Linear Minimum Mean Square Error Estimator Research Articles

A Best Linear Empirical Bayes Method for High-Dimensional Covariance Matrix Estimation

Covariance matrix estimation plays a significant role in both in the theory and practice of portfolio analysis and risk management. This paper deals with the available data prior to developing a factor model to enhance covariance matrix estimation. Our work has two main outcomes. First, for a general linear model with unknown prior parameters, a class of best linear empirical Bayes estimators is established through two kinds of architectures to improve the estimation accuracy by utilizing additional data prior. The theoretical results indicate two key points: the proposed estimators are equivalent to the linear minimum mean-square error estimator when complete or sufficient partial data prior are provided; and the proposed estimators perform better than the optimal weighted least squares method, which ignores the data prior in each situation. Second, the proposed estimators are used for calculating a high-dimensional covariance matrix through factor models. The numerical example and the simulation results verify the effectiveness of our methods.

Bayesian Over-the-Air Computation

As an important piece of the multi-tier computing architecture for future wireless networks, over-the-air computation (OAC) enables efficient function computation in multiple-access edge computing, where a fusion center aims to compute a function of the data distributed at edge devices. Existing OAC relies exclusively on the maximum likelihood (ML) estimation at the fusion center to recover the arithmetic sum of the transmitted signals from different devices. ML estimation, however, is much susceptible to noise. In particular, in the misaligned OAC where there are channel misalignments among received signals, ML estimation suffers from severe error propagation and noise enhancement. To address these challenges, this paper puts forth a Bayesian approach by letting each edge device transmit two pieces of statistical information to the fusion center such that Bayesian estimators can be devised to tackle the misalignments. Numerical and simulation results verify that, 1) For the aligned and synchronous OAC, our linear minimum mean squared error (LMMSE) estimator significantly outperforms the ML estimator. In the low signal-to-noise ratio (SNR) regime, the LMMSE estimator reduces the mean squared error (MSE) by at least 6 dB; in the high SNR regime, the LMMSE estimator lowers the error floor of MSE by 86.4%; 2) For the asynchronous OAC, our LMMSE and sum-product maximum a posteriori (SP-MAP) estimators are on an equal footing in terms of the MSE performance, and are significantly better than the ML estimator. Moreover, the SP-MAP estimator is computationally efficient, the complexity of which grows linearly with the packet length.

Surface EMG decomposition based on innervation zone mapping and an LMMSE framework

Muscle motor units (MUs) can provide valuable information on neuromuscular control and muscle diseases, and electromyogram (EMG) decomposition is an effective method for reconstructing MUs. In this paper, a novel decomposition method based on innervation zone mapping (IZM) and a linear minimum mean square error estimation (LMMSE) framework is proposed for high-density surface EMG (HD-sEMG) decomposition. First, initial discharges were selected according to the IZM at each discharge. Then, the LMMSE framework and multistep iterative process were employed to update and estimate the innervation pulse train (IPT). Each discharge in the IPT was then classified into an individual MU action potential train (MUAPT) according to the IZM at each discharge. Finally, the MUAPTs with the same IZMs at the initial discharge were merged. The method based on the IZM and the LMMSE framework (IZM-LMMSE) was validated on both simulated and experimental data. By comparing the decomposition results of IZM-LMMSE and K-means clustering-modified CKC (KmCKC), the IZM-LMMSE algorithm can obtain more MUs and higher accuracy in most cases. Moreover, the experimental results show that the ratio of all common discharges obtained by IZM-LMMSE and KmCKC is 0.94 ± 0.01 % (mean ± std), which proves the effectiveness of the IZM-LMMSE method. The decomposition method has wide application prospects in rehabilitation engineering and clinical diagnosis.

Detection-estimation strategies for delayed systems with unavailable packet sequence

In this paper, we study the problem of remote state estimation on networks with random delays and unavailable packet sequence due to malicious attacks. Two maximum a posteriori (MAP) schemes are proposed to detect the unavailable packet sequence. The first MAP strategy detects the packet sequence using data within a finite time horizon; the second MAP strategy detects the packet sequence by a recursive structure, which effectively reduces the computation time. With the detected packet sequence, we further design a linear minimum mean-squared error (LMMSE) estimation algorithm based on smoothing techniques, rather than using the classic prediction and update structure. A wealth of information contained in the combined measurements is utilized to improve the estimation performance. Finally, the effectiveness of the proposed algorithms is demonstrated by simulation experiments.

