Minimum Mean Square Error Estimator Research Articles

Linear precoder optimization of spectral efficiency of time division duplex hyper MIMO system with pilot contamination

Our work is developed in context of studing Massive MIMO in a 5G context. The aim is to optimize spectral efficiency of several users hyper MIMO system during Uplink communication in a multi-cell contaminated pilot environment, using a new type of precoders called single cell-minimum mean square eroor (S-MMSE) and multicell-minimum mean square eroor (MMMSE). Indeed, we address two key and well-known issues of massive multiuser MIMO (MU-MIMO) environments in a test-driven development (TDD) operation scheme, namely acquisition of uplink channel state information (UL) and optimisation of the bit stream per unit frequency, the spectral efficiency (SE). From a practical point of view, these two notions are inclusively linked. Indeed, a very good channel estimation leads to a better spectral efficiency. In our approcah, we derive from the minimum mean square error estimator (MMSE) to two new types of precoders that can operate in a multicell environment with a contaminated pilot sequence, namely the SMMSE and the M-MMSE. A comparative study performance of these classical precoders such as regulated zero forcing (RZF), ZF (Zero Forcing) and MR (Minimum Ratio) encountered in multi-antenna processing shows an improvement of nearly 51% in terms of system gain and spectral efficiency.

Quantum-Inspired Multi-Parameter Adaptive Bayesian Estimation for Sensing and Imaging

It is well known in Bayesian estimation theory that the conditional estimator attains the minimum mean squared error (MMSE) for estimating a scalar parameter of interest. In quantum, e.g., optical and atomic, imaging and sensing tasks the user has access to the quantum state that encodes the parameter. The choice of a measurement operator, i.e. a positive-operator valued measure (POVM), leads to a measurement outcome on which the aforesaid classical MMSE estimator is employed. Personick found the optimum POVM that attains the MMSE= over all possible physically allowable measurements and the resulting MMSE [1]. This result from 1971 is less-widely known than the quantum Fisher information (QFI), which lower bounds the variance of an unbiased estimator over all measurements without considering any prior probability. For multi-parameter estimation, in quantum Fisher estimation theory the inverse of the QFI matrix provides an operator lower bound on the covariance of an unbiased estimator, and this bound is understood in the positive semidefinite sense. However, there has been little work on quantifying the quantum limits and measurement designs, for multi-parameter quantum estimation in a Bayesian setting. In this work, we build upon Personick's result to construct a Bayesian adaptive (greedy) measurement scheme for multiparameter estimation. We illustrate our proposed measurement scheme with the application of localizing a cluster of point emitters in a highly sub-Rayleigh angular field-of-view, an important problem in fluorescence microscopy and astronomy. Our algorithm translates to a multi-spatial-mode transformation prior to a photon-detection array, with electro-optic feedback to adapt the mode sorter. We show that this receiver performs superior to quantum-noise-limited focal-plane direct imaging.

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.

Communication-Efficient Federated Learning via Quantized Compressed Sensing

In this paper, we present a communication-efficient federated learning framework inspired by quantized compressed sensing. The presented framework consists of gradient compression for wireless devices and gradient reconstruction for a parameter server (PS). Our strategy for gradient compression is to sequentially perform block sparsification, dimensional reduction, and quantization. By leveraging both dimension reduction and quantization, our strategy can achieve a higher compression ratio than one-bit gradient compression. For accurate aggregation of local gradients from the compressed signals, we put forth an approximate minimum mean square error (MMSE) approach for gradient reconstruction using the expectation-maximization generalized-approximate-message-passing (EM-GAMP) algorithm. Assuming Bernoulli Gaussian-mixture prior, this algorithm iteratively updates the posterior mean and variance of local gradients from the compressed signals. We also present a low-complexity approach for the gradient reconstruction. In this approach, we use the Bussgang theorem to aggregate local gradients from the compressed signals, then compute an approximate MMSE estimate of the aggregated gradient using the EM-GAMP algorithm. We also provide a convergence rate analysis of the presented framework. Using the MNIST dataset, we demonstrate that the presented framework achieves almost identical performance with the case that performs no compression, while significantly reducing communication overhead for federated learning.

Detection and Estimation of Randomly Phase Modulated Signals

The design of detectors for weak sinusoidal signals with random varying phase and frequency has been studied for many decades and continues to be a topic of significant theoretical and practical interest. For a random process, the likelihood ratio detector is known to have an estimator-correlator structure that requires generation of the causal minimum mean square error (MMSE) estimate of the random signal, which generally is very challenging for non-Gaussian signals. In this paper we present methods to project an infinite dimensional solution to the stochastic differential equations generating the required MMSE estimate onto a finite dimensional space, closely approximating the exact solution. Simulations of our proposed detectors are presented and compared with existing approaches, specifically optimized quadratic detectors and extended Kalman filter based methods.

