Minimum Mean Square Error Solution Research Articles

Dilated POCS: Minimax Convex Optimization

<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Alternating projection onto convex sets</i> (POCS) provides an iterative procedure to find a signal that satisfies two or more convex constraints when the sets intersect. For nonintersecting constraints, the method of simultaneous projections produces a minimum mean square error (MMSE) solution. In certain cases, a minimax solution is more desirable. Generating a minimax solution is possible using dilated POCS. The minimax solution uses morphological dilation of nonintersecting signal convex constraints. The sets are progressively dilated to the point where there is intersection at a minimax solution. Examples are given contrasting the MMSE and minimax solutions in problems of tomographic reconstruction of images. Dilated POCS adds a new imaging modality for image synthesis. Lastly, morphological erosion of signal sets is suggested as a method to shrink the overlap when sets intersect at more than one point.

ADMM-Based Infinity-Norm Detection for Massive MIMO: Algorithm and VLSI Architecture

In this article, we propose a novel data detection algorithm and a corresponding VLSI design for massive multiuser (MU) multiple-input-multiple-output (MIMO) wireless systems. Our algorithm uses alternating direction method of multipliers (ADMM)-based infinity-norm-constrained equalization and is called ADMIN. ADMIN is an iterative algorithm that outperforms linear detectors by a large margin when the ratio between the numbers of base-station (BS) and user antennas is small. In the first iteration, ADMIN computes the linear minimum mean-square error (MMSE) solution, which is sufficient when the ratio between the numbers of BS and user antennas is large. We develop time-shared and iterative VLSI architectures for LDL-decomposition-based soft-output ADMIN supporting 16- and 32-user systems. We present application-specific integrated circuit (ASIC) designs for 16-64 antenna base stations in 28-nm CMOS that supports up to 64 quadrature amplitude modulation (QAM). The 16-user ADMIN ASIC achieves 303 Mb/s while dissipating 85 mW. The 32-user ADMIN ASIC achieves 287 and 241 Mb/s while dissipating 121 and 135 mW for 32 and 64 BS antennas, respectively. ADMIN has also been implemented on a Xilinx Virtex-7 field-programmable gate array (FPGA) and is compared with state-of-the-art massive MIMO data detectors.

FDD Channel Estimation via Covariance Estimation in Wideband Massive MIMO Systems.

A method for channel estimation in wideband massive Multiple-Input Multiple-Output systems using hybrid digital analog architectures is developed. The proposed method is useful for Frequency-Division Duplex at either sub-6 GHz or millimeter wave frequency bands and takes into account the beam squint effect caused by the large bandwidth of the signals. To circumvent the estimation of large channel vectors, the posed algorithm relies on the slow time variation of the channel spatial covariance matrix, thus allowing for the utilization of very short training sequences. This is possibledue to the exploitation of the channel structure. After identifying the channel covariance matrix, the channel is estimated on the basis of the recovered information. To that end, we propose a novel method that relies on estimating the tap delays and the gains as sociated with each path. As a consequence, the proposed channel estimator achieves low computational complexity and significantly reduces the training overhead. Moreover, our numerical simulations show better performance results compared to the minimum mean-squared error solution.

Low Complexity Linear Detection for Uplink Multiuser MIMO SC-FDMA Systems

In this article, the computational complexity reduction of zero forcing (ZF) and minimum mean square error (MMSE) detection is presented for the uplink multiple input multiple output (MIMO) single carrier frequency division multiple access (SC-FDMA) systems. MIMO SC-FDMA structure with detection of user’s data by the well-known multiuser linear detection approaches such as MMSE and ZF often use single input single output detection systems due to its simple detection and significant performance. Although, these approaches involve inversion of a matrix computation whose matrix dimension depend on number of subcarriers used in the system, mainly, it can be few thousands. In practical detection, the computational complexity of matrix inversion becomes very high. Also, the complexity of the receiver is raised because of the superposition of all the transmitted signals at each antenna received in the systems. The proposed conjugate gradient approach reduces the higher computational overhead of linear detectors, which updates iteratively the ZF and MMSE solution and avoids the direct computation of matrix inverse operation. The analysis of the proposed algorithm reveals the superior performance and the low complexity detection in spatial multiplexing SC-FDMA system. Simulations have investigated that the computational complexity of the proposed method has been greatly reduced and bit-error-rate performance is closer to matched filter bound.

