An EM algorithm for linear distortion channel estimation based on observations from a mixture of Gaussian sources

This paper discusses the problem of Non Data Aided (NDA) Signal to Noise Ratio (SNR) estimation of Binary Phase Shift keying (BPSK) modulated signals using the Expectation Maximization (EM) Algorithm. In addition, the Cramer-Rao Lower Bounds (CRLB) for the estimation of Data Aided (DA) and Non Data Aided (NDA) Signal to Noise Ratio (SNR) estimation is derived. Multiple Input Single Output (MISO) channels with Space Time Block Codes (STBC) is used. The EM algorithm is a method that finds the Maximum Likelihood (ML) solution iteratively when there are unobserved (hidden or missing) data. Extension of the proposed approach to other types of linearly modulated signals in estimating SNR is straight forward. The performance of the estimator is assessed using the NDA CRLBs. Alamouti coding technique is used in this paper with two transmit antennas and one receive antenna. The authors’ assumption is that the received signal is corrupted by additive white Gaussian noise (AWGN) with unknown variance, and scaled by fixed unknown complex channel gain. Monte Carlo simulations are used to show that the proposed estimator offers a substantial improvement over the conventional Single Input Single Output (SISO) NDA SNR estimator due to the use of the statistical dependences in space and time. Moreover, the proposed NDA SNR estimator works close to the NDA SNR estimator over Single Input Multiple Output (SIMO) channels.

This paper considers the problem of the joint channel and carrier frequency offset (CFO) estimation for asynchronous amplify-and-forward (AF) two-way relay networks (TWRNs). This problem has not been solved to the best of our knowledge, and the presence of timing offsets between the two terminals, in addition to the frequency offsets, makes it necessary to develop signal sampling and joint estimation schemes that explicitly take into account both offsets. As potential solutions, we investigate two estimation strategies. The first strategy is pilot-based estimation, and in this context, we develop the maximum-likelihood joint CFO and channel estimator. Then, we consider the semi-blind estimation strategy, proposing a novel estimator based on expectation maximization (EM). As an alternative to the EM, we also investigate decision-directed (DD) semi-blind estimation. The semi-blind Cramer-Rao bound (CRB) is also obtained as a benchmark. Extensive experiments are used to compare the different estimation methods, showing that the proposed EM algorithm yields significantly lower mean-squared error than both the pilot-based estimator and the DD estimator and converges to the obtained CRB. It also yields much better symbol error rates. This is achieved at a reasonable computational complexity, and only a limited number of EM iterations are sufficient for convergence. The main contribution of our work is the development of the EM estimator, the corresponding CRB, and the use of simulations to demonstrate the superior performance of the proposed EM algorithm compared to other potential solutions.

Blind separation of non-stationary sources using continuous density hidden Markov models

The expectation maximization (EM) algorithm has received considerable attention in the area of positron emitted tomography (PET) as a restoration and reconstruction technique. In this paper, the restoration capabilities of the EM algorithm when applied to radiographic images is investigated. This application does not involve reconstruction. The performance of the EM algorithm is quantitatively evaluated using a "perceived" signal-to-noise ratio (SNR) as the image quality metric. This perceived SNR is based on statistical decision theory and includes both the observer's visual response function and a noise component internal to the eye-brain system. For a variety of processing parameters, the relative SNR (ratio of the processed SNR to the original SNR) is calculated and used as a metric to compare quantitatively the effects of the EM algorithm with two other image enhancement techniques: global contrast enhancement (windowing) and unsharp mask filtering. The results suggest that the EM algorithm's performance is superior when compared to unsharp mask filtering and global contrast enhancement for radiographic images which contain objects smaller than 4 mm.

Low complexity generalized EM algorithm for blind channel estimation and data detection in optical communication systems

This paper focuses on multi-target parameter estimation of multiple-input multiple-output (MIMO) radar with widely separated antennas on moving platforms. Aiming at the superimposed signals caused by multi-targets, the well-known expectation maximization (EM) is used in this paper. Target’s radar cross-section (RCS) spatial variations, different path losses and spatially-non-white noise appear because of the widely separated antennas. These variables are collectively referred to as signal-to-noise ratio (SNR) fluctuations. To estimate the echo delay/Doppler shift and SNR, the Q function of EM algorithm is extended. In addition, to reduce the computational complexity of EM algorithm, the gradient descent is used in M-step of EM algorithm. The modified EM algorithm is called generalized adaptive EM (GAEM) algorithm. Then, a weighted iterative least squares (WILS) algorithm is used to jointly estimate the target positions and velocities based on the results of GAEM algorithm. This paper also derives the Cramér-Rao bound (CRB) in such a non-ideal environment. Finally, extensive numerical simulations are carried out to validate the effectiveness of the proposed algorithm.

