Linear Minimum Mean Square Error Research Articles

Performance Analysis of a SAGE-Based Semi-Blind Channel Estimator for Pilot Contaminated MU Massive MIMO Systems

Massive multiple-input multiple-output (M-MIMO) is one of the key ingredients in the upcoming 5G technology. The benefits associated with M-MIMO rely largely on the accuracy of channel state information (CSI) available at the base station (BS). The existing literature mostly employs pilot-aided schemes which require additional pilots for improving their CSI accuracy; additionally, pilot contamination degrades their performance to a large extent. In this paper, we design a Space-Alternating Generalized Expectation-Maximization (SAGE) based semi-blind estimator for pilot contaminated multi-user (MU) M-MIMO systems. It utilizes both pilot and a few data symbols for CSI estimation through two stages, namely initialization and iterative SAGE (ISAGE). We obtain an initial channel estimate with the help of a pilot-aided linear minimum mean squared error (LMMSE) estimator in the initialization stage. The acquired initial estimate from the former stage is then iteratively updated by SAGE algorithm with the joint usage of pilot and a few data symbols in ISAGE stage. The inclusion of data information in ISAGE stages' estimation process aids in simultaneous improvement of CSI accuracy and spectral efficiency (SE) of a M-MIMO system; which is unlikely for a pilot based estimation scheme. Through simulations, we show that our estimator obtains a considerable improvement over the existing pilot-aided schemes in terms of mean squared error (MSE), bit error rate (BER), SE, and energy efficiency (EE) at a nominal increase in complexity. Besides, on average, it achieves convergence in almost two iterations. We also derive modified Cramer-Rao lower bound (MCRLB) to validate the estimation efficacy of our estimator. We evaluate a closed-form expression for lower bound on UL achievable rate of MU M-MIMO systems under both perfect and imperfect CSI scenario. We also discuss the trade-off between SE and EE.

Sherman-Morrison Formula Aided Adaptive Channel Estimation for Underwater Visible Light Communication With Fractionally-Sampled OFDM

In this paper, we investigate the channel estimation (CE) problem in an underwater visible light communication (UVLC) system invoking fractionally-sampled optical orthogonal frequency division multiplexing (FS-OOFDM). In practical UVLC scenarios, the communication links inevitably suffer from many stochastic channel effects including multi-path dispersion, scattering, turbulence, etc., and/or from the mobility of the transceiver, therefore resulting in a time-varying, location-dependent non-stationary propagation environment. Naturally, compared with the indoor visible light communication (VLC) scenario with a typical assumption being the time-flat channel models, it becomes a notable challenge for designing a low-complexity adaptive CE in the much more complicated UVLC scenarios. To solve this problem, we derive a class of Bayesian CE algorithms referred to as the Sherman-Morrison formula (SMF) based CE (SMF-CE), by exploiting the property of rank-one structure of the second-order channel statistics in the delay domain. Furthermore, an adaptive version of SMF-CE (ASMF-CE) can be obtained through updating the imperfect a priori knowledge of the channel and the noise's statistics. Simulation results demonstrate the superior performances of the proposed algorithms in comparison to existing methods, while maintaining a reduced computational complexity in comparison to the conventional linear minimum mean square error (LMMSE) scheme.

In-Band Full-Duplex Discriminatory Channel Estimation Using MMSE

This paper proposes full-duplex transmissions from legitimate nodes to achieve channel estimation performance deterioration at an eavesdropper as compared to the legitimate receiver. The proposed discriminatory channel estimation (DCE) technique comprises of two stages where, in the first stage, the self-interference channel is estimated by the respective legitimate nodes. Followed by in-band full-duplex transmission from both legitimate nodes for channel estimation at legitimate nodes, while providing equivocation at the eavesdropper due to the superposition of two signals. The discrimination of channel estimation performance provides secrecy against the passive eavesdropper while delivering information to the legitimate receiver. We provide the mean square error (MSE) to indicate the performance achieved by linear minimum mean square error (LMMSE) estimators. We have also provided bit error rate (BER), and secrecy capacity analysis to indicate the performance of secure communication achieved by securing the channel estimates from the eavesdropper. The BER analysis shows that for proposed DCE, BER at the eavesdropper is close to 0.1 while the legitimate node is able to robustly decode the information. Finally, simulation results show that the proposed DCE outperforms existing DCE techniques for the considered scenario.