Low-Complexity Linear Equalization for OTFS Modulation

In this letter, we propose low-complexity linear equalizers for orthogonal time frequency space (OTFS) modulation that exploit the structure of the effective channel matrix in OTFS. The proposed approach exploits the block circulant nature of the OTFS channel matrix to achieve significant complexity reduction. For an $N\times M$ OTFS system, where $N$ and $M$ are the number of Doppler and delay bins, respectively, the proposed approach gives exact minimum mean square error (MMSE) and zero-forcing (ZF) solutions with just $\mathcal {O}(MN \log MN)$ complexity, while MMSE and ZF solutions using the traditional matrix inversion approach require $\mathcal {O}(M^{3}N^{3})$ complexity. The proposed approach can provide low complexity initial solutions for local search techniques to achieve enhanced bit error performance.

Hyper-power zero forcing detector for massive MIMO systems

In massive multiple-input multiple-output (MIMO) systems when the number of base station antennas is much higher than the number of users, linear detectors, such as zero forcing (ZF) and minimum mean-square error (MMSE), are able to achieve the near-optimal performance due to the favorable massive MIMO channel propagation. But, these detectors employ, in general, exact matrix inversion which is computationally complex for such systems. In this paper, we affirm that computing the exact matrix inversion by direct methods is not necessary to find ZF or MMSE solution. An iterative matrix inversion procedure would yield similar performance. Thus, an efficient iterative matrix inversion based on the hyper-power (HP) method is proposed for massive MIMO detection. The computing efficiency of the iterative matrix inversion is further improved by optimizing the number of terms from the infinite series used in the HP method. Analytical results show that the optimum choice for the number of terms is three from the HP method. Simulation results show that the HP method with the optimum number of terms achieves the near-optimal ZF performance in a small number of iterations. Finally, the Coppersmith–Winograd algorithm for matrix multiplication is employed in order to reduce the computational complexity from \(O(K^{3})\) to \(O(K^{2.373})\), where K represents the number of users.

Computational Complexity Reduction of MMSE-IC MIMO Turbo Detection

High data rates and error-rate performance approaching close to theoretical limits are key trends for evolving digital wireless communication applications. To address the first requirement, multiple-input multiple-output (MIMO) techniques are adopted in emergent wireless communication standards and applications. On the other hand, turbo concept is used to alleviate the destructive effects of the channel and ensure error-rate performance close to theoretical limits. At the receiver side, the incorporation of MIMO techniques and turbo processing leads to increased complexity that has a severe impact on computation speed, power consumption and implementation area. Because of its increased complexity, the detector is considered critical among all receiver components. Low-complexity algorithms are developed at the cost of decreased performance. Minimum mean-squared error (MMSE) solution with iterative detection and decoding shows an acceptable tradeoff. In this paper, the complexity of the MMSE algorithm in turbo detection context is investigated thoroughly. Algorithmic computations are surveyed to extract the characteristics of all involved parameters. Consequently, several decompositions are applied leading to enhanced performance and to a significant reduction of utilized computations. The complexity of the algorithm is evaluated in terms of real-valued operations. The proposed decompositions save an average of [Formula: see text] and [Formula: see text] of required operations for 2 [Formula: see text] 2 and 4 [Formula: see text] 4 MIMO systems, respectively. In addition, the hardware implementation designed applying the devised simplifications and decompositions outperforms available state-of-the-art implementations in terms of maximum operating frequency, execution time, and performance.

Sum-Rate Enhancement in Multiuser MIMO Decode-and-Forward Relay Broadcasting Channel With Energy Harvesting Relays

In this paper, we consider a multiuser multiple-input multiple-output (MIMO) decode-and-forward (DF) relay broadcasting channel (BC) with single source, multiple energy harvesting (EH) relays, and multiple destinations. All the nodes are equipped with multiple antennas. The EH and information decoding tasks at the relays and destinations are separated over the time, which is termed as the time switching scheme. As optimal solutions for the sum-rate maximization problems of BC channels and the MIMO interference channels are hard to obtain, the end-to-end sum rate maximization problem of a multiuser MIMO DF relay BC channel is even harder. In this paper, we propose to tackle a simplified problem, where we employ the block diagonalization (BD) procedure at the source, and we mitigate the interference between the relay-destination channels using an algorithm similar to the BD method. In order to show the relevance of our low complex proposed solution, we compare it with the minimum mean-square error (mmse) solution that was shown in the literature to be equivalent to the solution of the sum-rate maximization in the MIMO broadcasting interfering channels. We also investigate the time division multiple access (TDMA) solution, which separates all the information transmissions from the source to the relays and from the relays to the destinations over time. We provide the numerical results to show the relevance of our proposed solution, in comparison with the no co-channel interference case, the TDMA-based solution, and the mmse-based solution.