Deep MMSE Estimation for Data Detection

This letter develops minimum mean-squared error (MMSE) estimators based on deep neural networks for data detection. Since the optimal MMSE is analytically intractable, researchers usually resort to linear MMSE approximations, which often incur performance degradation. To overcome this loss, we develop a near-optimal estimator by exploiting the Donsker-Varadhan representation of mutual information (MI) and the recently discovered derivative relationship between MI and MMSE. The near-optimal MMSE estimator can be computed with a deep neural network (NN). We can train the NN using the mini-batches of input and output samples. We validate this estimator using two examples where the closed-from MMSE is available. We then use this estimator to design an end-to-end communication system. We compare this setup with several conventional techniques and show promising performance.

Event-Triggered Sensor Scheduling for Remote State Estimation With Error-Detecting Code

This letter addresses the problem of remote state estimation subject to packet dropouts, focusing on the use of an event-triggered sensor scheduler to conserve communication resources. However, packet dropouts introduce significant challenges, as the remote estimator cannot distinguish between packet loss caused by poor channel conditions and the event trigger. To overcome this issue, we propose a novel formulation that incorporates error-detecting codes. We prove that the Gaussian property of the system state, commonly utilized in the literature, does not hold in this scenario. Instead, the system state follows an extended Gaussian mixture model (GMM). We present an exact minimum mean-squared error (MMSE) estimator and an approximate estimator, which significantly reduces algorithm complexity without sacrificing performance. Our simulation results show that the approximate estimator achieves nearly the same performance as the exact estimator while requiring much less computation time. Moreover, the proposed event trigger outperforms existing schedulers in terms of estimation accuracy.

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.

Widely-Linear MMSE Estimation of Complex-Valued Graph Signals

In this paper, we consider the problem of recovering random graph signals with complex values. For general Bayesian estimation of complex-valued vectors, it is known that the widely-linear minimum mean-squared-error (WLMMSE) estimator can achieve a lower mean-squared-error (MSE) than that of the linear minimum MSE (LMMSE) estimator for the estimation of improper complex-valued signals. Inspired by the WLMMSE estimator, in this paper we develop the graph signal processing (GSP)-WLMMSE estimator, which minimizes the MSE among estimators that are represented as a two-channel output of a graph filter, i.e. widely-linear GSP estimators. We discuss the properties of the proposed GSP-WLMMSE estimator. In particular, we show that the MSE of the GSP-WLMMSE estimator is always equal to or lower than the MSE of the GSP-LMMSE estimator. The GSP-WLMMSE estimator is based on <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">diagonal</i> covariance matrices in the graph frequency domain, and thus has reduced complexity compared with the WLMMSE estimator. This property is especially important when using the sample-mean versions of these estimators that are based on a training dataset. We state conditions under which the low-complexity GSP-WLMMSE estimator coincides with the WLMMSE estimator. In simulations, we investigate a synthetic linear estimation problem and the nonlinear problem of state estimation in power systems. For these problems, it is shown that the GSP-WLMMSE estimator outperforms the GSP-LMMSE estimator and achieves similar performance to that of the WLMMSE estimator. Moreover, the sample-mean version of the GSP-WLMMSE estimator outperforms the sample-mean WLMMSE estimator for a limited training dataset and is more robust to topology changes.

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.

A Statistical Framework to Investigate the Optimality of Signal-Reconstruction Methods

We present a statistical framework to benchmark the performance of reconstruction algorithms for linear inverse problems, in particular, neural-network-based methods that require large quantities of training data. We generate synthetic signals as realizations of sparse stochastic processes, which makes them ideally matched to variational sparsity-promoting techniques. We derive Gibbs sampling schemes to compute the minimum mean-square error estimators for processes with Laplace, Student's t, and Bernoulli-Laplace innovations. These allow our framework to provide quantitative measures of the degree of optimality (in the mean-square-error sense) for any given reconstruction method. We showcase our framework by benchmarking the performance of some well-known variational methods and convolutional neural network architectures that perform direct nonlinear reconstructions in the context of deconvolution and Fourier sampling. Our experimental results support the understanding that, while these neural networks outperform the variational methods and achieve near-optimal results in many settings, their performance deteriorates severely for signals associated with heavy-tailed distributions.