Neural Network-Optimized Channel Estimator and Training Signal Design for MIMO Systems With Few-Bit ADCs

This paper is concerned with channel estimation in MIMO systems with few-bit ADCs. In these systems, a linear minimum mean-squared error (MMSE) channel estimator obtained in closed-form is not an optimal solution. We first consider a deep neural network (DNN) and train it as a nonlinear MMSE channel estimator for few-bit MIMO systems. We then present a first attempt to use DNN in optimizing the training signal and the MMSE channel estimator concurrently. Specifically, we propose an autoencoder with a specialized first layer, whose weights embed the training signal matrix. Consequently, the trained autoencoder prompts a new training signal designed specifically for the MIMO channel model under consideration.

Self-Interference Cancellation and Channel Estimation in Multicarrier-Division Duplex Systems With Hybrid Beamforming

Design of full-duplex (FD) wireless systems faces many challenges, including self-interference cancellation (SIC), capability to provide high capacity, high flexibility for operation, best usage of resources, etc. In this paper, we propose and investigate a multicarrier-division duplex (MDD) based hybrid beamforming system operated in FD mode, which is endowed with the advantages of both time-division duplex and frequency-division duplex. It also shares some merits of FD and allows to be free of self-interference (SI) in digital domain, but faces the same challenge of SI as the FD in analog domain. Hence in this paper, we first propose an adaptive beamforming assisted SI cancellation scheme with taking into account the practical requirement of analog-to-digital conversion (ADC). It can be shown that the proposed approach is capable of jointly coping with the desired signals' transmission and SI suppression. Then, channel estimation (CEst) in MDD/MU-MIMO system is proposed by exploiting the reciprocity between the uplink and downlink subcarrier channels that is provided by MDD. Correspondingly, the orthogonality-achieving pilot symbols are designed, and the least-square (LS) CEst as well as linear minimum mean-square error (LMMSE) CEst are derived. Finally, the performance of MDD/MU-MIMO systems employing the proposed SIC method is investigated, with respect to the SI cancellation capability, sum-rate potential, CEst performance, and the effect of CEst on the achievable performance. Our studies show that MDD/MU-MIMO provides an effective option for design of future wireless transceivers.

Channel Estimation in Massive MIMO Under Hardware Non-Linearities: Bayesian Methods Versus Deep Learning

This paper considers the joint impact of non-linear hardware impairments at the base station (BS) and user equipments (UEs) on the uplink performance of single-cell massive MIMO (multiple-input multiple-output) in practical Rician fading environments. First, Bussgang decomposition-based effective channels and distortion characteristics are analytically derived and the spectral efficiency (SE) achieved by several receivers are explored for third-order non-linearities. Next, two deep feedforward neural networks are designed and trained to estimate the effective channels and the distortion variance at each BS antenna, which are used in signal detection. We compare the performance of the proposed methods with state-of-the-art distortion-aware and -unaware Bayesian linear minimum mean-squared error (LMMSE) estimators. The proposed deep learning approach improves the estimation quality by exploiting impairment characteristics, while LMMSE methods treat distortion as noise. Using the data generated by the derived effective channels for general order of non-linearities at both the BS and UEs, it is shown that the deep learning-based estimator provides better estimates of the effective channels also for non-linearities more than order three.

