Linear Minimum Mean Square Error Estimator Research Articles

Nonlocal linear minimum mean square error methods for denoising MRI

Abstract The presence of noise results in quality deterioration of magnetic resonance (MR) images and thus limits the visual inspection and influence the quantitative measurements from the data. In this work, an efficient two stage linear minimum mean square error (LMMSE) method is proposed for the enhancement of magnitude MR images in which data in the presence of noise follows a Rician distribution. The conventional Rician LMMSE estimator determines a closed-form analytical solution to the aforementioned inverse problem. Even-though computationally efficient, this approach fails to take advantage of data redundancy in the 3D MR data and hence leads to a suboptimal filtering performance. Motivated by this observation, we put forward the concept of nonlocal implementation with LMMSE estimation method. To select appropriate samples for the nonlocal version of the LMMSE estimation, the similarity weights are computed using Euclidean distance between either the gray level values in the spatial domain or the coefficients in the transformed domain. Assuming that the signal dependent component of the noise is optimally suppressed by this filtering and the rest is a white and uncorrelated noise with the image, we adopt a second stage LMMSE filtering in the principal component analysis (PCA) domain to further enhance the image and the noise variance is adaptively adjusted. Experiments on both simulated and real data show that the proposed filters have excellent filtering performance over other state-of-the-art methods.

Optimal Power Allocation for Channel Estimation in MIMO-OFDM System with Per-Subcarrier Transmit Antenna Selection

A novel hybrid channel estimator is proposed for multiple-input multiple-output orthogonal frequency- divi- sion multiplexing (MIMO-OFDM) system with per-subcar- rier transmit antenna selection having optimal power allo- cation among subcarriers. In practice, antenna selection information is transmitted through a binary symmetric control channel with a crossover probability. Linear mini- mum mean-square error (LMMSE) technique is optimal technique for channel estimation in MIMO-OFDM system. Though LMMSE estimator performs well at low signal to noise ratio (SNR), in the presence of antenna-to-subcar- rier-assignment error (ATSA), it introduces irreducible error at high SNR. We have proved that relaxed MMSE (RMMSE) estimator overcomes the performance degrada- tion at high SNR. The proposed hybrid estimator combines the benefits of LMMSE at low SNR and RMMSE estimator at high SNR. The vector mean square error (MSE) expres- sion is modified as scalar expression so that an optimal power allocation can be performed. The convex optimiza- tion problem is formulated and solved to allocate optimal power to subcarriers minimizing the MSE, subject to transmit sum power constraint. Further, an analytical expression for SNR threshold at which the hybrid estimator is to be switched from LMMSE to RMMSE is derived. The simulation results show that the proposed hybrid estimator gives robust performance, irrespective of ATSA error.

On the Capacity of Iteratively Estimated Channels Using LMMSE Estimators

This paper investigates the capacity of radio channels when iterative channel estimation, data detection, and decoding are employed. Knowing the capacity gain from iterative detection versus purely pilot-based channel estimation helps a designer compare the performance of an iterative receiver against a noniterative receiver and select the best balance between performance and cost. A bound is put on the linear minimum mean square error (LMMSE) channel estimation error, based on which a bound on the capacity is obtained. The attainable capacity is related to the channel estimation error of the receiver. The bounds take into account the uncertainty in symbol detection on channel estimation and incorporate the effect of channel estimation error on channel capacity. The interaction between the symbol detector and the decoder is analytically characterized and depicted in an extrinsic information transfer (EXIT) chart, where a bound on the detector curve is found. With optimal LMMSE pilot-based channel estimation, the results of this paper demonstrate that iterative channel estimation provides insignificant capacity advantage at fading rates below 1% of the symbol rate, although a computational-cost gain is still available. Iterative channel estimation provides a capacity benefit if suboptimal pilot signaling is used to provide initial channel estimates.

