Bayesian Perspective for Orientation Determination in Cryo-EM with Application to Structural Heterogeneity Analysis

Minimum mean square error estimation in linear regression

A Bayesian estimation approach for enhancing speech signals which have been degraded by statistically independent additive noise is motivated and developed. In particular, minimum mean square error (MMSE) and maximum a posteriori (MAP) signal estimators are developed using hidden Markov models (HMMs) for the clean signal and the noise process. It is shown that the MMSE estimator comprises a weighted sum of conditional mean estimators for the composite states of the noisy signal, where the weights equal the posterior probabilities of the composite states given the noisy signal. The estimation of several spectral functionals of the clean signal such as the sample spectrum and the complex exponential of the phase is also considered. A gain-adapted MAP estimator is developed using the expectation-maximization algorithm. The theoretical performance of the MMSE estimator is discussed, and convergence of the MAP estimator is proved. Both the MMSE and MAP estimators are tested in enhancing speech signals degraded by white Gaussian noise at input signal-to-noise ratios of from 5 to 20 dB. >

In this paper we investigate the block- type pilot channel estimation for orthogonal frequency division multiplexing (OFDM) systems. The estimation is based on the minimum mean square error (MMSE) estimator and the least square (LS) estimator. We derive the MMSE and LS estimators’ architecture and investigate their performances. We prove that the MMSE estimator performance is better but computational complexity is high, contrary the LS estimator has low complexity but poor performance. For reducing complexity we proposed two different solutions which are the Simplified Least Square (SLS) estimator and the modified MMSE estimator. In the SLS estimator, we apply an auto-correlation function with the LS estimator to remove the noise. In the modified MMSE estimator, we consider only the significant energy samples and ignore the remaining noisy samples. Based on this idea we introduce the modified MMSE estimator. We evaluate estimator’s performance on basis of mean square error and symbol error rate for 16 QAM systems using MATLAB.

An antenna array on the transmit side provides the system with an extra spatial dimension that can be utilized for coding both in the spatial as well as the temporal domain. The recent development of such space time codes shows that there are ways of exploiting multiple transmit antennas while completely avoiding traditional beam-forming techniques need of accurate channel state information. In this project, we generate a framework for training based channel estimation under different channel and interference statistics. The minimum mean square error (MMSE) estimator for channel matrix estimation in Rician fading multi-antenna systems is analyzed, and exclusively the proposal of mean square error (MSE) minimizing training structures. By considering Kronecker-structured systems with a grouping of noise and interference and random training sequence length, we gather and simplify numerous earlier results in the framework. We simplify the circumstances for attaining the optimal training sequence structure and show when the spatial training power distribution can be explained unambiguously. We also prove that spatial correlation improves the estimation performance and establish how it determines the optimal training sequence length. The analytic results for Kronecker-structured systems are used to derive a heuristic training sequence under general unstructured statistics. The MMSE estimator of the squared Frobenius norm of the channel matrix is also derived and shown to provide far better gain estimates than other approaches. It is shown under which conditions training sequences that minimize the non-convex MSE can be derived explicitly or with low complexity. Numerical examples are used to evaluate the performance of the two estimators for different training classifications and system statistics. We also elucidate how the finest length of the training sequence often can be shorter than the number of transmit antennas. Keyword: mse, mmse estimator, kronecker-structured systems.

An antenna array on the transmit side provides the system with an extra spatial dimension that can be utilized for coding both in the spatial as well as the temporal domain.The recent development of such space time codes shows that there are ways of exploiting multiple transmit antennas while completely avoiding traditional beam-forming techniques need of accurate channel state information.In this project, we generate a framework for training based channel estimation under different channel and interference statistics.The minimum mean square error (MMSE) estimator for channel matrix estimation in Rician fading multi-antenna systems is analyzed, and exclusively the proposal of mean square error (MSE) minimizing training structures.By considering Kronecker-structured systems with a grouping of noise and interference and random training sequence length, we gather and simplify numerous earlier results in the framework.We simplify the circumstances for attaining the optimal training sequence structure and show when the spatial training power distribution can be explained unambiguously.We also prove that spatial correlation improves the estimation performance and establish how it determines the optimal training sequence length.The analytic results for Kronecker-structured systems are used to derive a heuristic training sequence under general unstructured statistics.The MMSE estimator of the squared Frobenius norm of the channel matrix is also derived and shown to provide far better gain estimates than other approaches.It is shown under which conditions training sequences that minimize the non-convex MSE can be derived explicitly or with low complexity.Numerical examples are used to evaluate the performance of the two estimators for different training classifications and system statistics.We also elucidate how the finest length of the training sequence often can be shorter than the number of transmit antennas.