Regularized Linear Regression via Covariance Fitting

The linear minimum mean-square error estimator (LMMSE) can be viewed as a solution to a certain regularized least-squares problem formulated using model covariance matrices. However, the appropriate parameters of the model covariance matrices are unknown in many applications. This raises the question: how should we choose them using only the data? Using data-adaptive matrices obtained via the covariance fitting SPICE-methodology, we show that the empirical LMMSE is equivalent to tuned versions of various known regularized estimators – such as ridge regression, LASSO, and regularized least absolute deviation – depending on the chosen covariance structures. These theoretical results unify several important estimators under a common umbrella. Furthermore, through a number of numerical examples we show that the regularization parameters obtained via covariance fitting are close to optimal for a range of different signal conditions.

Performance Analysis and Optimization of Multicell Massive MIMO With Variable-Resolution ADCs Over Correlated Rayleigh Fading Channels

This article makes performance analysis and optimization for multicell massive multiple-input and multiple-output (MIMO) systems with variable-resolution analog-to-digital converters (ADCs). In such an architecture, every ADC uses arbitrary quantization resolution to save power and hardware cost. Along this direction, we first introduce a quantization-aware channel estimator based on linear minimum mean-squared error (LMMSE) estimate theory over spatially correlated Rayleigh fading channels. By leveraging on the estimated channel state information (CSI), we derive the theoretical expressions of the achievable uplink spectral efficiency (SE) for maximal ratio combining (MRC), quantization-aware multicell minimum mean-squared error (QA-M-MMSE) combining, and quantization-aware single-cell MMSE (QA-S-MMSE) combining, respectively. We consider the impacts of quantization errors and resort to random matrix theory to derive theoretical results. Afterwards, power and bit allocation problems are investigated under the variable-resolution architecture. Finally, simulation results demonstrate that our theoretical analyses are correct and that the proposed quantization-aware estimator and combiners are more beneficial than their quantization-unaware counterparts. Moreover, simulation results also verify the conclusions of the proposed power and bit allocation schemes.

Efficient Low-Complexity Optimized Channel Estimating Methods for OFDM-Based Low-Voltage Broadband Power Line Communication Systems

Broadband Power Line Communication (BB-PLC) technology enables data transmission for smart grid applications. Nevertheless, channel equalizers are required in the receiver to estimate and compensate for the nonlinear time-variant impulse response and noise interference effects introduced by the BB-PLC channel. In this paper, we append the Particle Swarm Optimization (PSO) algorithm and its proposed improved version to the commonly used Least-Square (LS) and Linear Minimum Mean Square Error (LMMSE) algorithms to advance four new blocktype pilot-aided hybrid channel estimation algorithms for low-voltage Orthogonal Frequency Division Multiplexing (OFDM)-based BB-PLC systems. Extensive numerical simulation results for four different M-QAM formats (M = 8, 16, 32, 64) show that the proposed algorithms significantly improve the performance of the traditional LS and LMMSE estimators, at least for the parameters of the BB-PLC system studied in this work. In addition, the computational load complexity of the PSO-inspired LMMSE algorithm is lower compared to the conventional LMMSE estimator.

Fast hyperparameter-free spectral approach for 2D seismic data reconstruction

Reconstruction of missing seismic data is a critical procedure for subsequent applications like multiple wave suppression, wave-equation migration imaging and so on. In this paper, a fast, hyperparameter-free and sparse iterative spectral estimation approach is proposed for the reconstruction of two-dimensional seismic data of randomly missing traces. The proposed approach is based on the harmonic structure of the frequency slice of seismic data and the weighted covariance fitting criterion. Specifically, the method first iteratively estimates the spectrum of the frequency slice by solving a weighted covariance fitting problem. Then, the missing data is reconstructed by using the estimated spectrum and a linear minimum mean-squared error estimator. However, the spectral estimation depends on matrix-vector multiplications for each iteration, which has a high computational cost when the data increase to a large size. To solve this problem, a fast iterative technology is proposed by using an inverse fast Fourier transform, which fully exploits the Hermitian–Toeplitz structure of the covariance matrix and the exponential form of the steering vector and it significantly reduces the computational complexity. The proposed algorithm is hyperparameter-free, can provide super spectral resolution, and thus obtain better reconstruction performance. The experimental results of synthetic and real seismic data show that the proposed algorithm has higher reconstruction accuracy and lower computational complexity compared to other commonly used reconstruction algorithms.

Distributed LMMSE Estimation for Large-Scale Systems Based on Local Information.

