Linear Minimum Mean Square Error Research Articles

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.

Probability-Constrained robust estimation for a random parameter with stochastic model uncertainties

This paper investigates the problem of estimating a random parameter in a linear observation model when the transformation matrix suffers stochastic uncertainties. In practice, the uncertainties can be either the bounded uncertainties or the Gaussian uncertainties. The existing minimax robust estimation method is only applicable to the bounded uncertainties, and minimizes the worst-case mean squared error (MSE) over the region of uncertainties, irrespective of the probability of such worst-case scenario, so it tends to be overly conservative. A probability-constrained (PC) robust linear estimation method is proposed in this paper to minimize the MSE with a guaranteed probability, which can handle bounded uncertainties as well as Gaussian uncertainties. However, the formulated problem is not analytically tractable due to the unspecified distribution of the uncertainties, so two approximate PC robust estimators are derived by tackling two upper bounds of the original optimization problem and converting them to semidefinite programming problems with the safe tractable approximation techniques. The comparative analysis reveals that there is a tradeoff between the MSE performance and computation complexity in the proposed two PC estimators. The Monte Carlo simulations are used to corroborate the theoretical results, which demonstrate that the proposed PC estimators are robust to the uncertainties compared to the nominal linear minimum mean squared error (LMMSE) estimator and the nominal best linear unbiased estimator (BLUE), and are less conservative compared to the traditional minimax robust estimator.

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.

ADMM-Based Infinity-Norm Detection for Massive MIMO: Algorithm and VLSI Architecture

In this article, we propose a novel data detection algorithm and a corresponding VLSI design for massive multiuser (MU) multiple-input-multiple-output (MIMO) wireless systems. Our algorithm uses alternating direction method of multipliers (ADMM)-based infinity-norm-constrained equalization and is called ADMIN. ADMIN is an iterative algorithm that outperforms linear detectors by a large margin when the ratio between the numbers of base-station (BS) and user antennas is small. In the first iteration, ADMIN computes the linear minimum mean-square error (MMSE) solution, which is sufficient when the ratio between the numbers of BS and user antennas is large. We develop time-shared and iterative VLSI architectures for LDL-decomposition-based soft-output ADMIN supporting 16- and 32-user systems. We present application-specific integrated circuit (ASIC) designs for 16-64 antenna base stations in 28-nm CMOS that supports up to 64 quadrature amplitude modulation (QAM). The 16-user ADMIN ASIC achieves 303 Mb/s while dissipating 85 mW. The 32-user ADMIN ASIC achieves 287 and 241 Mb/s while dissipating 121 and 135 mW for 32 and 64 BS antennas, respectively. ADMIN has also been implemented on a Xilinx Virtex-7 field-programmable gate array (FPGA) and is compared with state-of-the-art massive MIMO data detectors.

Performance improvement of data transmission using a hybrid underwater and terrestrial system

AbstractIn this paper, we investigate and enhance the performance of data transmission using a hybrid underwater and terrestrial system with low‐complexity linear equalization. Due to the low speed, salinity, water depth, pH degree, and water temperature variations, underwater acoustic communication has become one of the most challenging technologies in the world. The most common linear equalizers are the Linear Zero Forcing (LZF), and Linear Minimum Mean Square Error (LMMSE) equalizers. The LZF equalizer suffers from the noise enhancement problem, while the LMMSE equalizer requires the value of the operating Signal‐to‐Noise Ratio (SNR) to work, correctly, which increases the computational complexity. In addition, this paper presents a low‐complexity equalization scheme to overcome the drawbacks associated with the LZF, and LMMSE equalizers. The proposed equalizer is called the Joint Low‐Complexity Regularized LZF equalizer, which depends on a fixed regularization parameter to minimize the noise enhancement phenomenon caused by the LZF equalizer. The proposed equalizer does not need the estimation of the SNR, and the computational complexity is reduced due to the utilization of the banded‐matrix approximation. The proposed equalizer is implemented based on the Single‐Input‐Single‐Output configuration for Orthogonal Frequency Division Multiplexing (OFDM) using the Discrete Cosine Transform. The obtained simulation results show that the proposed equalizer outperforms different equalizers for the same channel conditions.