On the Convexity of Bit Depth Allocation for Linear MMSE Estimation in Wireless Sensor Networks

Energy efficiency is crucial for a wireless sensor network (WSN) since its nodes are generally powered by energy sources of limited capacity, such as batteries. The bit depth used to quantize the sensor signal samples heavily influences energy consumption, as it strongly impacts the amount of information to be transmitted between the sensor nodes. Bit depth allocation problems seek to assign a certain bit depth to each sensor signal such that energy consumption is minimized while respecting a performance constraint. For multi-channel signal estimation tasks these problems are generally non-convex, and they are often solved through simplifying assumptions or through convex relaxation. However, for linear minimum mean squared error (MMSE) estimation, we show how the matrix inversion lemma allows to transform the MMSE constraint into a convex constraint, which can then be interpreted as a constraint on the excess MMSE due to quantization. As a result, as long as the cost function representing energy consumption is convex, this class of bit depth allocation problems is convex, i.e., if the bit depth variable is relaxed to a real-valued variable. This guarantees global optimality up to discretization of the obtained solution.

Correlation Analysis and Link Gain Prediction for Optical Wireless Scattering Communication Over Broad Spectra

None-line-of-sight (NLOS) optical wireless scattering communication over broad spectra can further increase the transmission rate beyond single wavelength scattering communication. A fundamental question is to investigate the link gain correlation over multiple wavelengths, which may lead to more efficient communication design and optimization approaches. We investigate the link gain correlation over multiple wavelengths under varying transmission media environments and transceiver geometries, and show that the link gain vector can be represented in a reduced dimensional space using principle components analysis (PCA). We also justify such correlation properties theoretically through a simplified single-scattering model. The link gain prediction is formulated as a joint continuous and discrete optimization problem, and is transformed into tractable forms. Linear prediction approaches are proposed based on zero-forcing (ZF) and linear minimum mean square error (LMMSE) criteria. It is seen that the LMMSE-based approach generally leads to lower prediction error compared with the ZF-based one, especially for the scenario with large residual eigenvalues of the link gain covariance matrix. We also investigate the link rates on multiple wavelengths, which also exhibit strong correlation for negligible background radiation intensity.

Compression-Based LMMSE Channel Estimation With Adaptive Sparsity for Massive MIMO in 5G Systems

Massive multi-input multi-output (MIMO) has been regarded as one of the key technologies for fifth generation (5G) mobile communication systems, as it can significantly enhance the system capacity with high spectrum and energy efficiency. For massive MIMO systems, accurate channel estimation is a challenging problem, especially when the number of parameters to be estimated is large and the number of pilots is limited. In this paper, a compression-based linear minimum mean square error (CLMMSE) channel estimation algorithm is proposed for massive MIMO in 5G systems. Compared with the traditional linear minimum mean square error (LMMSE) algorithm, the proposed approach calculates the channel autocorrelation matrix by investigating the channel prior information based on compressive sensing (CS) theory, utilizing the block sparsity of massive MIMO channels, to reduce the complexity for obtaining autocorrelation matrix. Then it substitutes matrix inverse operation by singular value decomposition to further reduce the computational complexity. In addition, a block sparsity adaptive matching pursuit (BSAMP) method is also proposed to adaptively estimate the block sparsity of the channel in the first step of the proposed CLMMSE algorithm, which can make it more efficient. The sparsity-adaptive processing is achieved by setting a threshold and finding the position of the maximum backward difference, then using the regularized method to solve channel estimation as a convex optimization problem. Analyses and simulations indicate that with slight performance degradation, the proposed algorithm reduces the computational complexity significantly compared with the traditional LMMSE algorithm. And compared with pure CS methods, CLMMSE has an obviously better performance, which is beneficial to solve the pilot pollution problem of massive MIMO in 5G systems. Furthermore, the BSAMP-based CLMMSE algorithm has better performance and lower time complexity than the algorithm based on other CS methods, which further improves the system performance.