Design and analysis of direction of arrival using hybrid expectation-maximization and MUSIC for wireless communication

In the reconstruction surgery of Anterior Cruciate Ligament (ACL), how to locate the anatomical position is a very hard point to clinician occupational therapists. In this paper, we propose an Anatomical Position Locating (APL) approach based on Expectation Maximization (EM) algorithm. Firstly, the proposed intersection set operation algorithm is proposed to compute the attachment region between the injured ACL and femur or tibia. Then, the anatomical position is located by the 3D points cloud with the Gaussian spatial distribution. The last, the attachment spatial distribution and the barycenter, which are also viewed as the candidates of the anatomical position by a proposed EM algorithm, is partition. Experimental results verify our assumption and demonstrate that the located anatomical position has great serviceability.

In this paper, a computationally efficient algorithm is presented for joint maximum likelihood (ML) timing and carrier phase estimation of OFDM systems employing M-PSK modulation scheme with additive Gaussian noise, based on the expectation-maximization (EM) algorithm. A nondata-aided (NDA) scheme is considered for the joint timing and phase synchronizer which maximizes the low SNR limit of the likelihood function averaged over the M-PSK signal constellation. For this, an EM algorithm is derived which estimates the timing offset and the phase rotations of each subcarrier iteratively and which converges to the true ML estimation of the unknown timing and phase. It is shown that the algorithm becomes independent of the signal-to-noise ratio for both low and high SNR cases. The algorithm is applied to the QPSK modulated OFDM systems and it is concluded that for SNR values greater than 10 dB the convergence is achieved in first iteration and for SNR values less than 10 dB, at most in three iterations. It is also concluded that the convergence is independent of the initial starting points.

A novel multipath mitigation technique for Global Positioning System (GPS) receivers using the Expectation-Maximization (EM) algorithm is proposed. It is well-known that conventional propagation delay estimation using parallel sliding correlators is only optimal in additive white Gaussian noise channel. In practical positioning systems, the weak GPS line-of-sight signal is generally embedded in the multipath signals and other source of interference. Although the GPS direct sequence spread spectrum (DS-SS) signal has inherent resistance to interference, the received superimposed multipath signals, which are possibly coherent, are the dominant source of the propagation delay estimation errors. From the parameter estimation point of view, the problem of multipath mitigation is equivalent to estimating the unknown phases, propagation delays and amplitudes of the superimposed multipath signals. The joint maximum likelihood (ML) estimation of all the unknown parameters is optimal and asymptotically efficient. However it involves multi-dimensional search which is computationally expensive. The proposed coarse/acquisition (C/A) code acquisition system using the EM algorithm is an iterative maximum likelihood estimator which decomposes the multi-parameter estimation problem into a number of separate ML optimizations. The performance of the proposed EM algorithm has been tested by simulations. We have observed that the proposed acquisition system is significantly superior to the conventional correlating receiver in a multipath fading channel.

In this paper, we address the problem of non-data-aided SNR estimation in wireless SIMO channels. We derive the per-antenna ML SNR estimator using the expectation-maximization (EM) algorithm under constant channels and additive white Gaussian noise (AWGN). The new method is valid for any arbitrary constellation. It is NDA and, therefore, does not impinge on the hole throughput of the system. We obtain two non linear vector equations which are tackled by a less complex approach based on the EM algorithm. The noise components are assumed to be spatially uncorrelated over all the antenna elements and temporally white with equal power. Besides, in order to evaluate our EM-ML SNR estimator, we derive the Cramer-Rao lower bound (CRLB) in the DA case. Monte Carlo simulations show, that our new estimator offers, a substantial performance improvement over the SISO ML SNR estimator due to the optimal usage of the mutual information between the antenna branches, and that it reaches the derived DA CRLBs. To the best of our knowledge, we are the first to derive the ML per-antenna SNR estimators as well as the CRLBs in the NDA and the DA case, respectively, both over SIMO channels.

Advanced statistical methods have enabled trial-by-trial inference of the underlying excitatory and inhibitory synaptic conductances (SCs) of membrane-potential recordings. Simultaneous inference of both excitatory and inhibitory SCs sheds light on the neural circuits underlying the neural activity and advances our understanding of neural information processing. Conventional Bayesian methods can infer excitatory and inhibitory SCs based on a single trial of observed membrane potential. However, if multiple recorded trials are available, this typically leads to suboptimal estimation because they neglect common statistics (of synaptic inputs (SIs)) across trials. Here, we establish a new expectation maximization (EM) algorithm that improves such single-trial Bayesian methods by exploiting multiple recorded trials to extract common SI statistics across the trials. In this paper, the proposed EM algorithm is embedded in parallel Kalman filters or particle filters for multiple recorded trials to integrate their outputs to iteratively update the common SI statistics. These statistics are then used to infer the excitatory and inhibitory SCs of individual trials. We demonstrate the superior performance of multiple-trial Kalman filtering (MtKF) and particle filtering (MtPF) relative to that of the corresponding single-trial methods. While relative estimation error of excitatory and inhibitory SCs is known to depend on the level of current injection into a cell, our numerical simulations using MtKF show that both excitatory and inhibitory SCs are reliably inferred using an optimal level of current injection. Finally, we validate the robustness and applicability of our technique through simulation studies, and we apply MtKF to in vivo data recorded from the rat barrel cortex.