A Low-Complexity Double EP-Based Detector for Iterative Detection and Decoding in MIMO

We propose a new iterative detection and decoding (IDD) algorithm for multiple-input multiple-output (MIMO) based on expectation propagation (EP) with application to massive MIMO scenarios. Two main results are presented. We first introduce EP to iteratively improve the Gaussian approximations of both the estimation of the posterior by the MIMO detector and the soft output of the channel decoder. With this novel approach, denoted by double-EP (DEP), the convergence is very much improved with a computational complexity just two times the one of the linear minimum mean square error (LMMSE) based IDD, as illustrated by the included experiments. Besides, as in the LMMSE MIMO detector, when the number of antennas increases, the computational cost of the matrix inversion operation required by the DEP becomes unaffordable. In this work we also develop approaches of DEP where the mean and the covariance matrix of the posterior are approximated by using the Gauss-Seidel and Neumann series methods, respectively. This low-complexity DEP detector has quadratic complexity in the number of antennas, as the low-complexity LMMSE techniques. Experimental results show that the new low-complexity DEP achieves the performance of the DEP as the ratio between the number of transmitting and receiving antennas decreases.

Damped Jacobi Methods Based on Two Different Matrices for Signal Detection in Massive MIMO Uplink

Abstract For massive multiple-input multiple-output (m-MIMO) uplink, the performances of the linear minimum mean-square error (MMSE) detector are considered near optimal, and they occupy benchmark place for most linear iterative detectors. However, the MMSE algorithm is known by its load computational complexity due to the implication of large-scale matrix inversions, and in other hand, iterative methods are often preferred in signal detection because of its low complexity. In this paper, we propose a New Damped Jacobi (NDJ) detector in order to improve the performance of the classical Jacobi linear algorithm. Starting from the classical Jacobi technique to our new proposal, we go through the development of two variants; one uses a damping factor and the other uses a stair-matrix. However, the NDJ incorporates a damping factor in its construction and basing also on stair matrix instead of diagonal matrix. The performances in terms of convergence and low complexity of each Jacobi variant studied in this paper are analyzed. Finally, some simulation examples are given to illustrate the advantages of the new proposed algorithm.

On the general conditions of existence for linear MMSE filters: Wiener and Kalman

State estimation is a fundamental task in a plethora of applications. Two important classes of linear minimum mean square error (MMSE) estimators are Wiener filters (WFs) and Kalman filters (KFs), the latter being the recursive form of the former for linear discrete state-space (LDSS) dynamic models. The minimum set of uncorrelation conditions regarding states and measurements which allows to formulate a general KF form has been recently introduced in the literature, but for both WFs and KFs a standard key assumption is that the measurement noise covariance is invertible. In this contribution we provide i) an alternative derivation of the WF in the case of a non-invertible measurement covariance matrix, and ii) the proof that under the minimum set of uncorrelation conditions lately released, the KF can always be formulated irrespective of the measurement covariance matrix rank. Notice that these results also apply for Kalman predictors and smoothers, linearly constrained WFs and minimum variance distortionless response (MVDR) filters, therefore being a remarkable result of broad interest.

Performance Analysis of a Linear MMSE Receiver in Time-Variant Rayleigh Fading Channels

The performance of the uplink of single and multiuser multiple input multiple output (MIMO) systems depends crucially on the receiver architecture and the quality of channel state information at the receiver. Therefore, several previous works have developed minimum mean squared error (MMSE) receivers and proposed balancing the resources spent on acquiring channel state information and transmitting the payload of data packets. Somewhat surprisingly, the most popular MIMO linear MMSE receivers do not exploit the correlation structure that is present in autoregressive Rayleigh fading environments. Therefore, in this article we first develop a new linear receiver that not only takes channel state information errors into account in minimizing the MSE of the received data symbols, but it also utilizes that the subsequent noisy channel coefficients are correlated. For this new linear MMSE receiver, we derive the achieved MSE as a function of the number of receive antennas and the pilot-to-data power ratio. Interestingly, we find that the pilot power that minimizes the MSE of the data symbols does not depend on the number of antennas and that the new linear MMSE receiver outperforms previously proposed MIMO receivers when the autocorrelation coefficient of the channel is high.

