Minimum Mean-square Estimation Research Articles

Kalman filter for radio source power and direction of arrival estimation

Images are an important source of information for spacecraft navigation. Based on an image and a known attitude, triangulation techniques (intersection or resection) are often used for positioning and navigation. In the resection problem, an observer estimate its unknown location by using angle measurements to points at known locations (i.e., landmarks), the localization performance depending on the accuracy of the angle measurements. As a contribution to resection for spacecraft navigation, we considers the dynamic image estimation problem based on radio interferometry, i.e., image of radio source power, where the measurements are sample covariance matrices (SCMs). Considering the case where several measurements are available as well as a known dynamic linear model of image evolution, a.k.a a linear state model, the minimum mean-squared error image estimator (MMSE) is given by the Kalman filter (KF) or one of its variants. However standard Kalman-like filters are not a priori suitable for the problem at hand since the measurements (i.e., SCMs) cannot be formulated analytically as a function of state parameters to be estimated (i.e., radio source power). In fact, this lack of analytical formulation can be circumvented by a statistical linear fitting allowing the SCMs to be expressed in terms of the state. This linear fitting introduces an additive residual noise, equivalent to a measurement noise, whose covariance matrix depends on the current state, a non-standard case for a measurement model. The covariance matrix of the residual noise is derived whatever the distributions of the radio sources and of the additive noise at the samples level, unveiling the contribution of their multivariate kurtosis. The proposed method is evaluated on simulated data representative of a dynamic radio interferometry framework. The results show that the proposed method is capable of effectively tracking moving radio sources in complex scenes with theoretical guaranties when the signal multivariate kurtosis is known.

Optimal Design of Eigenvalues for the Full-Spectrum Modulated Nonlinear Frequency Division Multiplexing Transmission System

Nonlinear frequency division multiplexing (NFDM) system is an optional candidate to overcome the fiber nonlinearity limit. A full-spectrum modulated NFDM system, modulating data on combined continuous spectrum (CS) and discrete spectrum (DS) together, was proposed in recent years to improve the data rates and spectral efficiency (SE) by exploiting all the degrees of freedom offered in the nonlinear spectrum. However, the selection of discrete eigenvalues greatly affects the performance of both CS and DS. Designing appropriate eigenvalues of DS is an important issue to ensure the high SE and excellent performance of the system. In this paper, we discussed the selcection principle of eigenvalues and analyzed it from multiple perspectives, 11 eigenvalues with 64-quadrature amplitude modulation (64QAM) are selected for DS. Besides optimizing the eigenvalues at the transmitter, the linear minimum mean-square estimate (LMMSE) method was used at the receiver to furtherly improve the performance of DS. Through the numerical simulation, a 113 Gb/s (SE of 2.8 bits/s/Hz) full-spectrum modulated NFDM system was set up and transmitted 1120 km distance, where the Q-factors of both CS and DS are above the hard-decision forward error correction (HD-FEC) threshold. The results provide a way to design an efficiently full-spectrum modulated NFDM system.

Distributed Beamforming Techniques for Cell-Free Wireless Networks Using Deep Reinforcement Learning

In a cell-free network, a large number of mobile devices are served simultaneously by several base stations (BSs)/access points(APs) using the same time/frequency resources. However, this creates high signal processing demands (e.g., for beamforming) at the transmitters and receivers. In this work, we develop centralized and distributed deep reinforcement learning (DRL)-based methods to optimize beamforming at the uplink of a cell-free network. First, we propose a fully centralized uplink beamforming method (i.e., centralized learning) that uses the Deep Deterministic Policy Gradient algorithm (DDPG) for an offline-trained DRL model. We then enhance this method, in terms of convergence and performance, by using distributed experiences collected from different APs based on the Distributed Distributional Deterministic Policy Gradients algorithm (D4PG) in which the APs represent the distributed agents of the DRL model. To reduce the complexity of signal processing at the central processing unit (CPU), we propose a fully distributed DRL-based uplink beamforming scheme. This scheme divides the beamforming computations among distributed APs. The proposed schemes are then benchmarked against two common linear beamforming schemes, namely, minimum mean square estimation (MMSE) and the simplified conjugate symmetric schemes. The results show that the D4PG scheme with distributed experience achieves the best performance irrespective of the network size. Furthermore, although the proposed distributed beamforming technique reduces the complexity of centralized learning in the DDPG algorithm, it performs better than the DDPG algorithm only for small-scale networks. The performance superiority of the fully centralized DDPG model becomes more evident as the number of APs and/or UEs increases. The codes for all of our DRL implementations are available at <monospace xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><uri>https://github.com/RayRedd/Distributed_beamforming_rl</uri></monospace> .