Introduction: In real-world datasets, outliers are a common occurrence that can have a significant impact on the accuracy and reliability of statistical analyses. Detecting these outliers and developing robust models to handle their presence is a crucial challenge in data analysis. For instance, natural images may have complex distributions of values due to environmental factors like noise and illumination, resulting in objects with overlapping regions and non-trivial contours that cannot be accurately described by Gaussian mixture models. In many real life applications, observed data always fall in bounded support regions. This leads to the idea of bounded support mixture models. Motivated by the aforementioned observations, we introduce a bounded multivariate cntaminated normal distribution for fitting data with non-Gaussian distributions, asymmetry, and bounded support which makes finite mixture models more robust to fitting, since rare observations are given less importance in calculations. Methods: A family of finite mixtures of bounded multivariate contaminated normal distributions is introduced. The model is well-suited for computer vision and pattern recognition problems due to its heavily-tailed and bounded nature, providing flexibility in modeling data in the presence of outliers. A feasible expectation- maximization algorithm is developed to compute the maximum likelihood estimates of the model parameters using a selection mechanism. Results: The proposed methodology is validated by conducting experiments on both simulated data and two real natural skin cancer images. We estimate the parameters by the proposed expectation-maximization algorithm. The obtained results shown that the proposed model has successfully enhanced accuracy in segmenting skin lesions. Conclusion: The reliable model-based clustering using finite mixtures of bounded multivariate contaminated normal distributions is introduced. An expectation-maximization algorithm was created to estimate parameters, with closed-form expressions utilized at the E-step. Practical tests on images for skin cancer detection showed enhanced accuracy in delineating skin lesions.

In this article, we present two clustered sparse Bayesian learning (Cluster-SBL) channel estimation algorithms for millimeter-wave (mmWave) massive multiple-input-multiple-output (MIMO) systems. Different from prior literature, channel path correlations at both the transmitter and the receiver are considered and exploited for estimation performance enhancement. Specifically, we first propose a Kronecker product-based multivariate Gaussian (KP-Gaussian) prior model for mmWave massive MIMO channels, which captures not only sparsity but also the widely existing clustering channel property in mmWave communications. Then, a channel approximation method is derived and leveraged to overcome the challenge caused by channel priors' uncertainty due to the varying propagation environments and improve the robustness of the channel estimation algorithm against channel sparsity. After that, the Cluster-SBL channel estimation algorithm, which is developed based on the expectation maximization (EM) framework and has superior estimation accuracy and high robustness, is proposed. Moreover, to reduce the computation complexity in the channel estimation for practical applications, we also propose an efficient Cluster-SBL (eCluster-SBL) channel estimation algorithm. It is highly beneficial to mmWave massive MIMO systems, especially when the angular resolution is relatively high. Numerical results reveal that, compared with the existing compressed sensing-based and Bayesian criterion-based algorithms, the proposed two channel estimation algorithms exhibit the better performance from both the aspect of estimation accuracy and the aspect of robustness.

We have developed a three-dimensional (3-D) reconstruction procedure for positron emission tomography (PET) using stochastic sampling techniques-the SS3D algorithm. A forward- and back-projector pair has been developed that eliminates the need to explicitly calculate the system response matrix (SRM). This is advantageous as it eliminates the memory requirements for storing this matrix. As compared to on-the-fly methods, our method is more accurate in its assumptions and can readily incorporate various physical effects of attenuation, scatter, and detector characteristics. This method can be efficiently used with simultaneous or row action algorithms like image space reconstruction algorithm (ISRA), expectation maximization (EM), and ordered subsets expectation maximization (OSEM). We demonstrate the successful implementation of this technique with the maximum likelihood expectation maximization (ML-EM), maximum a posteriori expectation maximization (MAP-EM), and maximum a posteriori ordered subsets expectation maximization (MAP-OS-EM) algorithms by evaluating the images on the basis of uniformity of noise, contrast recovery, signal-to-noise ratio (SNR), point-spread function, structural accuracy, and computational time.