A NOVEL LMMSE BASED OPTIMIZED PEREZ-VEGA ZAMANILLO PROPAGATION PATH LOSS MODEL IN UHF/VHF BANDS FOR INDIA

Cognitive radio is the enabling technology for license-exempt access to the TV White Spaces (TVWS). There is ever increasing demand of users in the broadcasting and communication services. Large portions of unused spectrum in the UHF/VHF bands exist in India which can be used on geographical basis. This paper describes a study on path loss variation in UHF/VHF bands in India. The aim of this study is to develop and optimize a path loss model based on Linear minimum mean square error estimation (LMMSE) for India. We propose the LMMSE based Optimized Perez-Vega Zamanillo propagation path loss model. The measured path loss values, collected across India, are compared with proposed Optimized Perez-Vega Zamanillo path loss model and other existing path loss models. It is found that Optimized Perez-Vega Zamanillo propagation path loss model has the least root mean square Error (RMSE) of 13.98 dB. Other existing path loss models have root mean square Error(RMSE) value greater than 24 dB. Therefore, Optimized Perez-Vega Zamanillo propagation path loss model is best suited for predicting coverage area, interference analysis in India for TVWS

Exploiting a priori information for iterative channel estimation in block-fading amplify-and-forward cooperative networks

In an amplify-and-forward cooperative network, a closed-form expression of the a priori distribution of the complex-valued gain of the global relay channel is intractable, so that a priori information is often not exploited for estimating this gain. Here, we present two iterative channel gain and noise variance estimation algorithms that make use of a priori channel information and exploit the presence of not only pilot symbols but also unknown data symbols. These algorithms are approximations of maximum a posteriori estimation and linear minimum mean-square error estimation, respectively. A substantially reduced frame error rate is achieved as compared to the case where only pilot symbols are used in the estimation.

LMMSE Filtering in Feedback Systems With White Random Modes: Application to Tracking in Clutter

A generalized state space representation of dynamical systems with random modes switching according to a white random process is presented. The new formulation includes a term, in the dynamics equation, that depends on the most recent linear minimum mean squared error (LMMSE) estimate of the state. This can model the behavior of a feedback control system featuring a state estimator. The measurement equation is allowed to depend on the previous LMMSE estimate of the state, which can represent the fact that measurements are obtained from a validation window centered about the predicted measurement and not from the entire surveillance region. The LMMSE filter is derived for the considered problem. The approach is demonstrated in the context of target tracking in clutter and is shown to be competitive with several popular nonlinear methods.

Low-Sampling-Rate Ultra-Wideband Channel Estimation Using Equivalent-Time Sampling

In this paper, a low-sampling-rate scheme for ultra-wideband channel estimation is proposed. The scheme exploits multiple observations generated by transmitting multiple pulses. In the proposed scheme, $P$ pulses are transmitted to produce channel impulse response estimates at a desired sampling rate, while the ADC samples at a rate that is $P$ times slower. To avoid loss of fidelity, the number of sampling periods (based on the desired rate) in the inter-pulse interval is restricted to be co-prime with $P$ . This condition is affected when clock drift is present and the transmitted pulse locations change. To handle this case, and to achieve an overall good channel estimation performance, without using prior information, we derive an improved estimator based on the bounded data uncertainty (BDU) model. It is shown that this estimator is related to the Bayesian linear minimum mean squared error (LMMSE) estimator. Channel estimation performance of the proposed sub-sampling scheme combined with the new estimator is assessed in simulation. The results show that high reduction in sampling rate can be achieved. The proposed estimator outperforms the least squares estimator in almost all cases, while in the high SNR regime it also outperforms the LMMSE estimator. In addition to channel estimation, a synchronization method is also proposed that utilizes the same pulse sequence used for channel estimation.