Spatial Multiplexing Superimposed Pilots for Multicell Multiuser MIMO Uplink Systems

For a multiple-input multiple-output (MIMO) system, or a multiuser MIMO system, it is important to acquire the accurate channel state information. In this article, a superimposed-pilot-based channel estimation scheme is proposed for multicell multiuser MIMO uplink systems. In this scheme, the pilot sequences and data sequences are superimposed, but the space generated by the pilot sequences and the space spanned by data sequences are separated far enough in order to reduce interference. Based on this scheme, the minimum mean-square error channel estimation is deduced, and the optimal power allocation between the pilots and the data signals is derived to maximize throughput. These results show that our proposed scheme can overcome the pilot contamination and achieve better performance than the traditional time-multiplexing scheme with mutually orthogonal pilot sequences for different users when the coherent time slot is limited.

Examination of the Non-Orthogonal Multiple Access System Using Long Short Memory Based Deep Neural Network

This paper investigates deep learning (DL) non-orthogonal multiple access (NOMA) receivers based on long short-term memory (LSTM) under Rayleigh fading channel circumstances. The performance comparison between the DL NOMA detector and the traditional NOMA method is established, and results have shown that the DL-based NOMA detector performance is far better in comparison with conventional NOMA detectors. Simulation curves are compared with the performance of the DL detector in terms of minimum mean square estimate (MMSE) and least square error (LSE) estimate, taking all realistic circumstances, except the cyclic prefix (CP), and clipping distortion into account. The simulation curves demonstrate that the performance of the DL-based detector is exceptionally good when it equals 1 when the noise signal ratio (SNR) is more than 15 dB, assuming that the DL method is more resilient to clipping distortion.

Impact of the learning rate and batch size on NOMA system using LSTM-based deep neural network

In this work, the deep learning (DL)-based fifth-generation (5G) non-orthogonal multiple access (NOMA) detector is investigated over the independent and identically distributed (i.i.d.) Nakagami- m fading channel conditions. The end-to-end system performance comparisons are given between the DL NOMA detector with the existing conventional successive interference cancelation (SIC)-based NOMA detector and from results, it has been proved that the DL NOMA detector performance is better than the convention SIC NOMA detector. In our analysis, the long-short term memory (LSTM) recurrent neural network (RNN) is employed, and the results are compared with the minimum mean square estimation (MMSE) and least square estimation (LS) detector’s performance considering all practical conditions such as multipath fading and nonlinear clipping distortion. It has been shown that with the increase in the relay to destination (RD) channel gain, the bit error rate (BER) improves. Also, with the increase in fading parameter m, the BER performance improves. The simulation curves demonstrate that when the clipping ratio (CR) is unity, the performance of the DL-based detector significantly improves as compared to the MMSE and LS detector for the signal-to-noise ratio (SNR) values greater than 15 dB and it proves that the DL technique is more robust to the nonlinear clipping distortion.

A New Optimization Technique in Massive MIMO and LSAS using Hybrid Architecture and Channel Estimation Algorithm for 5G Networks

Massive Multiple Input Multiple Output (m-MIMO) or Large Scale Antenna System (LSAS) is the latest version of cellular network technology with the purpose to send data from the base station to different users. The huge determined expansion in the mobile equipment and data rate requirement has created the rigid necessity for future wireless communication networks. The m-MIMO envisages a significant increase in the capacity but at the verge of excessive hardware complication. In this paper, we put forward a less complex precoding scheme based on the hybrid architecture to achieve a performance higher than the conventional baseband Minimum Mean Square Estimation (MMSE) precoding. In this paper, we propose a precoding schema namely amplitude and phase MMSE (APMMSE) to attain the performance of traditional MMSE precoding. The informed greedy best-first search has offered to estimate the baseband precoding matrix. It formulates the amplitude of the baseband precoder. Radio Frequency Precoder modifies the phase and provides an optimal signal-to-interference -noise ratio SINR and thus improves spectral efficiency of hybrid architecture.

Kalman--Bucy Filtering and Minimum Mean Square Estimator under Uncertainty

In this paper, we study a generalized Kalman--Bucy filtering problem under uncertainty. The drift uncertainty for both signal process and observation process is considered, and the attitude to uncertainty is characterized by a convex operator (convex risk measure). The optimal filter or the minimum mean square estimator (MMSE) is calculated by solving the minimum mean square estimation problem under a convex operator. In the first part of this paper, this estimation problem is studied under $g$-expectation which is a special convex operator. For this case, we prove that there exists a worst-case prior $P^{\theta^{\ast}}$. Based on this $P^{\theta^{\ast}}$ we obtained the Kalman--Bucy filtering equation under $g$-expectation. In the second part of this paper, we study the minimum mean square estimation problem under general convex operators. The existence and uniqueness results of the MMSE are deduced.