Rician Denoising Based on Correlated Local Features LMMSE Approach.

In this study we propose a novel correction scheme that filters Magnetic Resonance Images data, by using a modified Linear Minimum Mean Square Error (LMMSE) estimator which takes into account the joint information of the local features. A closed-form analytical solution for our estimator is presented and it proves to make the filtering process far simpler and faster than other estimation techniques that rely on iterative optimization scheme and require multiple data samples. An experimental validation of our correction scheme was carried out through large scale experiments using both clinical and synthetic MR images, artificially corrupted with rician noise of σ varying from 1 to 40. These noisy images were filtered using our proposed method against the classical LMMSE, the Non-Local Means filter and the Nonlocality-Reinforced Convolutional Neural Networks (NRCNN) techniques. The results show an outstanding performance of our proposed method, given the fact that from σ ≈ 12 onwards, the proposed method outperforms all other methods. Another attention-grabbing feature of our method is that its Structural Similarity does not vary sharply [0.87, 0.95] across the σ spectrum as the other three techniques, which implies that this method can work on a wider range of deteriorated images than the rest of the techniques.

LMMSE Channel Estimation in OFDM Systems: A Vector Quantization Approach

The linear minimum mean square error (LMMSE) channel estimation yields the optimal performance in terms of the mean square error (MSE). However, the high computational complexity limits its application in practical systems. To overcome this issue, this letter proposes a vector quantization approach to the LMMSE channel estimation. The LMMSE filtering matrices corresponding to some typical wireless channels are calculated off-line. When performing the LMMSE channel estimation on-line, an appropriate LMMSE filtering matrix is selected according to the MSE criterion. At the expense of a negligible performance degradation, the computational complexity of the LMMSE channel estimation is reduced.

Performance enhancement of audio transmission based on LMMSE method

<p class="Abstract"><span>The research in wireless communication has developed rapidly for the last decades as a result of raising the demand for efficient data transmission with more security and accuracy. This paper proposed a system based on the special multiplexing (SM) technique and linear minimum mean square error (LMMSE) detection method with the assistance of the hamming code as well as the interleaving techniques for a better enhanced performance of an audio transmission. Moreover, the comparison was done between the two systems for different antenna configurations and with the presence of two types of modulation: binary phase shift key and quateradure phase shift key. These systems are employed by Matlab simulation to show significant results in terms of enhancing the Rayleigh fading channel capacity, bit error rate (BER) and security as well as in recovering the transmitting audio signals. Each system has advantages than the others in one performance term respect to the other terms. The Simulation results have provided to prove and discuss our analysis.</span></p>

Hybrid LMMSE Transceiver Optimization for Distributed IoT Sensing Networks With Different Levels of Synchronization

In this article, we investigate the analog–digital hybrid transceiver optimization for distributed Internet-of-Things (IoT) sensing networks consisting of a multiantenna fusion center (FC) and several multiantenna sensor nodes. Analog–digital hybrid transceiver is an economic way to realize tradeoffs between hardware cost and performance for multiantenna communications. Under the nonconvex unit modulus constraints and transmit power constraint at each sensor, two synchronization schemes are considered for the hybrid linear minimum mean-square error (LMMSE) transceiver optimization. First, a centralized algorithm is proposed, in which the hybrid transceivers are computed at the FC. Based on the framework of alternating direction method of multipliers (ADMMs), the unit modulus constraints can be satisfied by projecting the elements of analog transceivers onto the unit modulus circle. However, the centralized algorithm usually suffers from strict synchronous requirements and high communication overhead. In order to accommodate the inevitable computing and communication delays in distributed IoT sensing networks, an asynchronous distributed ADMM (AD-ADMM) algorithm is proposed. By using the aged information, the hybrid transceivers are computed at the sensors without the coordination of the FC. Thus, the AD-ADMM algorithm can greatly reduce the computation overhead of the FC and improve the scalability of IoT sensing networks. Simulation results are presented to show that both the centralized ADMM and AD-ADMM algorithms perform closely to the fully digital counterpart.

