Classical Minimum Mean Square Error Research Articles

Deep Learning OFDM Receivers for Improved Power Efficiency and Coverage

In this article, we propose multiple machine learning (ML) based physical-layer receiver solutions for demodulating orthogonal frequency-division multiplexing (OFDM) signals that are subject to high level of nonlinear distortion. Specifically, three novel deep learning based convolutional neural network receivers are devised, containing layers in time- and/or frequency-domains, allowing to demodulate and decode the transmitted bits reliably despite the high error vector magnitude (EVM) in the transmit signal. Applicable training procedures are also described, such that the learned layers in the receiver processing properly generalize over different nonlinear distortion and multipath channel characteristics. Extensive set of numerical results is provided, in the context of 5G NR uplink (UL) incorporating also measured terminal power amplifier (PA) characteristics. The obtained results show that the proposed receiver systems are able to clearly outperform the classical linear minimum mean-squared error (LMMSE) receiver as well as the existing ML receiver approaches, especially when the EVM is high compared to modulation order. This is particularly so when the devised ML receiver is of hybrid nature with layers both in time and frequency. The proposed ML receivers can thus facilitate pushing the terminal PA systems deeper into saturation, and thereon improve the terminal power-efficiency, radiated power and network coverage. Through combining the obtained radio link performance results with link budget calculations, all carried out at the 28 GHz mmWave band, it is shown that the proposed ML receivers can enhance the network coverage in terms of maximum UL link distances by close to 100%, when compared to classical LMMSE receiver based networks.

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

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

A Low-Complexity Equalizer for Video Broadcasting in Cyber-Physical Social Systems Through Handheld Mobile Devices

In Digital Video Broadcasting-Handheld (DVB-H) devices for cyber-physical social systems, the Discrete Fractional Fourier Transform-Orthogonal Chirp Division Multiplexing (DFrFT-OCDM) has been suggested to enhance the performance over Orthogonal Frequency Division Multiplexing (OFDM) systems under time and frequency-selective fading channels. In this case, the need for equalizers like the Minimum Mean Square Error (MMSE) and Zero-Forcing (ZF) arises, though it is excessively complex due to the need for a matrix inversion, especially for DVB-H extensive symbol lengths. In this work, a low complexity equalizer, Least-Squares Minimal Residual (LSMR) algorithm, is used to solve the matrix inversion iteratively. The paper proposes the LSMR algorithm for linear and nonlinear equalizers with the simulation results, which indicate that the proposed equalizer has significant performance and reduced complexity over the classical MMSE equalizer and other low complexity equalizers, in time and frequency-selective fading channels.

Analysis of Detection Methods in Massive MIMO Systems

Large-scale Multiple Input Multiple Output (MIMO) system is significant in our daily life such as used in hundreds of antennas at the base station work together at the same time. Matrix inversion can be used to solve this problem and calculate the result of formulas in this system, but this method is difficult for some people, and it may easy to make mistakes. In this paper uses some low-complexity signal detection algorithm, which are the Gauss-Seidel (GS) method, successive over relaxation (SOR) method, Minimum mean square error (MMSE) signal detection, in order to avoid the complicated matrix inversion. Can also use a systolic array and show reference FPGA implementation results for various system configurations. Steepest descent algorithm is employed to obtain an efficient searching direction for the following Jacobi iteration to speed up convergence is also a method to solve this problem. Firstly, we have to prove a special property that the MMSE filtering matrix. Then, we prove the convergence of a low-complexity iterative signal detection algorithm. The result shows that the method can reduce the computational complexity from K3 to K2 (K is the number of users). Finally, the result of the experiment prove that the method is more advantage than the recently proposed Neumann series approximation algorithm, and achieved the near-optimal performance of the classical MMSE algorithm.