Low-Density Spreading Codes for NOMA Systems and a Gaussian Separability-Based Design

Improved low-density spreading (LDS) code designs based on the Gaussian separability criterion are conceived. We show that the bit error rate (BER) hinges not only on the minimum distance criterion, but also on the average Gaussian separability margin. If two code sets have the same minimum distance, the code set having the highest Gaussian separability margin will lead to a lower BER. Based on the latter criterion, we develop an iterative algorithm that converges to the best known solution having the lowest BER. Our improved LDS code set outperforms the existing LDS designs in terms of its BER performance both for binary phase-shift keying (BPSK) and for quadrature amplitude modulation (QAM) transmissions. Furthermore, we use an appallingly low-complexity minimum mean-square estimation (MMSE) and parallel interference cancellation (PIC) (MMSE-PIC) technique, which is shown to have comparable BER performance to the potentially excessive-complexity maximum-likelihood (ML) detector. This MMSE-PIC algorithm has a much lower computational complexity than the message passing algorithm (MPA).Code sets for MPA are designed similar to low-density parity-check (LDPC) codes to avoid cycles and to increase girth of the Tanner graph, code sets that are “optimal” for MMSE-PIC might not be optimal for MPA.

Reliability-Based Signal Detection for Universal Filtered Multicarrier

In this letter, the reliability-based signal detection is proposed for universal filtered multicarrier (UFMC) systems. In the proposed detector, first, the initial subcarrier-wise estimation utilizing the even-component of frequency-domain (FD) received signal is performed, and the reliabilities of the estimated signals are measured by the corresponding FD channel magnitudes. Second, the estimated signals with high reliabilities are accepted and peeled off from the received signal. Third, the signals with low reliabilities are re-estimated by two different methods, symbol-flipping (SF) and joint minimum mean square estimation (MMSE), utilizing the additional odd-component of FD received signal after peeling off. Specifically, one or two of these unreliable signals are flipped simultaneously in turn from the initial estimates to their neighbors in the SF re-estimation, and all the unreliable signals are jointly estimated by MMSE in the joint MMSE re-estimation. Simulation results show that the proposed detection can achieve better performance-complexity tradeoff than conventional UFMC detections.

Spectral and Energy Efficiency in Cell-Free Massive MIMO Systems Over Correlated Rician Fading

In this article, we investigate the downlink (DL) of a cell-free massive multiple-input multiple-output (MIMO) system over spatially correlated Rician fading, in which many distributed access points (APs) equipped with multiple antennas serve single-antenna users. The APs apply minimum mean-square error channel estimation to obtain the uplink channel state information (CSI). Furthermore, in order to obtain DL CSI at users, this article considers the use of maximum-ratio transmission to beamform DL pilots in the DL beamforming training (BT) phase. For such a system, we derive the closed-form expressions of the sum spectral efficiency (SE) and total energy efficiency (EE). Based on the obtained closed-form expressions, we develop two successive approximation algorithms to improve the sum SE and total EE by optimizing the power control coefficients of DL data and pilot. Numerical results are provided to demonstrate the superiority of the proposed algorithms in improving the sum SE and total EE. In addition, the numerical results also show that the sum SE of a cell-free massive MIMO system with exploiting the BT scheme can be significantly improved over the system without employing the BT scheme.

Coexistence of Direct and Relayed Transmission Users in Multi-Cell Massive MIMO Systems

In this paper, we investigate the uplink performance of a multi-cell massive MIMO relay network, where the base stations (BSs) and the relay stations (RSs) are equipped with large scale antennas and conventional number of antennas, respectively. In each cell, a portion of users communicate with the BS directly while others are aided by the RSs, which are called direct transmission users (DTUs) and relayed transmission users (RTUs), respectively. The two types of users and RSs of all cells share the same time-frequency resource and thus interfere with each other. After describing the system model, we investigate linear minimum mean-squared error channel estimation for the DTUs and RTUs and derive tractable expressions of channel estimation accuracy. With maximal ratio combining and zero-forcing detections based on the estimated channel state information (CSI), we derive the closed-form lower bound and approximation of the achievable rates for DTUs and RTUs to reveal their mutual impact, including intra- and inter-cell interference and pilot contamination. Then, we propose the power scaling law, which proves to be a simple and energy-efficient way to enable the coexistence of DTUs and RTUs in all cells. Furthermore, an optimal power allocation (OPA) scheme is designed based on the proposed power scaling law to maximize the system sum rate. Numerical results verify the accuracy of our theoretical analysis and the effectiveness of the OPA scheme to guarantee performance improvement and fairness simultaneously.