This article studies the distributed linear minimum mean square error (LMMSE) estimation problem for large-scale systems with local information (LSLI). Large-scale systems are composed of numerous subsystems. Each subsystem only transmits information to its neighbors. Thus, only the local information is available to each subsystem. This implies that the information available to different subsystems is different. Using local information to design an LMMSE estimator, the gains of the estimator must satisfy the sparse structure constraint, which makes the estimator design challenging and complicates the boundedness analysis of the estimation error covariance (EEC). In this article, a framework of the distributed LMMSE estimation for LSLI is established. The gains of the LMMSE estimator are effectively constructed by solving linear matrix equations. A gradient descent algorithm is exploited to design the gains of the LMMSE estimator numerically. Sufficient conditions are derived to ensure the boundedness of the EEC. Also, a gradient-based search algorithm is developed to verify whether the sufficient conditions hold or not. Finally, an example is used to illustrate the effectiveness of the proposed results.

Channel Estimation and Data Detection Analysis of Massive MIMO With 1-Bit ADCs

We present an analytical framework for the channel estimation and the data detection in massive multiple-input multiple-output uplink systems with 1-bit analog-to-digital converters (ADCs) and i.i.d. Rayleigh fading. First, we provide closed-form expressions of the mean squared error (MSE) of the channel estimation considering the state-of-the-art linear minimum MSE estimator and the class of scaled least-squares estimators. For the data detection, we provide closed-form expressions of the expected value and the variance of the estimated symbols when maximum ratio combining is adopted, which can be exploited to efficiently implement minimum distance detection and, potentially, to design the set of transmit symbols. Our analytical findings explicitly depend on key system parameters such as the signal-to-noise ratio (SNR), the number of user equipments, and the pilot length, thus enabling a precise characterization of the performance of the channel estimation and the data detection with 1-bit ADCs. The proposed analysis highlights a fundamental SNR trade-off, according to which operating at the right noise level significantly enhances the system performance.

Phase-Noise Compensation for OFDM Systems Exploiting Coherence Bandwidth: Modeling, Algorithms, and Analysis

Phase-noise (PN) estimation and compensation are crucial in millimeter-wave (mmWave) communication systems to achieve high reliability. The PN estimation, however, suffers from high computational complexity due to its fundamental characteristics, such as spectral spreading and fast-varying fluctuations. In this paper, we propose a new framework for low-complexity PN compensation in orthogonal frequency-division multiplexing systems. The proposed framework also includes a pilot allocation strategy to minimize its overhead. The key ideas are to exploit the coherence bandwidth of mmWave systems and to approximate the actual PN spectrum with its dominant components, resulting in a non-iterative solution by using linear minimum mean squared-error estimation. The proposed method obtains a reduction of more than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2.5 \times $ </tex-math></inline-formula> in total complexity, as compared to the existing methods. Furthermore, we derive closed-form expressions for normalized mean squared-errors (NMSEs) as a function of critical system parameters, which help in understanding the NMSE behavior in low and high signal-to-noise ratio regimes. Lastly, we study a trade-off between performance and pilot-overhead to provide insight into an appropriate approximation of the PN spectrum.

High mobility enabled spatial and media‐based modulated orthogonal frequency division multiplexing systems for beyond 5G wireless communications

SummaryThis paper introduces the concept of spatial and media‐based modulated (SMBM) orthogonal frequency division multiplexing (OFDM) as a potential candidate for highly mobile next generation beyond 5G (B5G) wireless communications. The proposed SMBM‐OFDM technique utilizes not only the transmit antenna and channel state indices but also OFDM subcarriers to improve the system performance under high mobility. In addition, this study sheds light on challenging fast time‐varying channel estimation problem of MBM‐based systems by using the linear minimum mean square error (LMMSE) approach due to its optimality to investigate the achievable system performance with the aid of basis expansion modeling. The minimum lower bound on the channel estimation error (Bayesian Cramer–Rao bound) is derived theoretically and shown to be attainable by the considered LMMSE estimator. Moreover, symbol detection performance is provided for different modulation types and higher mobile velocities. Simulation results demonstrate that SMBM‐OFDM system under high mobility is able to provide around 12‐dB performance gains in terms of both channel estimation and symbol detection error compared to conventional spatial modulation (SM)‐OFDM systems without MBM. The presented framework is important due to addressing the high mobility support of SMBM‐OFDM systems for B5G wireless communications in terms of achievable channel estimation and data detection performance.

