (lSI). Equalizers are used to minimize Fading and interference effects. The analysis of the paper is studying the performance of Minimum mean square error method in terms of BER for MIMO (2×2) and (3×3) wireless channels. Binary Phase Shift Keying (BPSK) modulation technique was used in Rayleigh flat fading channel. Simulating in MATLAB results show that Minimum mean square error (MMSE) receivers is good for noise free channel and is successful in removing ISI. Successive interference method with optimal ordering outperforms the MMSE however a few problems arise because of iterative nature of the algorithm.

In orthogonal frequency-division multiplexing (OFDM) systems, it is generally assumed that the channel response is static in an OFDM symbol period. However, the assumption does not hold in high-mobility environments. As a result, intercarrier interference (ICI) is induced and the system performance is degraded. A simple remedy for this problem is the application of the zero-forcing (ZF) and minimum mean square error (MMSE) equalizers. Unfortunately, the direct ZF method requires the inversion of an NxN ICI matrix, where N is the number of subcarriers. When N is large, the computational complexity can become prohibitively high. As for the direct MMSE method, in addition to an NxN matrix inverse, it requires an extra NxN matrix multiplication, making the required computational complexity higher compared to the direct ZF method. In this dissertation, we first propose a low-complexity ZF method to solve the problem in single-input-single-output (SISO) OFDM systems. The main idea is to explore the special structure inherent in the ICI matrix and to apply Newton's iteration for matrix inversion. With our formulation, fast Fourier transforms (FFTs) can be used in the iterative process, reducing the complexity from O(N^3) to O(Nlog_2 N) . Also, the required number of the iteration is typically one or two. We also analyze the convergence behavior of the proposed method and derive the theoretical output signal-to-interference-noise-ratio (SINR). For the MMSE method, we first reformulate the MMSE solution in a way that the extra matrix multiplication can be avoided. Similar to the ZF method, we then exploit the structure of the ICI matrix and apply Newton's iteration to reduce the complexity of the matrix inversion. For a multiple-input-multiple-output (MIMO) OFDM system, the required complexity of the ZF and MMSE methods becomes more intractable. We then manage to extend the proposed ZF and MMSE methods for SISO-OFDM systems to MIMO-OFDM systems. It turns out that the computational complexity can be reduced even more significantly. Simulation results show that the performance of the proposed methods is almost as good as that of the direct ZF and MMSE methods, while the required computational complexity is reduced dramatically. Finally, we explore the application of the proposed methods in mobility-induced ICI mitigation for OFDM multiple access (OFDMA) systems, and in carrier frequency offset (CFO) induced ICI mitigation for OFDMA uplink systems. As that in OFDM systems, the proposed methods can reduce the required computational complexity, effectively.

Visible light communication (VLC) has been proposed as a promising way for next generation wireless communication networks to mitigate the scarcity of the radio frequency (RF) spectrum, and has consequently attracted much attention. This paper introduces a single-input single-output (SISO) VLC system under the joint effects of statistical random channel and signal-dependent shot noise (SDSN). Moreover, it estimates the channel of the considered system using maximum likelihood (ML), least square (LS), linear minimum mean square error (LMMSE), maximum posteriori probability (MAP) and minimum mean square error (MMSE) estimators. Furthermore, a Bayesian Cramér-Rao lower bound (BCRLB) is derived for the proposed system and it is compared to the mean square error (MSE) of the proposed estimators. The problem of unknown SDSN factor, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\zeta ^{2}$</tex-math></inline-formula> , at the receiver side is discussed and two solutions are investigated. The receiver of a VLC system under SDSN and random channel gain <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$h$</tex-math></inline-formula> is designed and its BER is studied. Finally, Monte Carlo simulation results of the proposed estimators, which show the dramatic effect of the SDSN on the considered system, are provided. In particular, the presence of noise variance, as well as the SDSN factor, causes an increase in the MSE of the system, while increasing the power reinforces the system performance.