This paper proposes a new precoding approach that utilizes two-dimension (2D) Angle of Arrival (AOA) based precoding (beamforming) approach for multi-cell massive Multiple Input Multiple Output (m-MIMO) systems. The proposed approach enhances the performance of m-MIMO systems by overcoming pilot contamination that is caused by corrupted Channel State Information. It exploits the spatial properties of the propagated signals to distinguish between various users in the multi-cell m-MIMO system and selects the partial space for beamforming based on AOA information to suppress inter-cell and intra-cell interference. We mathematically analyze the impact of pilot contamination in the uplink training and determine the closed-form approximation of uplink Minimum Mean Square Error (MMSE) estimator. By using MMSE estimator as well as 2D AOA and channel estimation, a corresponding MMSE precoding which minimizes the effect of pilot contamination and improves the downlink achievable rate of the system is proposed. Furthermore, a comparison is made of the resolution between the various 2D AOA estimations for an m-MIMO and the optimum values closest bound for the proposed precoding was chosen to reduce pilot contamination. The performance of the proposed MMSE precoding based on 2D AOA estimation methods: 2D Unitary Estimation of Signal Parameters via Rotational Invariance Techniques (2D UESPRIT), 2D Fourier Domain Line Search MUSIC (2D FDLSM), and 2D Propagator Method (2D PM)) were evaluated and compared with pilot contaminated system and deterministic MMSE precoding in terms of their respective achievable rate. Simulation results reveal that the achievable sum rate gains of 2D UESPRIT, 2D FDLSM, and 2D PM based precoding which achieved 97.6%, 92.4%, 86.3% of the desired deterministic performance, respectively, compared to pilot contaminated system, which achieved only 90% even when in the best case of having only one contaminated user out of all users within the activated cell. Findings from our research indicate the feasibility of utilizing the AOA based beamforming to reduce pilot contamination effect that is inherent in a larger order m-MIMO system for the Fifth Generation (5G) wireless transmissions.

In this paper we present an optimal estimator of magnitude spectrum for speech enhancement when the clean speech DFT coefficients are modeled by a Laplacian distribution and the noise DFT coefficients are modeled by a Gaussian distribution. Chen has already introduced a Minimum Mean Square Error (MMSE) estimator of the magnitude spectrum. However, the proposed estimator, namely LapMMSE, does not have a closed form and is computationally extensive. We use his formulation for the MMSE estimator, employ some approximations and propose a computationally effective estimator for the magnitude spectrum. Experimental studies demonstrate better performance of our proposed estimator, Improved LapMMSE (ImpLapMMSE) Compared to LapMMSE and previous estimators in which Laplacian and Gaussian assumptions were made.

Human body orientation estimation from microinertial/magnetic sensor units is highly important for synthetic environments, robotics, and other human-computer interaction applications. In practice, the main challenge is how to deal with linear acceleration interference and magnetic disturbance which always cause significant attitude-estimation errors. In this paper, we present a novel quaternion-based Kalman filter with vector selection scheme for accurate human body orientation estimation using an inertial/magnetic sensor unit. In the proposed algorithm, the gyroscope measurement is used as an input to construct the linear process equation, and the accelerometer and magnetometer measurements are manipulated to establish the linear pseudomeasurement equation. A linear Kalman filter is then deployed to estimate the body orientation. In the Kalman filter framework, a vector selection scheme is designed to protect the algorithm against undesirable conditions such as temporary intensive movement and magnetic disturbance and enable it to acquire more accurate orientation estimation. The experimental results have shown that the proposed algorithm can provide accurate attitude estimations with regard to the ground truth.

This paper considers multi-cell massive multiple-input multiple-output systems, where the channels are spatially correlated Rician fading. The channel model is composed of a deterministic line-of-sight path and a stochastic non-line-of-sight component describing a practical spatially correlated multipath environment. We derive the statistical properties of the minimum mean squared error (MMSE), element-wise MMSE, and least-square channel estimates for this model. Using these estimates for maximum ratio combining and precoding, rigorous closed-form uplink (UL) and downlink (DL) achievable spectral efficiency (SE) expressions are derived and analyzed. The asymptotic SE behavior, when using the different channel estimators, are also analyzed. The numerical results show that the SE is higher when using the MMSE estimator than that of the other estimators, and the performance gap increases with the number of antennas.