Performance Analysis of Hardware Impairment Aware MMSE Receiver With Channel and CFO Estimation Error Statistics for a Large MIMO-OFDM System

In this treatise, we conceive a hardware impairment aware receiver design of a large multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) system, taking into account the error statistics of parameter estimations. The hardware impairment includes the quantization noise, phase noise etc. There are two key contributions in this work. Firstly, consideration of error statistics of residual parameters in the receiver design improves receiver performance. In this work, we propose a linear equalizer based data detector at the receiver for a large MIMO-OFDM system and the parameters of interest are the channel impulse response (CIR) and carrier frequency offset (CFO) along with the hardware impairments awareness both at transmitter and receiver. The design criterion chosen for designing the detector equalizer is based on the linear minimum mean square error (LMMSE) one and it will consider the statistics of estimation error and hardware impairments. However, in any practical communication system, the availability of statistics for these parameter errors is inconceivable. Hence, we propose further to consider the Bayesian Crammer Rao lower bound (BCRLB) as the variance of the error with the key assumption of zero mean. Numerical simulation demonstrates that, consideration of such statistics attains an SNR gain of $2$ dB with respect to a fixed symbol error ratio (SER). It also shows that the SER performance with BCRLB matches very closely with the case, when exact error variance is used. For the second contribution, an extensive theoretical analysis has been performed to evaluate the MSE improvement of data detection with the LMMSE criterion with parameter estimation error statistics consideration in the large MIMO-OFDM system. This theoretical analysis of improved MSE has been conceived with the usage of large dimension random matrix theory and a closed-form expression has been obtained. Numerical simulation demonstrates that the theoretical analysis for MSE improvement matches closely with the simulated one.

On Orthogonal AMP in Coded Linear Vector Systems

Linear minimum mean square error (LMMSE) estimation based turbo detection has been extensively studied for coded linear systems since the seminal work of Wang and Poor (WP). The WP algorithm operates iteratively between a linear detector (LD) and a nonlinear detector (NLD): the LD suppresses the interference based on LMMSE filtering, and the NLD decodes the data by treating the output of the LD as an observation from an additive white Gaussian noise (AWGN) channel. In WP, the messages exchanged between LD and NLD are required to be extrinsic . For the NLD, the extrinsic message comes from the constraint imposed on feedforward error correction (FEC) codes. Therefore, WP does not work in an un-coded linear system. Recently, we proposed an orthogonal approximate message passing (OAMP) algorithm, which only requires the input/output error terms of LD and NLD to be orthogonal . We conjectured that for un-coded linear systems that involve certain large random matrices, the dynamics of OAMP can be accurately characterized by state evolution (SE). In this paper, we consider a coded linear system and develop an extrinsic message aided OAMP (EMA-OAMP) algorithm. Similar to the un-coded case, EMA-OAMP relaxes the requirements on output messages to be orthogonal instead of extrinsic. We derive an SE procedure to characterize the performance of OAMP in coded systems. We conjecture that this SE procedure is accurate, which is verified by simulation results. Under this conjecture, we show that EMA-OAMP can outperform WP under certain standard assumptions for iterative decoding. Extensive simulations results are provided to verify the advantages of OAMP in coded MIMO systems.

Optimum Distributed Estimation of a Spatially Correlated Random Field

The distributed estimation of a spatially correlated random field with decentralized sensor networks is studied in this paper. Nodes in the network take spatial samples of the random field, then each node estimates the values of arbitrary points on the random field by iteratively exchanging information with each other. The objective is to minimize the estimation mean squared error (MSE) while ensuring all nodes reach a distributed consensus on the estimation results. We propose a distributed iterative linear minimum mean squared error (LMMSE) algorithm that contains a state consensus stage and a local estimation stage in each iteration. The proposed algorithm requires the knowledge of the second-order statistics of the random field, and they are estimated by using a distributed learning algorithm with the help of distributed consensus. The key parameters of the algorithm, including an edge weight matrix and a sample weight matrix, are designed to minimize an MSE upper bound at all nodes when the number of iterations is large. It is shown that the optimum performance can be achieved by distributively mapping the high dimension measurement samples from all nodes into a low dimension subspace related to the covariance matrices of data and noise samples. The low-dimension mapping is achieved in a distributed manner through iterative information propagation. The low dimension mapping can significantly reduce the amount of data exchanged in the network, thus improve the convergence speed of the iterative algorithm.