A low-complexity signal detection utilizing AOR iterative method for massive MIMO systems

Massive multiple-input multiple-output (MIMO) system is capable of substantially improving the spectral efficiency as well as the capacity of wireless networks relying on equipping a large number of antenna elements at the base stations. However, the excessively high computational complexity of the signal detection in massive MIMO systems imposes a significant challenge for practical hardware implementations. In this paper, we propose a novel minimum mean square error (MMSE) signal detection using the accelerated overrelaxation (AOR) iterative method without complicated matrix inversion, which is capable of reducing the overall complexity of the classical MMSE algorithm by an order of magnitude. Simulation results show that the proposed AOR-based method can approach the conventional MMSE signal detection with significant complexity reduction.

Time-Domain Turbo Equalization for Single-Carrier Generalized Spatial Modulation

In this paper, low-complexity time-domain turbo equalization (TDTE) detectors based upon soft-interference-cancellation (SIC)-aided minimum mean-square error (MMSE) criterion are proposed for single carrier generalized spatial modulation (SC-GSM) systems. First, a symbol-by-symbol-aided TDTE detector for application to the small-scale GSM systems is proposed, where the zero symbols are considered as constellation points when performing SIC. Then, vector-by-vector-aided TDTE (VV-TDTE) detectors for application to larger-scale antenna systems are introduced, where the GSM symbol is treated as an entire vector when performing SIC. As for the proposed VV-TDTE detectors, in addition, different time-varying filter coefficients are designed, in order to strike a flexible tradeoff between complexity and performance. By relying upon extrinsic information transfer chart analysis, we show that the proposed TDTE detectors are capable of providing considerable bit error rate performance gains over existing TDTE detectors and over the classic frequency-domain equalization-based MMSE detector, especially for the unbalanced antenna configurations.

Low-Complexity Soft-Output Signal Detection Based on Gauss–Seidel Method for Uplink Multiuser Large-Scale MIMO Systems

For uplink large-scale multiple-input–multiple-output (MIMO) systems, the minimum mean square error (MMSE) algorithm is near optimal but involves matrix inversion with high complexity. In this paper, we propose to exploit the Gauss–Seidel (GS) method to iteratively realize the MMSE algorithm without the complicated matrix inversion. To further accelerate the convergence rate and reduce the complexity, we propose a diagonal-approximate initial solution to the GS method, which is much closer to the final solution than the traditional zero-vector initial solution. We also propose an approximated method to compute log-likelihood ratios for soft channel decoding with a negligible performance loss. The analysis shows that the proposed GS-based algorithm can reduce the computational complexity from ${{\mathcal O}({K^3})}$ to ${{\mathcal O}({K^2})}$ , where ${K}$ is the number of users. Simulation results verify that the proposed algorithm outperforms the recently proposed Neumann series approximation algorithm and achieves the near-optimal performance of the classical MMSE algorithm with a small number of iterations.

Low-Complexity Signal Detection for Large-Scale MIMO in Optical Wireless Communications

Optical wireless communication (OWC) has been a rapidly growing research area in recent years. Applying multiple-input multiple-output (MIMO), particularly large-scale MIMO, into OWC is very promising to substantially increase spectrum efficiency. However, one challenging problem to realize such an attractive goal is the practical signal detection algorithm for optical MIMO systems, whereby the linear signal detection algorithm like minimum mean square error (MMSE) can achieve satisfying performance but involves complicated matrix inversion of large size. In this paper, we first prove a special property that the filtering matrix of the linear MMSE algorithm is symmetric positive definite for indoor optical MIMO systems. Based on this property, a low-complexity signal detection algorithm based on the successive overrelaxation (SOR) method is proposed to reduce the overall complexity by one order of magnitude with a negligible performance loss. The performance guarantee of the proposed SOR-based algorithm is analyzed from the following three aspects. First, we prove that the SOR-based algorithm is convergent for indoor large-scale optical MIMO systems. Second, we prove that the SOR-based algorithm with the optimal relaxation parameter can achieve a faster convergence rate than the recently proposed Neumann-based algorithm. Finally, a simple quantified relaxation parameter, which is independent of the receiver location and signal-to-noise ratio, is proposed to guarantee the performance of the SOR-based algorithm in practice. Simulation results verify that the proposed SOR-based algorithm can achieve the exact performance of the classical MMSE algorithm with a small number of iterations.