Sequential Minimum Risk Point Estimation of the Parameters of an Inverse Gaussian Distribution

SYNOPTIC ABSTRACTIn the first part of this article, a minimum risk estimation procedure is developed for estimating the mean μ of an inverse Gaussian distribution having an unknown scale parameter λ. A weighted squared-error loss function is assumed, and we aim at controlling the associated risk function. First and second-order asymptotic properties are also established for our stopping rule. The second part deals with developing a minimum risk estimation procedure for estimating the scale parameter λ of an inverse Gaussian distribution. We make use of a squared-error loss function here. The failure of a fixed sample size procedure is established and, hence, some sequential procedures are proposed to deal with this situation. For this estimation problem, we make use of the uniformly minimum variance unbiased estimator (UMVUE) and the minimum mean square estimator (MMSE) of the associated parameters. Second-order approximations are derived for the sequential procedures and improved estimators are proposed.

Optimal Batch Distributed Asynchronous Multisensor Fusion With Feedback

Asynchronous multisensor systems have been widely equipped on various host platforms to meet the requirements of modern navigation campaigns. However, the existing asynchronous fusion algorithms are not efficient in many challenged environments, due to the limitations of communication condition, out-of-sequence observation, and computation affordance. To solve these problems, this paper proposes a batch state asynchronous fusion algorithm with feedback for distributed architecture systems. It is optimal in the sense of minimum mean-squared estimation and its fusion solution is rigorously derived by fully considering the correlations of sequential/cross channels and one-step prediction information. The final fusion solution is straightforward expressed with the combinations of multiple local Kalman filtering estimates outputs during one fusion period. Moreover, no additional interpolation, extrapolation or synchronization procedures are required. Furthermore, to alleviate the computational burden of our proposed algorithm, its approximation form is also given by replacing the weighting matrix by the weighting scalar. Simulations are carried out to illustrate the efficiency and the performance of the proposed algorithm.

Multiuser Massive MIMO Systems With Time-Offset Pilots and Successive Interference Cancellation

This paper proposes time-offset pilots for a single-cell multiuser massive multiple-input multiple-output (MIMO) system and studies its performance under the minimum mean-squared error channel estimator and successive interference cancellation. With the proposed time-offset pilots, users are divided into two groups and the uplink pilots from one group are transmitted simultaneously with the uplink data of the other group, which allows the system to accommodate more users for a given number of pilots. Successive interference cancellation is developed to ease the effect of pilot contamination and enhance data detection. Closed-form expressions for lower bounds of the uplink spectral efficiencies in both the training and data phases are derived when the maximum-ratio combining receiver is used at the base station. The power control problem is formulated with the objective of either maximizing the quality of service that can be equally provided to all users, or minimizing the total transmit power. Since the original power control problems are NP-hard, we also propose algorithms based on the bisection method to solve the problems separately in training and data phases. Analysis and numerical results show that the effect of pilot contamination can be mitigated by successive interference cancellation and proper power control.

Source compression in two-way two-relay network using compute-and-forward relaying

In this paper, the performance of relaying network that employs vector quantization with Compute-and-Forward (CF) relaying technique in AWGN and Rayleigh fading channel is analyzed. Radio relaying is needed when the source and destination cannot directly communicate with each other. Source compression techniques of optimum vector quantization and lattice quantization is employed at source nodes. Physical layer network coding (PNC) with compute and forward relaying is an ingenious relaying scheme that exploits the broadcast nature of wireless channel and forwards the linear combination of signals received from multiple nodes. On the reception of linear combination of messages from the relay nodes, source nodes are capable of extracting the data intended for them. The end-to-end error performance at the source nodes is analyzed. As the performance of CF relaying is sensitive to channel estimation error, we estimate the channel using Least Square Estimation (LSE) and Minimum Mean Square Estimation (MMSE) channel estimation algorithms. The sum-rate of nearly 7 bits/s/Hz for AWGN channel at 20 dB SNR in compute-and-forward relaying is achieved which is only 3.75 bits/s/Hz for decode-and-forward relaying (DF). Thus source compression techniques with efficient CF relaying in PNC is a promising technique to improve the efficiency in wireless relaying networks.

