Minimum Mean Square Error Approach Research Articles

Joint Transmit and Receive Beamforming Design for DPC-Based MIMO DFRC Systems

This paper proposes an optimal beamforming strategy for a downlink multi-user multi-input–multi-output (MIMO) dual-function radar communication (DFRC) system with dirty paper coding (DPC) adopted at the transmitter. We aim to achieve the maximum weighted sum rate of communicating users while adhering to a predetermined transmit covariance constraint for radar performance assurance. To make the intended problem trackable, we leverage the equivalence of the weighted sum rate and the weighted minimum mean squared error (MMSE) to reframe the issue and devise a block coordinate descent (BCD) approach to iteratively calculate transmit and receive beamforming solutions. Through this methodology, we demonstrate that the optimal receive beamforming aligns with the traditional MMSE approach, whereas the optimal transmit beamforming design can be cast into a quadratic optimization problem defined on a complex Stiefel manifold. Based on the majorization–minimization (MM) method, an iterative algorithm is then developed to compute the optimal transmit beamforming design by solving a series of orthogonal Procrustes problems (OPPs) that admit closed-form optimal solutions. Numerical findings serve to validate the efficacy of our scheme. It is demonstrated that our approach can achieve at least 73% higher spectral efficiency than the existing methods in a high signal-to-noise ratio (SNR) regime.

Developing analytical models for the post‐detection SINR of TSVD‐based techniques with imperfect CSI

SummaryIn massive multiple input multiple output (m‐MIMO) uplink, the performance gap between zero forcing (ZF) or direct pseudo‐inverse and minimum mean square error (MMSE) detection schemes increases with an increase in the load factor. To report this issue, truncated singular value decomposition (TSVD) is used in both ZF and direct pseudo‐inverse detection techniques using dynamic threshold. Moreover, the analytical models for the post‐detection signal‐to‐interference‐plus‐noise ratio (PDSINR) of TSVD‐based detection schemes are derived with imperfect channel state information (CSI), and subsequently, bit error rate (BER) is evaluated. Later, an optimum hard threshold for the TSVD‐based detection schemes is deduced from the empirical analysis. Further, the tightness of the analysis is verified through Monte Carlo trails. The simulation result shows that at lower values of average signal‐to‐noise ratio (SNR), TSVD‐based detection schemes with imperfect CSI offer comparable performance and low complexity when compared with the MMSE technique. However, the BER of TSVD‐based detection schemes decreases and gets saturated with an increase in the average SNR, whereas the BER of the MMSE detection scheme increases and approaches the BER of ZF or direct pseudo‐inverse detection schemes.

Image super-resolution with PCA reduced generalized Gaussian mixture models in materials science

Single Image Super-Resolution algorithms based on patches have been noticed and widely used over the past decade. Recently, generalized Gaussian mixture models (GGMMs) have been shown to be a suitable tool for many image processing problems because of the flexible shape parameter. In this work, we introduce a supervised GGMM-based approach for super-resolution of two- and three-dimensional images, in particular materials images. We first propose to use a joint GGMM learned from concatenated vectors of high- and low-resolution training patches. For each low-resolution patch, we compute the minimum mean square error (MMSE) estimator and generate the high-resolution image by averaging these estimates. We select the MMSE approach using GGMM as the method is invariant to affine contrast change and also invariant to a linear super-resolution operator. Unfortunately, the large dimension of the concatenated high- and low-resolution patches leads to instabilities and an intractable computational effort when estimating the parameters of the GGMM. Thus, we propose to combine a GGMM with a principal component analysis and derive an EM algorithm for estimating the parameters of the arising model. We demonstrate the performance of our model by numerical examples on synthetic and real images of material microstructure.

Max-Min Fair Precoder Design and Power Allocation for MU-MIMO NOMA

In this correspondence, transmit precoders and their power allocation coefficients are designed jointly for a downlink non-orthogonal multiple access (NOMA) wireless communication system. The objective is to provide max-min fairness (MMF) among the strongest users in each group, while maintaining minimum target rates for all the other users. The proposed solutions are respectively based on semi-definite relaxation (SDR) and successive convex approximation (SCA) and on the minimum mean square error (MMSE) approaches. For the latter approach, a simplification is also suggested to lower complexity. It is shown that while rate-splitting (RS) has the best MMF rates, the low-complexity MMSE approach has the least complexity. Moreover, the SDR/SCA approach offers an excellent tradeoff between the two.

Spectral distortion level resulting in a just-noticeable difference between an a priori signal-to-noise ratio estimate and its instantaneous case.