Low‐complexity near‐optimal signal detection for uplink large‐scale MIMO systems

The minimum mean square error (MMSE) signal detection algorithm is near-optimal for uplink multi-user large-scale multiple-input-multiple-output (MIMO) systems, but involves matrix inversion with high complexity. It is firstly proved that the MMSE filtering matrix for large-scale MIMO is symmetric positive definite, based on which a low-complexity near-optimal signal detection algorithm by exploiting the Richardson method to avoid the matrix inversion is proposed. The complexity can be reduced from O(K 3 ) to O(K 2 ), where K is the number of users. The convergence proof of the proposed algorithm is also provided. Simulation results show that the proposed signal detection algorithm converges fast, and achieves the near-optimal performance of the classical MMSE algorithm.

MMSE based map estimation for image denoising

Denoising of a natural image corrupted by the additive white Gaussian noise (AWGN) is a classical problem in image processing. The NeighShrink [17] , [18] , LAWML [19] , BiShrink [20] , [21] , IIDMWT [23] , IAWDMNC [25] , and GIDMNWC [24] denoising algorithms remove the noise from the noisy wavelet coefficients using thresholding by retaining only the large coefficients and setting the remaining to zero. Generally the threshold depends mainly on the variance, image size, and image decomposition levels. The performances of these methods are not very effective as they are not spatially adaptive i.e., the parameters considered are not smoothly varied in the neighborhood window. Our proposed method overcomes this weakness by using minimum mean square error (MMSE) based maximum a posterior (MAP) estimation. In this paper, we modify the parameters such as variance of the classical MMSE estimator in the neighborhood window of the noisy wavelet coefficients to remove the noise effectively. We demonstrate experimentally that our method outperforms the NeighShrink, LAWML, BiShrink, IIDMWT, IAWDMNC, and GIDMNWC methods in terms of the peak signal-to-noise ratio (PSNR) and structural similarity index measure (SSIM). It is more effective particularly for the highly corrupted natural images. • Image denoising method based on minimum mean square error (MMSE) with maximum a posterior (MAP). • Removing the high density additive white Gaussian noise (AWGN). • Propose four different cases for locally estimating the parameters. • Propose method recovers the noise free image from the noisy one.

A practical two-stage MMSE based MIMO detector for interference mitigation with non-cooperative interferers

Wireless Multiple Input Multiple Output systems provide system designers with additional degrees of freedom. These can be used to increase throughput, reliability, or even combat spatial interference. The classical Minimum Mean Squared Error (MMSE) solution is the optimal linear estimator for these systems. Its primary drawback is that it requires an estimate of the channel response. This is generally not an issue when interference is absent. However, in environments where interference power is stronger than the desired signal power, this can become difficult to estimate. The problem is even worse in packet-based systems, which rely on training data to estimate the channel before estimating the signal. A strong interference will hinder the receiver's ability to detect the presence of the packet. This makes it impossible to estimate the channel, a critical component for the classical MMSE estimator. For this reason, the classical solution is infeasible in real environments with stronger interferences. We propose a two-stage system that uses practically obtainable channel state information. We will show how this approach significantly improves packet detection, and how the overall solution approaches the performance of the classical MMSE estimator.

Managing the interference structure of MIMO HSDPA: A multi-user interference aware MMSE receiver with moderate complexity

It is known that Wideband Code-Division Multiple Access (W-CDMA) networks are limited by interference more than by any other single effect. Due to the frequency selectivity of the wireless channel, intra-cell interference composed of selfand other-user interference occurs. The intra-cell interference as a result of Transmit Antenna Array (TxAA) HSDPA multi-user transmissions, however, shows a special structure that can be exploited to allow for an efficient interference suppression. The contributions of this article are the following: we (a) propose a suitable system model to derive an intra-cell interference aware Minimum Mean Squared Error (MMSE) equalizer with moderate complexity and investigate its capabilities to suppress the multi-user intra-cell interference, (b) propose solutions to estimate the pre-coding state of the cell both with and without available training, and (c) investigate the resulting throughput performance with physical layer as well as system-level simulations. Our proposed solution can be interpreted as a multi-user extension of the classical MMSE equalizer for HSDPA systems without decoding the undesired users, resulting in only slightly increased complexity.