Well-Conditioned Linear Minimum Mean Square Error Estimation

Linear minimum mean square error (LMMSE) estimation is often ill-conditioned, suggesting that unconstrained minimization of the mean square error is an inadequate approach to filter design. To address this, we first develop a unifying framework for studying constrained LMMSE estimation problems. Using this framework, we explore an important structural property of constrained LMMSE filters involving a certain prefilter. Optimality is invariant under invertible linear transformations of the prefilter. This parameterizes all optimal filters by equivalence classes of prefilters. We then clarify that merely constraining the rank of the filter does not suitably address the problem of ill-conditioning. Instead, we adopt a constraint that explicitly requires solutions to be well-conditioned in a certain specific sense. We introduce two well-conditioned filters and show that they converge to the unconstrained LMMSE filter as their truncation-power loss goes to zero, at the same rate as the low-rank Wiener filter. We also show extensions to the case of weighted trace and determinant of the error covariance as objective functions. Finally, our quantitative results with historical VIX data demonstrate that our two well-conditioned filters have stable performance while the standard LMMSE filter deteriorates with increasing condition number.

Optimum DCT type-I based transceiver model and effective channel estimation for uplink NB-IoT system

Single-tone and multi-tone transmissions in the narrowband Internet of Things (NB-IoT) uplink system can be accomplished by employing different orthogonal basis. In this paper, we develop an uplink NB-IoT baseband system model and derive the corresponding analytical signal model based on the discrete cosine transform type-I (DCT-I) domain. Basically, the circular convolution and channel matrix diagonalization issues for designing DCT-based systems are addressed without exploiting any redundancy (a prefix and a suffix sequences, and zero-padding (ZP)) into each transmitted data symbol blocks. In our proposed DCT-I based transceiver, we employ a standard cyclic prefix (CP) and apply discrete Fourier transform (DFT) and inverse DFT (IDFT) operations directly to make the channel matrix circulant and diagonal, resulting in the use of one-tap frequency-domain channel equalizer without incorporating extra front-end prefilter at the receiver. Furthermore, we propose novel least squares (LS) and linear minimum mean square error (LMMSE) channel estimators by considering the energy concentration and spectral compaction properties of DCT-I for the uplink NB-IoT system. The expressions for the DCT-I domain LS and LMMSE estimators are derived for channel frequency response estimation and mean square error computation. Finally, the viability of the proposed CP-DCT-I-NB-IoT uplink system, as well as the channel estimators, are tested compared with the standardized CP-DFT-NB-IoT and ZP-DCT type-2 even based single-carrier frequency-division multiple access (SC-FDMA) systems through rigorous computer simulations.

LMMSE channel estimation for OFDM systems with channel correlation function selection

In the linear minimum mean square error (LMMSE) estimation for orthogonal frequency division multiplexing (OFDM) systems, the problem about the determination of the algorithm's parameters, especially those related with channel frequency response (CFR) correlation, has not been readily solved yet. Although many approaches have been proposed to determine the statistic parameters, it is hard to choose the best one within those approaches in the design phase, since every approach has its own most suitable application conditions and the real channel condition is unpredictable. In this paper, we propose an enhance LMMSE estimation capable of selecting parameters by itself. To this end, sampled noise MSE is first proposed to evaluate the practical performance of interpolation. Based on this evaluation index, a novel parameter comparison scheme is proposed to determine the parameters which can endow LMMSE estimation best performance within a parameter set. After that, the structure of the enhanced LMMSE is illustrated, and it is applied in OFDM systems. Besides, the issues about theoretical analysis on accuracy of the parameter comparison scheme, the parameter set design and algorithm complexity are explained in detail. At last, our analyses and performance of the proposed estimation method are demonstrated by simulation experiments.

BL-ALM: A Blind Scalable Edge-Guided Reconstruction Filter for Smart Environmental Monitoring Through Green IoMT-UAV Networks

Today, low-cost radar/optical visual sensing for monitoring of environment and remote sensing through a network of unmanned aerial vehicles (UAVs) has received more attention such that many different intelligent and pervasive computing techniques are being used to make vision-assisted UAV Networks more reliable and dependable in order to distributed surveillance. On the other hand, the issue of visual information reconstruction has been an interesting topic of machine vision and UAV-borne remote sensing in recent years. To do this, a non-linear blind edge-guided spatial filter based on linear minimum mean square error-estimation (LMMSE) theory has recently been proposed for video sequence reconstruction problems in green monitoring with radars. Although performance of this technique is acceptable compared to the conventional linear techniques in the area, the strategy behind this visual estimator is not generally flexible on which it is not applicable in scalable zooming and its corresponding reconstruction templates. The main objective of this research is to introduce a new version of this filter towards scalable zooming and flexibility in different templates with arbitrary scales. Thus, the proposed algorithm can simultaneously benefit from both edge-directed and distance-weighted adaptation. The proposed approach is theoretically proven, and with using experiments through real UAV visual data, its suitability for low-cost sensing in green Internet of multimedia things (G-IoMT) is shown.