Low-Complexity MIMO Detection Based on Belief Propagation Over Pairwise Graphs

This paper considers a belief propagation algorithm over pairwise graphical models to develop low-complexity iterative multiple-input multiple-output (MIMO) detectors. The pairwise graphical model is a bipartite graph where a pair of variable nodes are related by an observation node represented by the bivariate Gaussian function obtained by marginalizing the posterior joint probability density under the Gaussian input assumption. Specifically, we consider two types of pairwise models: the fully connected and ring-type. The pairwise graphs are sparse, compared with the conventional graphical model introduced by Bickson et al., insofar as the number of edges connected to an observation node (edge degree) is only two. Consequently, the computations are much easier than those of maximum likelihood (ML) detection, which are similar to the belief propagation (BP) that is run over the fully connected bipartite graph. The link level performance for non-Gaussian input is evaluated via simulations, and the results show the validity of the proposed algorithms. We also customize the algorithm with Gaussian input assumption to obtain the Gaussian BP run over the two pairwise graphical models, and for the ring-type, we prove its convergence to the linear minimum mean square error (MMSE) estimates. Since the maximum a posterior (MAP) estimator for Gaussian input is equivalent to the linear MMSE estimator, it shows the optimality of the scheme for Gaussian input.

Security-reliability performance of cognitive AF relay-based wireless communication system with channel estimation error

Abstract In this paper, both the security and the reliability performance of the cognitive amplify-and-forward (AF) relay system are analyzed in the presence of the channel estimation error. The security and the reliability performance are represented by the outage probability and the intercept probability, respectively. Instead of perfect channel state information (CSI) predominantly assumed in the literature, a certain channel estimation algorithm and the influence of the corresponding channel estimation error are considered in this study. Specifically, linear minimum mean square error estimation (LMMSE) is utilized by the destination node and the eavesdropper node to obtain the CSI, and the closed form for the outage probability and that for the intercept probability are derived with the channel estimation error. It is shown that the transmission security (reliability) can be improved by loosening the reliability (security) requirement. Moreover, we compare the security and reliability performance of this relay-based cognitive radio system with those of the direct communication system without relay. Interestingly, it is found that the AF relay-based system has less reliability performance than the direct cognitive radio system; however, it can lower the sum of the outage probability and the intercept probability than the direct communication system. It is also found that there exists an optimal training number to minimize the sum of the outage probability and the intercept probability.

Optimal State Estimation for Discrete-Time Markov Jump Systems with Missing Observations

This paper is concerned with the optimal linear estimation for a class of direct-time Markov jump systems with missing observations. An observer-based approach of fault detection and isolation (FDI) is investigated as a detection mechanic of fault case. For systems with known information, a conditional prediction of observations is applied and fault observations are replaced and isolated; then, an FDI linear minimum mean square error estimation (LMMSE) can be developed by comprehensive utilizing of the correct information offered by systems. A recursive equation of filtering based on the geometric arguments can be obtained. Meanwhile, a stability of the state estimator will be guaranteed under appropriate assumption.

Performance Analysis for Distributed Fusion with Different Dimensional Data

Different sensors or estimators may have different capability to provide data. Some sensors can provide a relatively higher dimensional data, while other sensors can only provide part of them. Some estimators can estimate full dimensional quantity of interest, while others may only estimate part of it due to some constraints. How is such kind of data with different dimensions fused? How do the common part and the uncommon part affect each other during fusion? To answer these questions, a fusion algorithm based on linear minimum mean-square error (LMMSE) estimation is provided in this paper. Then the fusion performance is analyzed, which is the main contribution of this work. The conclusions are as follows. First, the fused common part is not affected by the uncommon part. Second, the fused uncommon part will benefit from the common part through the cross-correlation. Finally, under certain conditions, both the more accurate common part and the stronger correlation can result in more accurate fused uncommon part. The conclusions are all supported by some tracking application examples.

Security and Reliability Performance Analysis for Cloud Radio Access Networks With Channel Estimation Errors

Physical layer (PHY) security is recently regarded as a promising technique to improve the security performance of wireless communication networks. Current developments in PHY security are often based on the assumption of perfect channel state information (CSI). In this paper, both security and reliability performance for the downlink cloud radio access network with optimal remote radio heads (RRHs) node selection are investigated in a practical scenario by considering channel estimation (CE) errors. In particular, a three-phase transmission scheme is proposed and the linear minimum mean-square error (MMSE) estimation method is utilized to obtain the CSI. Based on the CSI estimates and the statistics of CE errors, the outage probability and intercept probability are derived in closed-form expression to evaluate the security and reliability performance, respectively. In addition, two possible cases (with or without intercepting signals from baseband unit) are considered for the eavesdropper. It is found that the suggested optimal RRHs selection scheme outperforms the nonselection scheme, and that the increasing number of RRHs can lower the outage probability as well as the intercept probability. It is also shown that there exists an optimal training number to minimize the sum of the outage probability and intercept probability. Finally, simulation results are provided to corroborate our proposed studies.