Autonomous Underwater Vehicles (AUVs) need accurate navigation and positioning data to carry out their operations in challenging marine environments. However, non-Gaussian environmental noise and sensor outliers in underwater environments degrade the positioning accuracy and stability of traditional algorithms. In such circumstances, the typical Unscented Kalman Filter (UKF) performs worse according to the Minimum Mean Square Error (MMSE) criteria. To reduce non-Gaussian environmental disturbances, a Maximum Correntropy Criterion (MCC) is created. This study extends the MCUKF by utilizing a mixture correntropy framework and an effective iterative optimization technique, and an improved Sage-Husa filtering mechanism is integrated into the iterative algorithm to address time-varying noise dynamics in complex scenarios. These changes increase the algorithm's resilience and flexibility in demanding underwater conditions. The algorithm's capability is confirmed using simulation and sea experiments, which show notable gains in Root Mean Square Error and navigation accuracy. Experimental results show that the proposed method outperforms other existing algorithms in terms of navigation accuracy performance in our simulations and experiments. To guarantee algorithm performance and resilience for the non-Gaussian time-varying measurement noise issue that AUVs encounter in intricate maritime settings, and to additionally enhance the precision and dependability of localization, this study uses the MCC-UKF to combat the environment's non-Gaussian noise, and the Levenberg-Marguardt iterative method to improve reliability of numbers and estimation precision, while the Maximum Mixture Correntropy Criterion (MMCC) serves the purpose of improve the filter flexibility and convergence speed, and finally, the improved Sage-Husa algorithm is combined to improve the robustness to time-varying noise. The following are the study's primary contributions. (1) In our study, we suggest an enhanced UKF algorithm that combines the MCC and iterated cost function, and adopt a Levenberg-Marquardt optimization strategy to accelerate the convergence. The mixture correntropy criterion was added in the construction of the LMIMC-UKF framework in order to achieve the synergistic enhancement of convergence performance and calculate accuracy. (2) Targeting the issue of time-varying environment noise interference, a fusion approach of iterative tuning and improved Sage-Husa filtering is suggested to achieve the systematic improvement of algorithm robustness. (3) By combining the Monte Carlo simulation with maritime measurement data, the algorithm's technical viability is confirmed, and the parameter sensitivity is quantitatively examined by methodical simulation tests. According to the results, the suggested algorithm performs noticeably better than the current maximum correntropy filtering techniques in terms of anti-interference capability, iteration speed, and navigation accuracy.

Multiple-input multiple-output (MIMO) systems can achieve high data-rate and high capacity transmission. In MIMO systems, the scheme that weight substreams based on minimum mean square error (MMSE) criterion at the transmitter using feedback channel state information (CSI) achieves good performance (denoted hereafter as MMSE precoder). However, if the optimum number of substreams is not selected based on CSI and methods for reducing the performance degradation due to feedback delay is not used, MMSE precoder cannot achieve the full performance possible. In this paper, we propose an MMSE precoder with mode selection for MIMO systems, which selects both the number of substreams and the modulation scheme for each substream to minimize the average BER at a fixed rate. We evaluate the BER of the proposed MMSE precoder with two methods for reducing performance degradation due to feedback delay, channel prediction and receive weight robust to feedback delay. We also evaluate the BER of the proposed MMSE precoder with the method that combines channel prediction with receive weight robust to feedback delay. Simulation results show that the BER of the proposed MMSE precoder is improved compared to that of the conventional MMSE precoder using the fixed number of substreams. We also show that the method that combines channel prediction with receive weight robust to feedback delay can achieve good BER even when the large feedback delay exists.

A robust adaptive filtering algorithm based on the convex combination of two adaptive filters under the maximum correntropy criterion (MCC) is proposed. Compared with conventional minimum mean square error (MSE) criterion-based adaptive filtering algorithm, the MCC-based algorithm shows a better robustness against impulsive interference. However, its major drawback is the conflicting requirements between convergence speed and steady-state mean square error. In this letter, we use the convex combination method to overcome the tradeoff problem. Instead of minimizing the squared error to update the mixing parameter in conventional convex combination scheme, the method of maximizing the correntropy is introduced to make the proposed algorithm more robust against impulsive interference. Additionally, we report a novel weight transfer method to further improve the tracking performance. The good performance in terms of convergence rate and steady-state mean square error is demonstrated in plant identification scenarios that include impulsive interference and abrupt changes.

In this paper, the minimum mean square error (MSE) convergence of the RBF equalizer is evaluated and compared with the linear equalizer based on the theoretical minimum MSE. The basic idea of comparing these two equalizers comes from the fact that the relationship between the hidden and output layers in the RBF equalizer is also linear. As extensive studies of this research, various channel models are selected, which include linearly separable channel, slightly distorted channel, and severely distorted channel models. The theoretical minimum MSE for both RBF and linear equalizers were computed, compared and the sensitivity of minimum MSE due to RBF center spreads was analyzed.