Minimum mean-square error (MMSE) approaches to speech enhancement are widely used in the literature. The quality of enhanced speech produced by an MMSE approach is directly impacted by the accuracy of the employed a priori signal-to-noise ratio (SNR) estimator. In this paper, the a priori SNR estimate spectral distortion (SD) level that results in a just-noticeable difference (JND) in the perceived quality of MMSE approach enhanced speech is found. The JND SD level is indicative of the accuracy that an a priori SNR estimator must exceed to have no impact on the perceived quality of MMSE approach enhanced speech. To measure the JND SD level, listening tests are conducted across five SNR levels, five noise sources, and two MMSE approaches [the MMSE short-time spectral amplitude (MMSE-STSA) estimator and the Wiener filter]. A statistical analysis of the results indicates that the JND SD level increases with the SNR level, is higher for the MMSE-STSA estimator, and is not impacted by the type of background noise. Following the literature, a significant improvement in a priori SNR estimation accuracy is required to reach the JND SD level.

LSTM-convolutional-BLSTM encoder-decoder network for minimum mean-square error approach to speech enhancement

In recent years, deep learning models have been employed for speech enhancement. Most of the existing methods based on deep learning use fully Convolutional Neural Network (CNN) to capture time–frequency information of input features. Compared with CNNs, it is more reasonable to use Long Short-Term Memory (LSTM) network to capture contextual information on the time axis of features. However, the computation load of a fully LSTM structure is heavy. To balance the model complexity and the capability of capturing time–frequency features, we present an LSTM-Convolutional-BLSTM Encoder-Decoder (LCLED) network for speech enhancement. The LCLED additionally incorporates transpose convolution and skip connection. The key idea is that we use two LSTM parts and convolutional layers to model the contextual information and frequency dimension features, respectively. Furthermore, in order to achieve a higher quality of enhanced speech, a priori Signal-to-Noise Ratio (SNR) is applied as the learning target of LCLED. The Minimum Mean-Square Error (MMSE) approach is used for postprocessing. The results indicate that the proposed LCLED not only reduces the model complexity and training time but also improves the quality and the intelligibility of enhanced speech compared with the fully LSTM structure.

Implementation of Pilot Channel Estimation Techniques for OFDM System with Low Mean Square Error

Abstract: Orthogonal Frequency Division Multiplexing(OFDM) has grown to be the most popular multi carriercommunication system in high speed data communications.The available bandwidth is utilized efficiently, and more datarates can be achieved because of orthogonal carriers in OFDMsystem. OFDM system requires precise channel estimation andsynchronization to reduce Mean Square Error (MSE). In thiswork, various Pilot Channel Estimation (PCE) techniques areused to compute CSI (Channel State Information). Pilots aretransmitted along with the data and channel response iscomputed with various interpolation techniques such as linear,spline, cubic Interpolation techniques by extracting the pilotsat the receiver. The performance metrics like Mean SquareError (MSE) and computation time are computed andcompared for different PCE techniques. Discrete FourierTransform (DFT) based channel Estimation is combined withinterpolation techniques and channel response is computed andcompared with Least Squares (LS) and Minimum MeanSquare Error (MMSE) approaches. LS, MMSE, LMMSE(Linear MMSE) methods are combined with TimeDivision(TD) and Time Division Duplexing(TDD) techniquesto compute MSE. TD-LMMSE approach achieves least MSEwith more computation time. Index Terms: OFDM, channel estimation, interpolation, CSI,MSE, computation time.

Deep learning for minimum mean-square error approaches to speech enhancement

Recently, the focus of speech enhancement research has shifted from minimum mean-square error (MMSE) approaches, like the MMSE short-time spectral amplitude (MMSE-STSA) estimator, to state-of-the-art masking- and mapping-based deep learning approaches. We aim to bridge the gap between these two differing speech enhancement approaches. Deep learning methods for MMSE approaches are investigated in this work, with the objective of producing intelligible enhanced speech at a high quality. Since the speech enhancement performance of an MMSE approach improves with the accuracy of the used a priori signal-to-noise ratio (SNR) estimator, a residual long short-term memory (ResLSTM) network is utilised here to accurately estimate the a priori SNR. MMSE approaches utilising the ResLSTM a priori SNR estimator are evaluated using subjective and objective measures of speech quality and intelligibility. The tested conditions include real-world non-stationary and coloured noise sources at multiple SNR levels. MMSE approaches utilising the proposed a priori SNR estimator are able to achieve higher enhanced speech quality and intelligibility scores than recent masking- and mapping-based deep learning approaches. The results presented in this work show that the performance of an MMSE approach to speech enhancement significantly increases when utilising deep learning.Availability: The proposed a priori SNR estimator is available at: https://github.com/anicolson/DeepXi.