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.

A Low Complexity Optimal LMMSE Channel Estimator for OFDM System

Linear minimum mean square error (LMMSE) is the optimal channel estimator in the mean square error (MSE) perspective, however, it requires matrix inversion with cubic complexity. In this paper, by exploiting the circulant property of the channel frequency autocorrelation matrix RHH, an efficient LMMSE channel estimation method has been proposed for orthogonal frequency division multiplexing (OFDM) based on fast Fourier transformation (FFT) and circular convolution theorem. Finally, the computer simulation is carried out to compare the proposed LMMSE method with the classical LS and LMMSE methods in terms of performance measure and computational complexity. The simulation results show that the proposed LMMSE estimator achieves exactly same performance as conventional LMMSE estimator with much lower computational complexity.

Efficient Machine Learning-Enhanced Channel Estimation for OFDM Systems

Recently much research work has focused on employing deep learning (DL) algorithms to perform channel estimation in the upcoming 6G communication systems. However, these DL algorithms are usually computationally demanding and require a large number of training samples. Hence, this work investigates the feasibility of designing efficient machine learning (ML) algorithms that can effectively estimate and track time-varying, frequency-selective channels. The proposed algorithm is integrated with orthogonal frequency-division multiplexing (OFDM) to eliminate intersymbol interference (ISI) induced by the frequency-selective multipath channel and compared with the well-known least square (LS) and linear minimum mean square error (LMMSE) channel estimation algorithms. The obtained results have demonstrated that even when a small number of pilot samples, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{P}$ </tex-math></inline-formula> , is inserted before the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> subcarriers OFDM symbol, the introduced ML-based channel estimation is superior to the LS and LMMSE algorithms. This dominance is reflected in the bit-error-rate (BER) performance of the proposed algorithm, which attains a gain of 2.5 dB and 5.5 dB over the LMMSE and LS algorithms, respectively, when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N_{P}=\frac {N}{8}$ </tex-math></inline-formula> . Furthermore, the BER performance of the proposed algorithm is shown to degrade by only 0.2 dB when the maximum Doppler frequency is randomly varied. Finally, the number of iterations required by the proposed algorithm to converge to the smallest achievable mean-squared error (MSE) are thoroughly examined for various signal-to-noise ratio (SNR) levels.

Partially Linear Bayesian Estimation Using Mixed-Resolution Data

In this letter, we consider Bayesian parameter estimation using mixed-resolution data consisting of both analog and 1-bit quantized measurements. We investigate the use of the partially linear minimum mean-squared-error (PL-MMSE) estimator for this mixed-resolution scheme. The use of the PL-MMSE estimator, proposed for general models with ``straightforward and ``complicated parts, has not been demonstrated for quantized data. We derive closed-form analytic expressions for the linear minimum mean-squared-error (LMMSE) and for the PL-MMSE estimator for the mixed-resolution scheme with linear Gaussian orthonormal measurements. We discuss the properties of the proposed PL-MMSE estimator and show that in this case, the PL-MMSE is the sum of a linear function of the quantized measurements and a general Borel measurable function of the analog measurements. In the simulations, we show that the PL-MMSE estimator outperforms the LMMSE estimator for the problem of channel estimation in multiple-input-multiple-output (MIMO) communication systems with mixed analog-to-digital converters (ADCs).