Large random matrix‐based channel estimation for massive MIMO‐OFDM uplink

This study investigates the benefits offered by random matrix theory (RMT) towards the design of reliable channel estimation algorithms for a multi-user massive multiple-input multiple-out (MIMO)-orthogonal frequency-division multiplexing uplink. Assuming no a priori knowledge of channel statistics (KCS) at the massive base station, the authors propose RMT-aided minimum mean square estimation (MMSE) and RMT-aided sparse Bayesian learning (SBL) approaches for massive channel estimation. These approaches render efficient channel estimates, as illustrated through mean square error (MSE) performance, extracted via Monte-Carlo simulations. The results also show that with increasing antennas at the base station, MSE from the RMT-aided MMSE approach decreases, suggesting its aptness to massive MIMO systems. To further enhance the MSE performance, the MMSE and SBL estimated channel impulse responses are pruned using threshold computed from RMT analysis. The authors characterise MSE degradation due to the randomness in the threshold, with the help of the Marcenko-Pastur law-based non-asymptotic framework and concentration inequalities. Analysis results show that, for channels with approximate sparse common support, this MSE degradation is quite insignificant. Altogether, the study demonstrates that RMT analysis is competent in improving channel estimation at a massive MIMO system, when a priori KCS is completely unavailable.

Two-stage procedures for the bounded risk point estimation of the parameter and hazard rate in two families of distributions

ABSTRACTTwo families of distributions are considered that cover a large number of probability distributions useful in investigations involving reliability studies and survival analyses. The problem of bounded risk point estimation of the parameter and hazard rate function of the two families of distribution is handled. Motivated by Mukhopadhyay and Pepe (2006), Roughani and Mahmoudi (2015), and Mahmoudi and Lalehzari (2017), two-stage procedures are developed based on the maximum likelihood estimator (MLE) as well as uniformly minimum variance unbiased estimator (UMVUE). The estimation problem based on the minimum mean square estimator (MMSE) is also considered. We establish that the MMSE of the parameter and hazard rate provides a smaller risk.

On the Energy Efficient Multi-Pair Two-Way Massive MIMO AF Relaying With Imperfect CSI and Optimal Power Allocation

In this paper, we consider multipair two-way massive multiple-input and multiple-output (MIMO) communication system, in which multiple pairs of source nodes exchange messages with the help of one shared amplify-and-forward relay provisioned with large number of antennas. Unlike the perfect channel state information (CSI) assumption, the practical minimum mean-squared error-based channel estimation is considered. And the asymptotic signal to interference plus noise ratio (SINR) analysis of four power-scaling scenarios is presented to show that, even with the imperfect CSI, the interpair interference can still be eliminated, the small-scale fading as well as the noise at source nodes and the relay can be averaged out in the regime of very large number of antenna. It is unveiled that the impact of the channel estimation error will present in the effective large-scale fading coefficient, which explicates the degradation by the imperfect CSI estimate and consumes the results with perfect CSI as special cases. On the basis of the asymptotic SINR analysis results, the optimal power allocation problem is formulated to derive the energy efficient relaying scheme for the maximum ratio combining-maximum ratio transmission precoding scheme. Finally, numerical results are presented to validate that the energy efficient design can effectively reduce the power consumption while maintaining a reasonable sum rate in the multipair two-way massive MIMO relaying system, even in the presence of CSI error.

Power Allocation Optimization for Energy-Efficient Massive MIMO Aided Multi-Pair Decode-and-Forward Relay Systems

We investigate power allocation optimization for global energy efficiency (GEE) maximization in the massive multiple-input multiple-output technique aided multi-pair one-way decode-and-forward relay systems. Assuming that the minimum mean-square error channel estimator and zero-forcing transceivers are employed at the relay, we first derive an accurate closed-form expression of the GEE of this complex system. Based on our analytical results, a non-convex power allocation optimization problem with the objective of GEE maximization is formulated under specific quality-of-service (QoS) and transmit power constraints. To solve this challenging problem, the successive convex approximation technique is invoked to transform the original optimization problem into a concave fractional programming problem, which is then efficiently solved by Dinkelbach’s method and by the Charnes–Cooper transformation-based method. In addition, as a special case, the GEE maximization problem under the assumption of using the equal power allocation strategy at both the source users and the relay is also considered. Simulation results demonstrate the accuracy of our analytical results and the effectiveness of the proposed algorithms. Furthermore, the impact of several important system parameters (i.e., the QoS constraint, the transmit power constraints at both the source users and the relay, as well as the quality of channel estimation) on the maximum GEE achieved by the proposed algorithms is also illustrated.