Estimation Theoretic Secure Communication via Encoder Randomization

Estimation theoretic secure transmission of a scalar random parameter is investigated in the presence of an eavesdropper. The aim is to minimize the estimation error at the receiver under a secrecy constraint at the eavesdropper; or, alternatively, to maximize the estimation error at the eavesdropper for a given estimation accuracy limit at the receiver. In the considered setting, the encoder at the transmitter is allowed to use a randomized mapping between two one-to-one and continuous functions and the eavesdropper is fully aware of the encoding strategy at the transmitter. For small numbers of observations, both the eavesdropper and the receiver are modeled to employ linear minimum mean-squared error (LMMSE) estimators, and for large numbers of observations, the expectation of the conditional Cramer-Rao bound (ECRB) metric is employed for both the receiver and the eavesdropper. Optimization problems are formulated and various theoretical results are provided in order to obtain the optimal solutions and to analyze the effects of encoder randomization. In addition, numerical examples are presented to corroborate the theoretical results. It is observed that stochastic encoding can bring significant performance gains for estimation theoretic secrecy problems.

Generalized LMMSE Filtering with Out‐of‐Sequence Observations in Arbitrary Constant Delay

Focusing on the problem of state estimation in the presence of sensor faults and Out-of-sequence measurement (OOSM) observations synchronously, we derive a formulation of Linear minimum mean squared error (LMMSE) filter with the arbitrary time delay of OOSM, the generalization of the present work lies in simultaneous treatment of correlated faults together with OOSMs that arrive at an arbitrary delay in a linear-optimal manner. The approach is demonstrated in a numerical comparison.

Deep Learning-Based Joint Pilot Design and Channel Estimation for Multiuser MIMO Channels

In this paper, we propose a joint pilot design and channel estimation scheme based on the deep learning (DL) technique for multiuser multiple-input multiple output (MIMO) channels. To this end, we construct a pilot designer using two-layer neural networks (TNNs) and a channel estimator using deep neural networks (DNNs), which are jointly trained to minimize the mean square error (MSE) of channel estimation. To effectively reduce the interference among the multiple users, we also use the successive interference cancellation (SIC) technique in the channel estimation process. The numerical results demonstrate that the proposed scheme considerably outperforms the state-of-the-art linear minimum mean square error (LMMSE) based channel estimation scheme.

Spectral image recovery from Spectral Filter Array cameras using LMMSE

A hyperspectral camera can record a cube of data with both spatial 2D and spectral 1D dimensions. Spectral Filter Arrays (SFAs) overlaid on a single sensor allows a snapshot version of a hyperspectral camera. But acquired image is subsampled both spatially and spectrally, and a recovery method should be applied. In this paper we present a linear model of spectral and spatial recovery based on Linear Minimum Mean Square Error (LMMSE) approach. The method learns a stable linear solution for which redundancy is controlled using spatial neighborhood. We evaluate results in simulation using gaussian shaped filter's sensitivities on SFA mosaics of upto 9 filters with sensitivities both in visible and Near-Infrared (NIR) wavelength. We show by experiment that by using big neighborhood sizes in our model we can accurately recover the spectra from the RAW images taken by such a camera. We also present results on recovered spectra of Macbeth color chart from a Bayer SFA having 3 filters.