In recent years, multiple-input multiple-output (MIMO) systems, which use several transmit and receive antennas, have attracted much attention for high performance radio systems. In MIMO systems, the channel estimation is important to distinguish transmit signals from multiple transmit antennas. While the space-alternating generalized expectation-maximization (SAGE) algorithm is known to offer good channel estimation and data detection. We proposed earlier the minimum mean square error (MMSE)-based SAGE algorithm for MIMO systems where the MMSE estimation is used for channel estimation. We showed that the proposed MMSE-based SAGE algorithm can achieve the better bit error rate (BER) than the maximum likelihood (ML) detection with training symbols. The MMSE channel estimation needs the knowledge of the maximum Doppler frequency F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sub> for deriving the covariance matrix of the channel and the variance sigma <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> of additive white Gaussian noise (AWGN). Additionally, the computation of the MMSE channel estimation requires O(L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) operations where L is the transmitted frame length. Thus, its computational complexity is high. In this paper, we propose a zero-forcing (ZF)-based SAGE algorithm for channel estimation and data detection in MIMO systems that does not need the knowledge of Fd and sigma <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . Since the computation of the proposed ZF-based SAGE algorithm requires O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) operations where N is the number of transmit antennas, its computational complexity is low. We show that the proposed ZF-based tracking SAGE algorithm with less computational complexity can achieve almost the same BER as that of the MMSE-based tracking SAGE algorithm

Orthogonal frequency division multiplexing (OFDM) systems may experience significant intercarrier interference (ICI) when used in time- and frequency-selective, or doubly-selective, channels. In such cases, the classical symbol estimation schemes, e.g., minimum mean-squared error (MMSE) and zero-forcing (ZF) estimation, require matrix inversion that is prohibitively complex for large symbol lengths. An analysis of the ICI generation mechanism leads us to propose a novel two-stage equalizer whose complexity (apart from the FFT) is linear in the OFDM symbol length. The first stage applies optimal linear preprocessing to restrict ICI support and the second stage uses iterative MMSE estimation to estimate finite-alphabet frequency-domain symbols. Simulation results indicate that our equalizer has significant performance and complexity advantages over the classical linear MMSE estimator in doubly-selective channels.

Generalized mutual information (GMI) is used to compute achievable rates for fading channels with various types of channel state information at the transmitter (CSIT) and receiver (CSIR). The GMI is based on variations of auxiliary channel models with additive white Gaussian noise (AWGN) and circularly-symmetric complex Gaussian inputs. One variation uses reverse channel models with minimum mean square error (MMSE) estimates that give the largest rates but are challenging to optimize. A second variation uses forward channel models with linear MMSE estimates that are easier to optimize. Both model classes are applied to channels where the receiver is unaware of the CSIT and for which adaptive codewords achieve capacity. The forward model inputs are chosen as linear functions of the adaptive codeword's entries to simplify the analysis. For scalar channels, the maximum GMI is then achieved by a conventional codebook, where the amplitude and phase of each channel symbol are modified based on the CSIT. The GMI increases by partitioning the channel output alphabet and using a different auxiliary model for each partition subset. The partitioning also helps to determine the capacity scaling at high and low signal-to-noise ratios. A class of power control policies is described for partial CSIR, including a MMSE policy for full CSIT. Several examples of fading channels with AWGN illustrate the theory, focusing on on-off fading and Rayleigh fading. The capacity results generalize to block fading channels with in-block feedback, including capacity expressions in terms of mutual and directed information.

In many cases, one needs to estimate a target that is obscured by the sidelobes of a very large, adjacent scatterer. For example, the Earth&amp;apos;s surface can ruin nearsurface rain-rate measurements obtained from an airborne platform, or it can prevent a ground-penetrating radar from detecting land mines located at shallow depth. In these cases, the range sidelobes of the ground are larger than the targets near the surface. The range sidelobes cannot be optimally controlled because target information is not used. However, in most situations the location and magnitude of such a large scatterer can be predicted. In this paper, a minimum meansquared error (MMSE) estimator is proposed to improve the range sidelobe problem. The MMSE estimator uses information about the scenario to find the optimum processing filter. Simulated results are provided to demonstrate the effectiveness of this process.

An MMSE approach to nonlocal image denoising: Theory and practical implementation

In this paper, we create a framework for training-based channel estimation under different channel and interference statistics. The minimum mean square error (MMSE) estimator for channel matrix estimation in Rician fading multi-antenna systems is analyzed, and especially the design of mean square error (MSE) minimizing training sequences. By considering Kronecker-structured systems with a combination of noise and interference and arbitrary training sequence length, we collect and generalize several previous results in the framework. We clarify the conditions for achieving the optimal training sequence structure and show when the spatial training power allocation can be solved explicitly. We also prove that spatial correlation improves the estimation performance and establish how it determines the optimal training sequence length. The analytic results for Kronecker-structured systems are used to derive a heuristic training sequence under general unstructured statistics. The MMSE estimator of the squared Frobenius norm of the channel matrix is also derived and shown to provide far better gain estimates than other approaches. It is shown under which conditions training sequences that minimize the non-convex MSE can be derived explicitly or with low complexity. Numerical examples are used to evaluate the performance of the two estimators for different training sequences and system statistics. We also illustrate how the optimal length of the training sequence often can be shorter than the number of transmit antennas.