Large-Scale-Fading Decoding in Cellular Massive MIMO Systems With Spatially Correlated Channels

Massive multiple-input--multiple-output (MIMO) systems can suffer from coherent intercell interference due to the phenomenon of pilot contamination. This paper investigates a two-layer decoding method that mitigates both coherent and non-coherent interference in multi-cell Massive MIMO. To this end, each base station (BS) first estimates the channels to intra-cell users using either minimum mean-squared error (MMSE) or element-wise MMSE (EW-MMSE) estimation based on uplink pilots. The estimates are used for local decoding on each BS followed by a second decoding layer where the BSs cooperate to mitigate inter-cell interference. An uplink achievable spectral efficiency (SE) expression is computed for arbitrary two-layer decoding schemes. A closed-form expression is then obtained for correlated Rayleigh fading, maximum-ratio combining, and the proposed large-scale fading decoding (LSFD) in the second layer. We also formulate a sum SE maximization problem with both the data power and LSFD vectors as optimization variables. Since this is an NP-hard problem, we develop a low-complexity algorithm based on the weighted MMSE approach to obtain a local optimum. The numerical results show that both data power control and LSFD improves the sum SE performance over single-layer decoding multi-cell Massive MIMO systems.

Two-Stage Estimation in Inverse Problems using Combined Wavelet Thresholding and Penalized Maximum Likelihood

Inverse problems occur in a wide range of practical scientific investigations where the variables of interest are only observed indirectly, such as magnetic and seismic imaging in geophysics, electrical tomography in industrial process monitoring, or PET scanning in medicine. Linear inverse problems can be thought of as highly multivariate regression problems with strong multicollinearity where the aim is to interpret regression parameters-prediction is not of interest. Estimation, to give a fitted model, is known as an inverse problem which can be ill-posed and ill-conditioned, making estimation using least-squares or maximum likelihood unstable or even impossible. Instead, one approach is to introduce additional constraints through a penalty term and a penalized least-squares or penalized maximum likelihood approach taken. The major cause of numerical problems in the estimation is noise in the data and hence using a pre-processing which reduces noise may be helpful. Wavelet thresholding has proven to be highly efficient at separating useful information from noise but there has been very little work considering the use of wavelet methods for inverse problems. Hence it is of great interest to investigate the usefulness of this as an additional step in estimation for inverse problems. In particular a two stage process is proposed combining inversion and wavelet thresholding. The thresholding will be considered as either a pre-inversion or post-inversion filter and the results compared. A simulation investigation is described and reported which compares these two alternative, and also which uses a minimum mean-squared error approach to choose the penalty parameter, in the inversion, and the threshold, in the wavelet thresholding, either sequentially or jointly. The results demonstrate that a combined approach is worthwhile and that for the piecewise constant test function considered, it is better to post-process after the inversion step than it is to use the more intuitive wavelet thresholding pre-processing step for noise reduction before inversion. This new approach hence has the potential to enhance the estimation results in a wide range of applied inverse problems.

Low-Distortion MMSE Speech Enhancement Estimator Based on Laplacian Prior

The most well-known conventional speech enhancement algorithms introduce unwanted artifact noise and speech distortion to the enhanced signal. Reducing the effects of such issues require more robust linear and non-linear estimators. This paper proposes new optimum linear and non-linear Laplacian distribution-based estimators. The proposed estimators are derived based on a minimum mean squared error (MMSE) sense to minimize the distortion in different conditions of the underlying speech. Thus, artifact noise is reduced without compromising the noise reduction process. The analytical solutions of the Laplacian distribution-based estimators, linear bilateral Laplacian gain estimator (LBLG), and non-linear bilateral Laplacian gain estimator (NBLG), are presented. The proposed estimators are implemented in three steps. First, the observation signal is decorrelated through a real transform domain to obtain its transform coefficients. Second, the proposed estimators are applied to estimate the clean speech signal from the noisy signal in the decorrelated domain. Finally, the inverse of the real transform is applied to obtain the original speech signal in the time domain. Two conditions in these estimators account for interference events between the speech signal and noise coefficients in the decorrelated domain. Moreover, a mathematical aspect of mean square error of LBLG is evaluated, which presents a significant improvement over other methods. Furthermore, a comprehensive description of the whole variations of the LBLG and NBLG gains characteristics is presented. A comparative evaluation is performed with effective quality metrics, segmental signal-to-noise ratio and perceptual evaluation of speech quality, to demonstrate the advantage and effectiveness of the proposed estimators. The performance of the proposed estimators outperformed other methods, which are the traditional MMSE approach, perceptually motivated Bayesian estimator, dual gain Wiener estimator, and dual MMSE estimator in terms of different objective measurements.