Image Bit-Depth Enhancement via Maximum &lt;italic&gt;A Posteriori&lt;/italic&gt; Estimation of AC Signal

When images at low bit-depth are rendered at high bit-depth displays, missing least significant bits needs to be estimated. We study the image bit-depth enhancement problem: estimating an original image from its quantized version from a minimum mean squared error (MMSE) perspective. We first argue that a graph-signal smoothness prior-one defined on a graph embedding the image structure-is an appropriate prior for the bit-depth enhancement problem. We next show that directly solving for the MMSE solution is, in general, too computationally expensive to be practical. We then propose an efficient approximation strategy. In particular, we first estimate the ac component of the desired signal in a maximum a posteriori formulation, efficiently computed via convex programming. We then compute the dc component with an MMSE criterion in a closed form given the computed ac component. Experiments show that our proposed two-step approach has improved performance over the conventional bit-depth enhancement schemes in both objective and subjective comparisons.

Iterative MMSE equalization and CFO compensation for the uplink SC‐FDMA transmission

SummarySingle‐carrier frequency division multiple access is greatly sensitive to carrier frequency offset (CFO) between transceivers. This leads to the destruction of orthogonality among subcarriers, which in turn leads to inter‐carrier interference and multiple access interference between different users. Minimum mean square error (MMSE) equalizer that uses an inverse operation on an interference matrix with a dimension equal to the number of subcarriers is normally used to invalidate CFO effects. Hence, the terminal processing complexity is very high. The proposed conjugate gradient method attempts to mitigate the higher computational complexity by iteratively evaluating the MMSE solution without direct matrix inverse operation. To further mitigate the multiple access interference, MMSE combined with parallel interference cancellation is also implemented. The analysis of the proposed method shows better performance and fast convergence in single‐carrier frequency division multiple access systems. The maximum iteration number to formulate an accurate solution is almost equal to the number of active users in the uplink access. Simulation results bring out the effectiveness of the present method compared with the existing CFO compensation schemes in terms of computational complication and system performance with large frequency offsets. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Lattice Reduction Aided Detector for MIMO Communication Via Ant Colony Optimisation

In this work heuristic ant colony optimisation (ACO) procedure is deployed in conjunction with lattice reduction (LR) technique aiming to improve the performance-complexity tradeoff of detection schemes in MIMO communication. A hybrid LR-ACO MIMO detector using the linear minimum mean squared error (MMSE) criterion as initial guess is proposed and compared with two other traditional (non)linear MIMO detectors, as well as with heuristic MIMO detection approaches from the literature, in terms of both performance and complexity metrics. Numerical results show that the proposed LR-ACO outperforms the traditional ACO-based MIMO detectors and the ACO detector with the MMSE solution as initial guess, with a significant complexity reduction while is able to reach full diversity degree in all scenarios considered, including different channel correlation levels, modulation orders, and antennas configuration.

Tree pruning for MIMO sphere detection based on MMSE detection

Tree pruning is an effective algorithm to reduce the complexity of sphere detection (SD) for multiple-input multiple-output (MIMO) communication systems. How to determine the tree pruning rule, as well as by what the tradeoff between the performance and the complexity can be achieved, is still an open problem. In this paper, a tree pruning algorithm is proposed based on minimum mean square error (MMSE) detection. The proposed algorithm first preforms MMSE detection since the complexity of MMSE detection is very low. Then the pruning constraints will be set according to the scaled path metrics of the MMSE solution. The choice of the scale factors and their influences on the complexity and performance are also discussed. Through analysis and simulations, it is shown that the complexity is reduced significantly with negligible performance degradation and additional computations.