Estimation Theoretic Optimal Encoding Design for Secure Transmission of Multiple Parameters

In this paper, optimal deterministic encoding of a vector parameter is investigated in the presence of an eavesdropper. The objective is to minimize the expectation of the conditional Cramer–Rao bound at the intended receiver, while satisfying an individual secrecy constraint on the mean-squared error of estimating each parameter at the eavesdropper. The eavesdropper is modeled to employ the linear minimum mean-squared error estimator based on the noisy observation of the encoded parameter without being aware of encoding. First, the problem is formulated as a constrained optimization problem in the space of vector-valued functions. Then, two practical solution strategies are developed based on nonlinear individual encoding and affine joint encoding of parameters. Theoretical results on the solutions of the proposed strategies are provided for various scenarios on channel conditions and parameter distributions. Finally, numerical examples are presented to illustrate the performance of the proposed solution approaches.

Data Ridden Pilots Based Estimation of Sparse Multipath Massive MIMO Channels Using Orthogonal Matching Pursuit

Massive Multiple-Input Multiple-Output (MIMO) is envisioned to be a strong candidate technology for the upcoming 5th generation (5G) of wireless communication networks. This research work presents a novel Compressed Sensing (CS) and Superimposed Training (SiT) based technique for estimating the sparse uplink channels in massive MIMO systems. The proposed technique involves arithmetic addition of a periodic, but low powered training sequence with each user’s information sequence. Consequently, separately dedicated resources for the pilot symbols are not needed. Moreover, to attain the estimates of the Channel State Information (CSI) in the uplink, the sparsity exhibited by the MIMO channels is exploited by incorporating CS based Orthogonal Matching Pursuit (OMP) algorithm. For decoding the transmitted information symbols of each user, a Linear Minimum Mean Square Error (LMMSE) based equalizer is incorporated at the receiving Base Station (BS). Based on the obtained simulation results, the proposed SiT-OMP technique outperforms the existing Least Squares (SiT) channel estimation technique. The comparison is done using performance metrics of the Bit Error Rate (BER) and the Normalized Channel Mean Square Error (NCMSE).

Improve computational efficiency and estimation accuracy of multi-channel surface EMG decomposition via dimensionality reduction

Electromyography (EMG) decomposition serves as a powerful tool to provide valuable information regarding motor control strategy and muscle pathology. Various approaches have recently been developed to decompose surface electromyography (sEMG) signals into motor unit action potential (MUAP) trains, but there is still room to improve decomposition accuracy and efficiency. In this article, the singular value decomposition (SVD) method is employed to decompose extended multi-channel sEMG signals based on the linear minimum mean square error (LMMSE) estimation and convolution kernel compensation (CKC). The column dimension of the right unitary matrix obtained via SVD is compressed to reduce the run-time of the decomposition process. As such, an innervation pulse train (IPT) can be extracted only using the compressed right unitary matrix. We studied the effect of reduced dimensionality on the computational efficiency and accuracy of the proposed method. Results from both simulated and experimental data demonstrate that this algorithm can decompose multi-channel sEMG efficiently and accurately. In most cases, 20%–60% of run-time can be reduced. In addition, the number of extracted IPTs can also be increased, especially when the signal-to-noise ratio of the EMG data is low. The method developed in this study has potential applications in clinical diagnosis, rehabilitation engineering and research on human motion control systems.

Low-complexity joint channel estimation and symbol detection for OFDMA systems

In this paper, we propose a joint channel estimation and symbol detection (JCESD) algorithm relying on message-passing algorithms (MPA) for orthogonal frequency division multiple access (OFDMA) systems. The channel estimation and symbol detection leverage the framework of expectation propagation (EP) and belief propagation (BP) with the aid of Gaussian approximation, respectively. Furthermore, to reduce the computation complexity involved in channel estimation, the matrix inversion is transformed into a series of diagonal matrix inversions through the Sherman-Morrison formula. Simulation experiments show that the proposed algorithm can reduce the pilot overhead by about 50%, compared with the traditional linear minimum mean square error (LMMSE) algorithm, and can approach to the bit error rate (BER) performance bound of perfectly known channel state information within 0.1 dB.