A Shrinkage Linear Minimum Mean Square Error Estimator

The conventional linear minimum mean square error (LMMSE) estimator is commonly implemented through the sample covariance matrix. This estimator can only be implemented if the sample size N is higher than the observation dimension M. Moreover, this estimator performs poorly when the sample size is not sufficiently large. To address this problem, we propose a new shrinkage LMMSE estimator. The proposed estimator performs efficiently over a wide range of observation dimensions and sample sizes. In contrast to existing methods, the proposed estimator can be applied if M ≥ N. Even if M <; N, the proposed estimator performs more efficiently than existing estimators.

Modeling and State Estimation for Dynamic Systems With Linear Equality Constraints

The problem of modeling and estimation for linear equality constrained (LEC) systems is considered. The exact constrained dynamic model usually is not readily available or is too complicated, and hence in many studies an auxiliary dynamic model is employed in which the state does not necessarily obey the constraint strictly. Based on the understanding that the constraints, as prior information about the state, should be incorporated into the dynamics modeling, an LEC dynamic model (LECDM) is constructed first. The model optimally fuses the linear equality constraint (LEC) and the auxiliary dynamics. Some of its superior properties are presented. Next, the linear minimum mean squared error (LMMSE) estimate of the LEC state is proved to satisfy the constraint. The LMMSE estimator for linear systems, called the LEC Kalman filter (LECKF), and two approximate LMMSE estimators for nonlinear systems are presented. The LECKF is compared with other constrained estimators, and a sufficient condition is also provided under which the estimate projection method mathematically equals the LECKF. Furthermore, extensions of the LECDM for the LEC systems with uncertain or unknown constraint parameters are discussed. Finally, illustrative examples are provided to show the effectiveness and efficiency of the LECKF and to verify the theoretical results given in the paper.

Channel estimation and optimal training with the LMMSE criterion for OFDM-based two-way relay networks

Abstract In this paper, we consider the linear minimum mean square error (LMMSE) estimation of channel for the two-way relay network employing analog network coding and orthogonal frequency division multiplexing. The channel responses required for self-interference cancelation and coherent detection are estimated in the time domain with the preamble. We derive the optimal training condition for minimizing the mean square error (MSE) of the LMMSE estimator and obtain the corresponding MSE. It is observed that the LMMSE estimator has the optimal training condition equivalent to that of the least square estimator and that the former is less sensitive to the training sequences than the latter. We also propose new training sequences not only satisfying the optimal training condition but also providing the minimum peak-to-average power ratio.

Noise correction for HARDI and HYDI data obtained with multi-channel coils and Sum of Squares reconstruction: An anisotropic extension of the LMMSE

Parallel magnetic resonance imaging (MRI) yields noisy magnitude data, described in most cases as following a noncentral χ distribution when the signals received by the coils are combined as the sum of their squares. One well-known case of this noncentral χ noise model is the Rician model, but it is only valid in the case of single-channel acquisition. Although the use of parallel MRI is increasingly common, most of the correction methods still perform Rician noise removal, yielding an erroneous result due to an incorrect noise model. Moreover, the existence of noise correlations in phased array systems renders noise nonstationary and further modifies the noise description in parallel MRI. However, the noncentral χ model has been demonstrated to work as a good approximation as long as effective voxelwise parameters are used. A good correction step, adapted to the right noise model, is of paramount importance, especially when working with diffusion-weighted MR data, whose signal-to-noise ratio is low. In this paper, we present a noise removal technique designed to be fast enough for integration into a real-time reconstruction system, thus offering the convenience of obtaining corrected data almost instantaneously during the MRI scan. Our method employs the noncentral χ noise model and uses a simplified method to account for noise correlations; this leads to an efficient and rapid correction. The method consists of an anisotropic extension of the Linear Minimum Mean Square Error estimator (LMMSE) that is a far better edge-preserving method than the traditional LMMSE and addresses noncentral χ distributions along with empirically computed global effective parameters. The results on simulated and real data demonstrate that this anisotropic extended LMMSE outperforms the original LMMSE on images corrupted by noncentral χ noise. Moreover, in comparison with the existing LMMSE technique incorporating the estimation of voxelwise effective parameters, our method yields improved results.