Generative Bayesian Image Super Resolution With Natural Image Prior

We propose a new single image super resolution (SR) algorithm via Bayesian modeling with a natural image prior modeled by a high-order Markov random field (MRF). SR is one of the long-standing and active topics in image processing community. It is of great use in many practical applications, such as astronomical observation, medical imaging, and the adaptation of low-resolution contents onto high-resolution displays. One category of the conventional approaches for image SR is formulating the problem with Bayesian modeling techniques and then obtaining its maximum-a-posteriori solution, which actually boils down to a regularized regression task. Although straightforward, this approach cannot exploit the full potential offered by the probabilistic modeling, as only the posterior mode is sought. On the other hand, current Bayesian SR approaches using the posterior mean estimation typically use very simple prior models for natural images to ensure the computational tractability. In this paper, we present a Bayesian image SR approach with a flexible high-order MRF model as the prior for natural images. The minimum mean square error (MMSE) criteria are used for estimating the HR image. A Markov chain Monte Carlo-based sampling algorithm is presented for obtaining the MMSE solution. The proposed method cannot only enjoy the benefits offered by the flexible prior, but also has the advantage of making use of the probabilistic modeling to perform a posterior mean estimation, thus is less sensitive to the local minima problem as the MAP solution. Experimental results indicate that the proposed method can generate competitive or better results than state-of-the-art SR algorithms.

A new optimal MUD algorithm based on searching in the local space of sub-optimal MMSE solutions for DS-UWB communication systems

In this paper, we have studied the multiuser detection (MUD) technology applied in multiaccess (MA) direct sequence ultrawideband (DS-UWB) systems. Some typical MUD algorithms were analyzed and a new approximate optimal MUD algorithm based on searching in the local space of suboptimal minimum mean-squared error (MMSE) solution was proposed. In this new algorithm, the suboptimal solution of MMSE is considered as the initial value for the local search, and then a local solution space of this suboptimal solution is constructed by Euclidean distance. Within this space, some likely solutions for the classical optimal MUD algorithm (OMUD) are searched and the optimal one of them is chosen as the solution of this new optimal MUD algorithm. Theoretical analysis and simulation results indicate that, the BER performance and the Near-far Effect resistant ability of this proposed algorithm are much better than those of MMSE and adaptive MMSE detections, which are very close to those of OMUD. Furthermore, its time complexity is much lower than OMUD's. Compared to adaptive MMSE, this algorithm has no difficulty in choosing the suitable parameters for adaptive MMSE. Besides these, we examine the application of this new algorithm to the problem of suppressing the digital narrowband interference (NBI). It is clearly seen that this approximate optimal algorithm has perfect performance under strong NBI circumstance and coincides with the classical OMUD algorithm bound in the same condition.

MMSE-Based CFO Compensation for Uplink OFDMA Systems with Conjugate Gradient

In this paper, we present a low-complexity carrier frequency offset (CFO) compensation algorithm based on the minimum mean square error (MMSE) criterion for uplink orthogonal frequency division multiple access systems. CFO compensation with an MMSE filter generally requires an inverse operation on an interference matrix whose size equals the number of subcarriers. Thus, the computational complexity becomes prohibitively high when the number of subcarriers is large. To reduce the complexity, we employ the conjugate gradient (CG) method which iteratively finds the MMSE solution without the inverse operation. To demonstrate the efficacy of the CG method for our problem, we analyze the interference matrix and present several observations which provide insight on the iteration number required for convergence. The analysis indicates that for an interleaved carrier assignment scheme, the maximum iteration number for computing an exact solution is at most the same as the number of users. Moreover, for a general carrier assignment scheme, we show that the CG method can find a solution with far fewer iterations than the number of subcarriers. In addition, we propose a preconditioning technique which speeds up the convergence of the CG method at the expense of slightly increased complexity for each iteration. As a result, we show that the CFO can be compensated with substantially reduced computational complexity by applying the CG method.