On DCT-based MMSE estimation of short time spectral amplitude for single-channel speech enhancement

This paper proposes Discrete Cosine Transform (DCT) based speech enhancement algorithms. These algorithms utilize minimum mean square error (MMSE) estimator of clean short-time spectral amplitude, which respectively uses Gaussian, Laplace and Gamma probability density functions (PDF) as speech priors. We consider the noise process is additive and Gaussian. The proposed estimators are closed-form solutions, whereas the conventional Discrete Fourier Transform (DFT) based estimators derived under super-Gaussian speech priors have no closed-form solutions. We also examine the estimators with the Speech Presence Uncertainty (SPU) that addresses the speech or silence problem with probability. Compared to the alternative approaches, such as the Ephraim and Malah or the Erkelens et.al MMSE-STSA estimators, the proposed methods demonstrate superior performance in terms of Segmental SNR (SegSNR), Perceptual Evaluation of Speech Quality (PESQ), short-time objective intelligibility measure (STOI), and mean subjective preference score, while exhibiting an equal or lower complexity.

Low-complex Bayesian estimator for imperfect channels in massive muti-input multi-output system

&lt;span lang="EN-US"&gt;Motivated by the fact that the complexity of the computations is one of the main challenges in large multiple input multiple output systems, known as massive multiple-input multiple-output (MIMO) systems, this article proposes a low-complex minimum mean squared error (MMSE) Bayesian channel estimator for uplink channels of such systems. First, we have discussed the necessity of the covariance information for the MMSE estimator and how their imperfection knowledge can affect its accuracy. Then, two reduction phases in dimension and floating-point operations have been suggested to reduce its complexity: in phase 1, eigenstructure reduction for channel covariance matrices is implemented based on some truncation rules, while in phase 2, arithmetic operations reduction for matrix multiplications in the MMSE equation is followed. The proposed procedure has significantly reduced the complexity of the MMSE estimator to the first order O(M), which is less than that required for the conventional MMSE with O(M&lt;sup&gt;3&lt;/sup&gt;) in terms of matrix dimension. It has been shown that the estimated channels using our proposed procedure are asymptotically aligned and serve the same quality as the full-rank estimated channels. Our results are validated by averaging the normalized mean squared error (NMSE) over a length of 500 sample realizations through a Monte Carlo simulation using MATLAB R2020a.&lt;/span&gt;

Optimal deception attacks on remote state estimators equipped with interval anomaly detectors

This paper studies the problem of optimal deception attacks on remote state estimation, where an interval χ2 detector is deployed to reveal anomalies. The information-based attack policy that can bypass the anomaly detector and cause the maximum estimation quality degradation is derived. For both attacks with strict and relaxed stealthiness, the optimal compromised measurements can be designed with three steps: obtain the minimum mean-square error estimation of the prediction error, de-correlate the estimate with historical compromised innovations, and design the compromised innovation as an optimal linear transformation. All available information for attackers is fully utilized for performance maximization while the stealthiness constraint is satisfied precisely to deceive the anomaly detector. The attack effect depends on both the amount of online information and the duration of detection interval. Contrary to well-studied innovation-based attacks using static linear combinations, the information-based deception policy is shown to be generated by a linear time-varying system, whose coefficients can be completely determined offline. The optimality of the proposed attack is verified with numerical examples and comparative studies.

An approximate minimum mean-square error estimator for linear discrete time-varying systems: Handling Try-Once-Discard protocol

This paper is concerned with the remote state estimation problem for a class of linear discrete time-varying stochastic systems under communication constraints. In order to mitigate the undesired phenomena of data collisions, a Try-Once-Discard (TOD) protocol is utilized to regulate the signal transmissions over the sensor-to-estimator communication channel, where the TOD protocol is based on the scheduling rule described by a specific switching function. Under the introduced TOD protocol, the approximate minimum mean-square error (MMSE) state estimation problem is investigated and a recursive algorithm is developed for the MMSE estimator design whose computational complexity is comparable to that of the conventional Kalman filter. Furthermore, some conditions are established for the boundedness of the estimation error covariance. A numerical example is presented to illustrate the effectiveness of the proposed MMSE estimator.

Deep Learning Channel Estimation for OFDM 5G Systems with Different Channel Models

At cellular wireless communication systems, channel estimation (CE) is one of the key techniques that are used in Orthogonal Frequency Division Multiplexing modulation (OFDM). The most common methods are Decision‐Directed Channel Estimation, Pilot-Assisted Channel Estimation (PACE) and blind channel estimation. Among them, PACE is commonly used and has a steadier performance. Applying deep learning (DL) methods in CE is getting increasing interest of researchers during the past 3 years. The main objective of this paper is to assess the efficiency of DL-based CE compared to the conventional PACE techniques, such as least-square (LS) and minimum mean-square error (MMSE) estimators. A simulation environment to evaluate OFDM performance at different channel models has been used. A DL process that estimates the channel from training data is also employed to get the estimated impulse response of the channel. Two channel models have been used in the comparison: Tapped Delay Line and Clustered Delay Line channel models. The performance is evaluated under different parameters including number of pilots (64 pilots or 8 pilots), number of subcarriers (64), the length of cyclic prefix (16 or 0 samples) and carrier frequency (4 GHz) through computer simulation using MATLAB. From the simulation results, the trained DL estimator provides better results in estimating the channel and detecting the transmitted symbols compared to LS and MMSE estimators although, the complexity of the proposed LSTM estimator exceeds the equivalent LS estimator. Furthermore, the DL estimator also demonstrates its effectiveness with various pilot densities and with different cyclic prefix periods.