Robust Symbol Detection in Large-Scale Overloaded NOMA Systems

We a framework for the design of low-complexity and high-performance receivers for multidimensional overloaded non-orthogonal multiple access (NOMA) systems. The framework is built upon a novel compressed sensing (CS) regularized maximum likelihood (ML) formulation of the discrete-input detection problem, in which the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> -norm is introduced to enforce adherence of the solution to the prescribed discrete symbol constellation. Unlike much of preceding literature,.g., (Assaf et al., 2020, Yeom et al., 2019, Nagahara, 2015, Naderpour and Bizaki, 2020, Hayakawa and Hayashi, 2017, Hayakawa and Hayashi, 2018, and Zeng et al., 2020), the method is not relaxed into the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm, but rather approximated with a continuous and asymptotically exact expression without resorting to parallel interference cancellation (PIC). The objective function of the resulting formulation is thus a sum of concave-over-convex ratios, which is then tightly convexified via the quadratic transform (QT), such that its solution can be obtained via the iteration of a simple closed-form expression that closely resembles that of the classic zero-forcing (ZF) receiver, making the method particularly suitable to large-scale set-ups. By further transforming the aforementioned problem into a quadratically constrained quadratic program with one convex constraint (QCQP-1), the optimal regularization parameter to be used at each step of the iterative algorithm is then shown to be the largest generalized eigenvalue of a pair of matrices which are given in closed-form. The method so obtained, referred to as the Iterative Discrete Least Square (IDLS), is then extended to address several factors of practical relevance, such as noisy conditions, imperfect channel state information (CSI), and hardware impairments, thus yielding the Robust IDLS algorithm. Simulation results show that the proposed art significantly outperforms both classic receivers, such as the linear minimum mean square error (LMMSE), and recent CS-based state-of-the-art (SotA) alternatives, such as the sum-of-absolute-values (SOAV) and the sum of complex sparse regularizers (SCSR) detectors. It is also shown via simulations that the technique can be integrated with existing iterative detection-and-decoding (IDD) methods, resulting in accelerated convergence.

Massive MIMO Channel Estimation With Low-Resolution Spatial Sigma-Delta ADCs

We consider channel estimation for an uplink massive multiple-input multiple-output (MIMO) system where the base station (BS) uses an array with low-resolution (1-2 bit) analog-to-digital converters and a spatial Sigma-Delta ( ΣΔ) architecture to shape the quantization noise away from users in some angular sector. We develop a linear minimum mean squared error (LMMSE) channel estimator based on the Bussgang decomposition that reformulates the nonlinear quantizer model using an equivalent linear model plus quantization noise. We also analyze the uplink achievable rate with maximal ratio combining (MRC), zero-forcing (ZF) and LMMSE receivers and provide a lower bound for the achievable rate with the MRC receiver. Numerical results show superior channel estimation and sum spectral efficiency performance using the ΣΔ architecture compared to conventional 1- or 2-bit quantized massive MIMO systems.

Resource Allocation and Dithering of Bayesian Parameter Estimation Using Mixed-Resolution Data

Quantization of signals is an integral part of modern signal processing applications, such as sensing, communication, and inference. While signal quantization provides many physical advantages, it usually degrades the subsequent estimation performance that is based on quantized data. In order to maintain physical constraints and simultaneously bring substantial performance gain, in this work we consider systems with mixed-resolution, 1-bit quantized and continuous-valued, data. First, we describe the linear minimum mean-squared error (LMMSE) estimator and its associated mean-squared error (MSE) for the general mixed-resolution model. However, the MSE of the LMMSE requires matrix inversion in which the number of measurements defines the matrix dimensions and thus, is not a tractable tool for optimization and system design. Therefore, we present the linear Gaussian orthonormal (LGO) measurement model and derive a closed-form analytic expression for the MSE of the LMMSE estimator under this model. In addition, we present two common special cases of the LGO model: 1) scalar parameter estimation and 2) channel estimation in mixed-ADC multiple-input multiple-output (MIMO) communication systems. We then solve the resource allocation optimization problem of the LGO model with the proposed tractable form of the MSE as an objective function and under a power constraint using a one-dimensional search. Moreover, we present the concept of dithering for mixed-resolution models and optimize the dithering noise as part of the resource allocation optimization problem for two dithering schemes: 1) adding noise only to the quantized measurements and 2) adding noise to both measurement types. Finally, we present simulations that demonstrate the advantages of using mixed-resolution measurements and the possible improvement introduced with dithering and resource allocation.