The Gaussian Transform of Distributions: Definition, Computation and Application

This paper introduces the general-purpose Gaussian transform of distributions, which aims at representing a generic symmetric distribution as an infinite mixture of Gaussian distributions. We start by the mathematical formulation of the problem and continue with the investigation of the conditions of existence of such a transform. Our analysis leads to the derivation of analytical and numerical tools for the computation of the Gaussian transform, mainly based on the Laplace and Fourier transforms, as well as of the afferent properties set (e.g., the transform of sums of independent variables). The Gaussian transform of distributions is then analytically derived for the Gaussian and Laplacian distributions, and obtained numerically for the generalized Gaussian and the generalized Cauchy distribution families. In order to illustrate the usage of the proposed transform we further show how an infinite mixture of Gaussians model can be used to estimate/denoise non-Gaussian data with linear estimators based on the Wiener filter. The decomposition of the data into Gaussian components is straightforwardly computed with the Gaussian transform, previously derived. The estimation is then based on a two-step procedure: the first step consists of variance estimation, and the second step consists of data estimation through Wiener filtering. To this purpose, we propose new generic variance estimators based on the infinite mixture of Gaussians prior. It is shown that the proposed estimators compare favorably in terms of distortion with the shrinkage denoising technique and that the distortion lower bound under this framework is lower than the classical minimum mean-square error bound

Low-Complexity Equalization of OFDM in Doubly Selective Channels

Orthogonal frequency division multiplexing (OFDM) systems may experience significant inter-carrier 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.

Performance analysis for the improved linear multiuser detectors in bpsk-modulated ds-cdma systems

Recently, a new class of linear multiuser receivers for direct-sequence code-division multiple-access (CDMA) systems employing binary phase-shift keying modulation has been introduced. Unlike classical decorrelating and minimum mean-square error linear multiuser detectors, the new receivers exploit the information contained in the pseudo-autocorrelation of the observables, and are, thus, capable of achieving much better performance. We present new results on the performance analysis of this class of new receivers. In particular, with reference to a CDMA system with deterministic spreading codes, we show that the new receivers outperform the classical ones in terms of both error probability and near-far resistance. With regard, instead, to CDMA systems with random spreading, we compute the average system near-far resistance, showing that the new receivers can accommodate twice the number of users accommodated by the classical linear multiuser receivers.

A generalized minimum-mean-output-energy strategy for CDMA systems with improper MAI

It has been shown that in a direct-sequence/code-division multiple-access (DS/CDMA) system employing binary phase-shift keying (BPSK) modulation the baseband equivalent of the CDMA multiplex is, under very mild assumptions, an improper complex random process, i.e., it has a nonzero pseudoautocorrelation function. The problem of linear multiuser detection for asynchronous DS/CDMA systems with improper multiaccess interference (MAI) is considered. A new mean-output-energy (MOE) cost function is introduced, whose constrained minimization leads to two new linear multiuser detectors, exploiting the information contained in the pseudoautocorrelation of the observables, and which generalize the classical decorrelating and minimum mean-square error (MMSE) receivers. The problem of blind adaptive receiver implementation based on subspace tracking is also tackled. Finally, the superiority of the new detectors with respect to the classical linear detection structures present in the literature is demonstrated through both theoretical considerations and computer simulations.

The use of block truncation coding in DPCM image coding

The results of using block truncation coding (BTC) in predictive differential pulse code modulation (DPCM) coding of images are presented. BTC is based on the use of a moment-preserving (MP) quantizer that is designed such that the quantizer preserves statistical moments of the input and output. It is shown that while it is theoretically impossible to preserve moments in the reconstructed image, as is normally done in BTC, the DPCM/BTC system works quite well at data rates of 1.18 b/pixel. The performance of the DPCM/BTC system is compared to classical minimum mean-square error quantizers. These methods are also compared in the presence of channel errors. >