Under-determined Training and Estimation for Distributed Transmit Beamforming Systems

Distributed transmit beamforming (DTB) can significantly boost the signal-to-noise ratio (SNR) of a wireless communication system. To realize the benefits of DTB, generating and feeding back beamforming vector are very challenging tasks. Existing schemes have either enormous overhead or weak robustness in noisy channels. In this paper, we investigate the design of training sequences and beamforming vector estimators in DTB systems. We consider an under-determined case, where the length of training sequence N sent from each node is smaller than the number of source nodes M. We derive the optimal estimation of the beamforming vector that maximizes the beamforming gain and show that it can be well approximated as the linear minimum mean square error (LMMSE) estimator. Based on the LMMSE estimator, we investigate the optimal design of training sequences and propose efficient DTB schemes. We analytically show that these schemes can achieve approximately N times increased SNR in uncorrelated channels, and even higher gain in correlated ones. We also propose a concatenated training scheme which optimally combines the training signals over multiple frames to obtain the beamforming vector. Simulation results demonstrate that the proposed DTB schemes can yield significant gains even at very low SNRs, with total feedback bits much less than those required in the existing schemes.

Weighted Linear Least Square Localization Algorithms for Received Signal Strength

Localization based on the received signal strength (RSS) is by far the cheapest and simplest option. By constructing the appropriate path-loss model, the RSS information can be converted to the distance estimates which can determine the position of the target node with Linear Least Square (LLS) algorithm. In this paper, the LLS algorithm can be directly applied by subtracting a reference node equation from the remaining ones rather than defining a new variable, which selects the reference node with the maximal RSS among all the RSS measurements. Furthermore, based on the latest research results about the estimates of the squared distance, we investigate the unbiased and linear minimum mean square error (MMSE) estimates of the squared distance, and the covariance matrixes about the two estimates with the proposed LLS algorithm are derived, respectively. Then the weighted LLS localization algorithms are presented by utilizing the covariance matrixes. Simulation results demonstrate that the localization accuracy of the linear MMSE estimate with the proposed LLS algorithm is superior to the unbiased estimate. Moreover, the proposed weighted LLS algorithms outperform the traditional LLS localization algorithms in terms of localization accuracy.

A modified Rician LMMSE estimator for the restoration of magnitude MR images

In this article two modified versions of a simple and recently proposed method which is known as linear minimum mean square error (LMMSE) estimator for denoising of magnetic resonance (MR) images are presented. In the introduced approaches the self-similarity and natural redundancy of the acquired MR image are considered to achieve the higher performance of unknown signal estimation. Since, the MR data are in a large majority 3D, the proposed methods are developed to deal with 3D volumes. The introduced methods are compared with related state-of-the-art schemes over both clinical and simulated MR data. The quantitative and qualitative results show their superior denoising ability.

Dual-Diagonal LMMSE Channel Estimation for OFDM Systems

We propose a low-complexity dual-diagonal (DD-) linear-minimum-mean-square-error (LMMSE) channel estimation algorithm for orthogonal frequency-division multiplexing (OFDM) systems involving iterative channel estimation and signal detection. Computational complexity and mean-square-error (MSE) analysis are presented to evaluate the efficiency of the proposed algorithm. A closed-form expression is derived for the asymptotic MSE of the DD-LMMSE channel estimator. Both analysis and numerical results show that DD-LMMSE performs close to the well-known LMMSE estimator with much lower complexity.