Energy-Efficient Resource Allocation Optimization for Multimedia Heterogeneous Cloud Radio Access Networks

The heterogeneous cloud radio access network (H-CRAN) is a promising paradigm that incorporates cloud computing into heterogeneous networks (HetNets), thereby taking full advantage of cloud radio access networks (C-RANs) and HetNets. Characterizing cooperative beamforming with fronthaul capacity and queue stability constraints is critical for multimedia applications to improve the energy efficiency (EE) in H-CRANs. An energy-efficient optimization objective function with individual fronthaul capacity and intertier interference constraints is presented in this paper for queue-aware multimedia H-CRANs. To solve this nonconvex objective function, a stochastic optimization problem is reformulated by introducing the general Lyapunov optimization framework. Under the Lyapunov framework, this optimization problem is equivalent to an optimal network-wide cooperative beamformer design algorithm with instantaneous power, average power, and intertier interference constraints, which can be regarded as a weighted sum EE maximization problem and solved by a generalized weighted minimum mean-square error approach. The mathematical analysis and simulation results demonstrate that a tradeoff between EE and queuing delay can be achieved, and this tradeoff strictly depends on the fronthaul constraint.

Range-Based Localization Algorithm for Next Generation Wireless Networks Using Radical Centers

Localization is an important issue in next generation wireless networks (NGWNs), such as 5G cellular networks, IEEE 802.11ax WiFi networks, and wireless ad hoc sensor network (WASNs). This paper proposes a 3-D localization algorithm for NGWNs by utilizing the concept of radical centers from analytic geometry, and takes WASNs as an example. Assume that an unknown node can measure its distances to four or more anchor nodes (reference nodes). By picking four distance measurements to four anchor nodes, a radical center is computed. Through analytic formulation, the radical center is shown to be able to be treated as an estimation of the unknown node location. Since every four distance measurements generates one radical center (in 3-D space), to further improve and fuse these estimations, effective filtering mechanisms are proposed to filter out the improper estimations. Afterward, the remaining radical centers are averaged, and the solution is the final estimation of the unknown node location. The location errors of the proposed algorithm and the conventional minimum mean square error (MMSE) method were analytically compared. It was shown that the proposed algorithm outperforms the conventional MMSE method both in accuracy and efficiency. Extensive computer simulations were carried out and the results verified the advantage of the proposed location algorithm over the MMSE approach.

An MMSE Approach to Channel Shorting for Underwater Acoustic FH-FSK Communication

Underwater acoustic channel is the biggest obstacle which causes the underwater acoustic communication falling behind the wireless communication. Generally speaking, underwater acoustic channel will represent doubly spread, i.e. multipath spread and Doppler spread. Also the multipath spread will be quite long in the underwater acoustic channel. For example, the multipath spread will be several milliseconds in the shallow sea and several seconds in the deep sea. Multipath spread will lead to Inter-Symbol Interference (ISI), and Doppler spread will cause the channel to be rapidly time-varying. Although ISI can be suppressed by channel equalizer, but because of the long multipath spread, it increases the complexity about the design of the equalizer. In this paper, channel shortening will be used to reduce the ISI. We present two methods of channel shorting, which are Time Reversal (TR) filter and the Minimum Mean Squared Error (MMSE) filter. When the two filters are used in FH-FSK, we found that the performance of MMSE filter is better than TR filter. Taking into account the characteristics of sparse underwater acoustic channel, Compressive Sensing (CS) algorithm is used for channel estimation. Finally, We perform a series of numerical simulations and the experiments on the lake to show that MMSE filter outperformances the TR filter under the FH-FSK system using the bit error rate (BER) as an evaluation criterion.