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.

Performance Evaluation of Spectral Efficiency for Uplink and Downlink Multi-Cell Massive MIMO Systems

Massive multiple-input and multiple-output (MIMO) systems have become the most persuasive technology for 5G as it increased the energy efficiency gigantically as compared to other wireless communication systems. Being the most vibrant research technology in the communication sector, this research work is based on the optimal model development of energy-efficient massive MIMO systems. The proposed model is a realistic model that augmented the spectral efficiency (SE) of massive MIMO systems where a multi-cell model scenario is considered. Channel estimation is carried out at the base stations (BSs) based on uplink (UL) transmission while the minimum mean-squared error (MMSE), Element-wise MMSE, and Least-square (LS) estimators are used for the estimation. We analyze the achievable SE of the UL based on the MMSE channel estimator with different receive combining schemes. Moreover, the downlink (DL) transmission model is also modelled with different precoding schemes by taking the same vectors used in combining schemes. The simulation results show a significant improvement in spectral efficiency by developing UL and DL transmission models and also realized that the average sum of SE per cell can be improved by optimized MMSE channel estimation, installing multiple BS antennas, and serving multiple UEs per cell. The findings of this work specify that the massive MIMO system can be developed by optimizing the channel estimation for the augmentation of SE in UL and DL transmissions. Conclusively, it can be summarized that some complex computations of MMSE channel estimators can enhance the average sum of SE per cell as per the results verified in this model.

Deep Neural Network: An Alternative to Traditional Channel Estimators in Massive MIMO Systems

Fifth-generation (5G) requires a highly accurate estimate of the channel state information (CSI) to exploit the benefits of massive multiple-input-multiple-output (MaMIMO) systems. 5G systems use pilot sequences to estimate channel behaviour using traditional methods like least squares (LS), or minimum mean square error (MMSE) estimation. However, traditional methods do not always obtain reliable estimations: LS exhibits a poor estimation when inadequate channel conditions (i.e., low- signal-to-noise ratio (SNR) region) and MMSE requires prior statistical knowledge of the channel and noise (complex to implement in practice). We present a deep learning framework based on deep neural networks (DNNs) for fifth-generation (5G) MaMIMO channel estimation. After a first preliminary scheme with which we verify the good estimation capacity of our DNN-based approach, we propose two different models, which differ in the information processed by the DNN and benefit from lower computational complexity or greater flexibility for any reference signal pattern, respectively. The results show that, compared to the LS-based channel estimation, the DNN approach decreases the mean square error (MSE) and the system’s spectral efficiency (SE) increases, especially in the low- SNR region. Our approach provides results close to optimal MMSE estimation but benefits from not requiring any prior channel statistics information.

Energy Efficiency Optimization Based on Power Allocation in Massive MIMO Downlink Systems

To solve the energy efficiency (EE) optimization in a multi-cell (MU) massive multiple-input multiple-output (MIMO) downlink system, of which channels are of symmetry in the Time-Division Duplex (TDD) protocol, we utilize a spatially correlated channel model and adopt the minimum mean-squared error (MMSE) estimator to tailor linear precoding vectors. Then, we derive the expression of downlink spectral efficiency (SE), taking interference into account. Subsequently, we establish the EE optimization function, which is defined as the average capacity divided by power consumption. In an interference-limited scenario, the EE optimization is of high complexity to solve globally as it is not jointly concave. To this end, we propose the Dinkelbach-like power allocation algorithm to obtain a suboptimal solution. We transform the EE problem in a fractional form into a subtractive optimization form called an auxiliary subproblem. Then, we relax the sub-problem to a concave problem by initializing the interference and omitting the dynamic power term about throughput. Lastly, we solve iteratively the Karush–Kuhn–Tucker (KKT) conditions by bisection search. Consequently, we obtain a sub-solution with modest complexity. The simulation results justify the rationality of the Dinkelbach-like algorithm and demonstrate that the proposal outperforms the reference schemes and effectively improves the performance metrics EE and SE.