Rank-1 Matrix Approximation-Based Channel Estimation for Intelligent Reflecting Surface-Aided Multi-User MISO Communications

The acquisition of channel state information is an essential step in enabling intelligent reflecting surfaces (IRSs) for wireless communications. In this letter, we introduce a novel procedure to estimate the cascaded channel between the base station (BS), IRS, and users in a multi-user multiple-input single-output system. The common BS-IRS channel, over which all users transmit, is leveraged to decompose the cascaded channel into a series of rank-1 matrices. Low-rank matrix recovery methods are utilized to improve upon the linear minimum mean-squared error estimate of the cascaded channel. A theoretical upper bound for the mean-square error (MSE) of the proposed estimator is derived. Numerical results reveal that the proposed techniques outperform the existing counterparts in terms of the MSE and scale with the number of BS antennas.

Inference of the Selective Auditory Attention Using Sequential LMMSE Estimation.

Attentive listening in a multispeaker environment such as a cocktail party requires suppression of the interfering speakers and the noise around. People with normal hearing perform remarkably well in such situations. Analysis of the cortical signals using electroencephalography (EEG) has revealed that the EEG signals track the envelope of the attended speech stronger than that of the interfering speech. This has enabled the development of algorithms that can decode the selective attention of a listener in controlled experimental settings. However, often these algorithms require longer trial duration and computationally expensive calibration to obtain a reliable inference of attention. In this paper, we present a novel framework to decode the attention of a listener within trial durations of the order of two seconds. It comprises of three modules: 1) Dynamic estimation of the temporal response functions (TRF) in every trial using a sequential linear minimum mean squared error (LMMSE) estimator, 2) Extract the N1 -P2 peak of the estimated TRF that serves as a marker related to the attentional state, and 3) Obtain a probabilistic measure of the attentional state using a support vector machine followed by a logistic regression. The efficacy of the proposed decoding framework was evaluated using EEG data collected from 27 subjects. The total number of electrodes required to infer the attention was four: One for the signal estimation, one for the noise estimation and the other two being the reference and the ground electrodes. Our results make further progress towards the realization of neuro-steered hearing aids.

Deep Learning at the Edge for Channel Estimation in Beyond-5G Massive MIMO

Massive multiple-input multiple-output (mMIMO) is a critical component in upcoming 5G wireless deployment as an enabler for high data rate communications. mMIMO is effective when each corresponding antenna pair of the respective transmitter-receiver arrays experiences an independent channel. While increasing the number of antenna elements increases the achievable data rate, at the same time computing the channel state information (CSI) becomes prohibitively expensive. In this article, we propose to use deep learning via a multi-layer perceptron architecture that exceeds the performance of traditional CSI processing methods like least square (LS) and linear minimum mean square error (LMMSE) estimation, thus leading to a beyond fifth generation (B5G) networking paradigm wherein machine learning fully drives networking optimization. By computing the CSI of all pairwise channels simultaneously via our deep learning approach, our method scales with large antenna arrays as opposed to traditional estimation methods. The key insight here is to design the learning architecture such that it is implementable on massively parallel architectures, such as GPU or FPGA. We validate our approach by simulating a 32-element array base station and a user equipment with a 4-element array operating on millimeter-wave frequency band. Results reveal an improvement up to five and two orders of magnitude in BER with respect to fastest LS estimation and optimal LMMSE, respectively, substantially improving the end-to-end system performance and providing higher spatial diversity for lower SNR regions, achieving up to 4 dB gain in received power signal compared to performance obtained through LMMSE estimation.

Fast iterative adaptive approach for 3D seismic data reconstruction

ABSTRACT In this paper, we introduce a fast iterative adaptive approach for the reconstruction of 3D seismic data with randomly missing traces. Our method starts by transforming 3D seismic volume to 2D harmonic signal for each frequency slice, and then the power spectrum of the frequency slice is iteratively estimated by a weighted least square fitting criterion. The missing data can be recovered with the obtained spectral estimate using a linear minimum mean-squared error estimator. However, estimation of the power spectrum depends on matrix-vector multiplications for each iteration that leads to high computation complexity when the data are large scale and high dimension. To solve the associated spectrum estimation problem above, a fast iterative adaptive scheme is adopted by utilising 2D fast discrete Fourier transform, which makes use of the block-Vandermonde structure and the 2D Fourier property of the steering matrix. Simulation experiments on synthetic and field data verify the effectiveness of the proposed algorithm. Compared with other reconstruction methods based on frequency slice, such as minimum-weighted norm interpolation approach, multi-channel singular spectrum analysis, and Curvelet transform method, the proposed method achieves better reconstruction performance and low computational complexity.