Adaptive Processing With Signal Contaminated Training Samples

We consider the adaptive beamforming or adaptive detection problem in the case of signal contaminated training samples, i.e., when the latter may contain a signal-like component. Since this results in a significant degradation of the signal to interference and noise ratio at the output of the adaptive filter, we investigate a scheme to jointly detect the contaminated samples and subsequently take this information into account for estimation of the disturbance covariance matrix. Towards this end, a Bayesian model is proposed, parameterized by binary variables indicating the presence/absence of signal-like components in the training samples. These variables, together with the signal amplitudes and the disturbance covariance matrix are jointly estimated using a minimum mean-square error (MMSE) approach. Two strategies are proposed to implement the MMSE estimator. First, a stochastic Markov Chain Monte Carlo method is presented based on Gibbs sampling. Then a computationally more efficient scheme based on variational Bayesian analysis is proposed. Numerical simulations attest to the improvement achieved by this method compared to conventional methods such as diagonal loading. A successful application to real radar data is also presented.

Linear MMSE-Optimal Turbo Equalization Using Context Trees

Formulations of the turbo equalization approach to iterative equalization and decoding vary greatly when channel knowledge is either partially or completely unknown. Maximum aposteriori probability (MAP) and minimum mean-square error (MMSE) approaches leverage channel knowledge to make explicit use of soft information (priors over the transmitted data bits) in a manner that is distinctly nonlinear, appearing either in a trellis formulation (MAP) or inside an inverted matrix (MMSE). To date, nearly all adaptive turbo equalization methods either estimate the channel or use a direct adaptation equalizer in which estimates of the transmitted data are formed from an expressly linear function of the received data and soft information, with this latter formulation being most common. We study a class of direct adaptation turbo equalizers that are both adaptive and nonlinear functions of the soft information from the decoder. We introduce piecewise linear models based on context trees that can adaptively approximate the nonlinear dependence of the equalizer on the soft information such that it can choose both the partition regions as well as the locally linear equalizer coefficients in each region independently, with computational complexity that remains of the order of a traditional direct adaptive linear equalizer. This approach is guaranteed to asymptotically achieve the performance of the best piecewise linear equalizer, and we quantify the MSE performance of the resulting algorithm and the convergence of its MSE to that of the linear minimum MSE estimator as the depth of the context tree and the data length increase.

A Leakage-Based MMSE Beamforming Design for a MIMO Interference Channel

We propose a low complexity design of the linear transmit filters for a MIMO interference channel. This design is based on a minimum mean squared error (MMSE) approach incorporating the signal and the interference leakage for each transmitter. Unlike the previous methods, it allows a closed-form expression of the regularization factor for the MMSE transmit filter. Hence, it requires a lower computational complexity compared to the conventional MMSE approach that is optimally achieved by solving a polynomial equation to find the regularization factor. Furthermore, the mean squared error (MSE) performance of the proposed design is verified by simulations to have nearly no loss compared to the conventional MMSE approach.

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

A nonlocal minimum mean square error (MMSE) image denoising algorithm is proposed in this work. Based on the Bayesian estimation theory, we first derive that the conventional nonlocal means filter is an MMSE estimator in the special case of noise-free nonlocal neighbors. Then, we develop the nonlocal MMSE denoising filter that can minimize the mean square error (MSE) of a denoised block in more general cases of noisy nonlocal neighbors. Furthermore, the proposed algorithm searches nonlocal neighbors from an external database as well as the entire input image to improve the performance even when a noisy block may not have similar blocks within the image. Since the extended search range demands a higher computational burden, we develop a probabilistic tree-based search method to reduce the computational complexity. Simulation results show that the proposed algorithm provides significantly better denoising performance than the conventional nonlocal means filter.

Sum-rate maximising in cognitive MIMO ad-hoc networks using weighted MMSE approach

This reported work focuses on weighted sum-rate maximisation (WSRM) of cognitive radio ad-hoc networks, where a K user multiple-input multiple-output interference network (K-MIMO-IFN) uses the same spectrum with a licenced primary user. Because the WSRM problem in K-MIMO-IFN is non-convex and it is difficult to get an optimal solution directly, the weighted minimum mean square error (MMSE) approach is used to make the problem easier to handle. This report then proposes a dual-MMSE-algorithm, which iteratively finds a local optimal solution and only needs the local channel knowledge. Simulation results show that the proposed algorithm outperforms other conventional algorithms.

Joint Design of Uplink-Downlink MIMO Relay Networks Using Duality

This letter introduces a joint design method for uplink-downlink multiple-input multiple-output (MIMO) relay communication systems in which the source nodes transmit information to the destination nodes with the help of a relay. We propose a signal forwarding schceme based on the minimum mean-square error (MMSE) approach in uplink relay systems. Exploiting the duality of relay systems, we also propose a relaying scheme for downlink relay systems. Simulation results confirm that the proposed joint design method improves the performance of the relay systems compared with that of conventional relaying schemes in uplink and downlink MIMO relay systems.