IDMA vs. CDMA: Analysis and Comparison of Two Multiple Access Schemes

This article presents comprehensive comparisons of interleave division multiple access (IDMA) and direct sequence code division multiple access (DS-CDMA) in terms of performance and complexity assuming iterative multiuser detection. IDMA can be seen as a special case of DS-CDMA with spreading gain of one using very low rate code and user-specific interleavers for user separation. We focus on three suboptimum linear detectors: minimum mean square error (MMSE), rake (or matched filter), and soft-rake detectors from practical concerns. We analytically prove that the three detectors are equivalent for asynchronous users of IDMA on frequency flat channels for complex modulation alphabets. Such equivalence has been shown only for binary phase shift keying (BPSK) in the literature. The equivalence guarantees the MMSE solution for IDMA without computationally expensive matrix inversions or matrix-vector multiplications. This is generally not the case for DS-CDMA since DS-CDMA is sensitive to user asynchronism. We also discuss complexity aspects when the MMSE detector is used where we focus on essential differences in complexity between IDMA and DS-CDMA, instead of discussing particular complexity reduction techniques. Computer simulations are performed in various scenarios and the performance is analyzed by bit error rate simulations as well as by extrinsic information transfer (EXIT) charts. The analysis reveals the advantages of IDMA over DS-CDMA in terms of performance and complexity under practical considerations, particularly in highly user loaded scenarios.

Blind equalizer for constant-modulus signals based on Gaussian process regression

A new blind equalization method for constant modulus (CM) signals based on Gaussian process for regression (GPR) by incorporating a constant modulus algorithm (CMA)-like error function into the conventional GPR framework is proposed. The GPR framework formulates the posterior density function for weights using Bayes' rule under the assumption of Gaussian prior for weights. The proposed blind GPR equalizer is based on linear-in-weights regression model, which has a form of nonlinear minimum mean-square error solution. Simulation results in linear and nonlinear channels are presented in comparison with the state-of-the-art support vector machine (SVM) and relevance vector machine (RVM) based blind equalizers. The simulation results show that the proposed blind GPR equalizer without cumbersome cross-validation procedures shows the similar performances to the blind SVM and RVM equalizers in terms of intersymbol interference and bit error rate.

Relationship Between Two Distortion Measures for Memoryless Nonlinear Systems

Memoryless nonlinear transforms of random signals with Gaussian statistics are encountered in a variety of signal processing applications. A well-known criterion for the description of distortion errors is the Mean Squared Error (MSE) between the input and output signals of the nonlinearity. A different criterion which has been found to be useful especially in the field of multicarrier communications is the Signal-to-Distortion Noise Ratio (SDNR), which is based on the BUSSGANG decomposition of the output of the nonlinearity. It is shown that the SDNR optimum coincides with the minimum MSE solution, which simplifies the optimization of memoryless nonlinear functions with respect to SDNR. The results are applied to quantization and dynamic range reduction of Gaussian signals.

Block-Based Transceivers With Minimum Redundancy

The standard design of multicarrier and single-carrier employing frequency-domain equalization transceivers requires, at least, L elements of redundancy, where L stands for the channel order. The redundancy eliminates the inherent interblock interference (IBI), which is part of all block-based transceivers, and turns the channel matrix circulant. The spectral decomposition of the circulant channel matrix through the discrete Fourier transform (DFT) allows the use of superfast algorithms for both the design of zero-forcing (ZF) and minimum mean squared error (MMSE) equalizers, and the equalization of received signals. However, it is well known that the minimum redundancy for IBI-free designs of block-based transceivers is [L/2] . This paper proposes practical ZF and MMSE solutions by using DFT, inverse DFT, and diagonal matrices. In particular, it is shown that, for some particular mild constraints on the channel model, the new designs may have similar bit error rate performance when compared to the standard ones, while keeping the same asymptotic complexity for the equalization process, that is, O(n log2 n) numerical operations. The key feature of the proposed transceivers is their higher throughput.

A pre-test like estimator dominating the least-squares method

We develop a pre-test type estimator of a deterministic parameter vector β in a linear Gaussian regression model. In contrast to conventional pre-test strategies, that do not dominate the least-squares (LS) method in terms of mean-squared error (MSE), our technique is shown to dominate LS when the effective dimension is greater than or equal to 4. Our estimator is based on a simple and intuitive approach in which we first determine the linear minimum MSE (MMSE) estimate that minimizes the MSE. Since the unknown vector β is deterministic, the MSE, and consequently the MMSE solution, will depend in general on β and therefore cannot be implemented. Instead, we propose applying the linear MMSE strategy with the LS substituted for the true value of β to obtain a new estimate. We then use the current estimate in conjunction with the linear MMSE solution to generate another estimate and continue iterating until convergence. As we show, the limit is a pre-test type method which is zero when the norm of the data is small, and is otherwise a non-linear shrinkage of LS.