Block Diagonalization Technique Research Articles

Failure of the simultaneous block diagonalization technique applied to complete and cluster synchronization of random networks.

We discuss here the application of the simultaneous block diagonalization (SBD) of matrices to the study of the stability of both complete and cluster synchronization in random (generic) networks. For both problems, we define indices that measure success (or failure) of application of the SBD technique in decoupling the stability problem into problems of lower dimensionality. We then see that in the case of random networks the extent of the dimensionality reduction achievable is the same as that produced by application of a trivial transformation.

An Efficient Fisher Matrix Approximation Method for Large-Scale Neural Network Optimization.

Although the shapes of the parameters are not crucial for designing first-order optimization methods in large scale empirical risk minimization problems, they have important impact on the size of the matrix to be inverted when developing second-order type methods. In this article, we propose an efficient and novel second-order method based on the parameters in the real matrix space [Formula: see text] and a matrix-product approximate Fisher matrix (MatFisher) by using the products of gradients. The size of the matrix to be inverted is much smaller than that of the Fisher information matrix in the real vector space [Formula: see text]. Moreover, by utilizing the matrix delayed update and the block diagonal approximation techniques, the computational cost can be controlled and is comparable with first-order methods. A global convergence and a superlinear local convergence analysis are established under mild conditions. Numerical results on image classification with ResNet50, quantum chemistry modeling with SchNet, and data-driven partial differential equations solution with PINN illustrate that our method is quite competitive to the state-of-the-art methods.

Characterizing the universal rigidity of generic tensegrities

A tensegrity is a structure made from cables, struts, and stiff bars. A d-dimensional tensegrity is universally rigid if it is rigid in any dimension d' with d'ge d. The celebrated super stability condition due to Connelly gives a sufficient condition for a tensegrity to be universally rigid. Gortler and Thurston showed that super stability characterizes universal rigidity when the point configuration is generic and every member is a stiff bar. We extend this result in two directions. We first show that a generic universally rigid tensegrity is super stable. We then extend it to tensegrities with point group symmetry, and show that this characterization still holds as long as a tensegrity is generic modulo symmetry. Our strategy is based on the block-diagonalization technique for symmetric semidefinite programming problems, and our proof relies on the theory of real irreducible representations of finite groups.

Optimal analysis for optimal design of cyclic symmetric structures subject to frequency constraints

In general, the design optimization of large-scale structural systems is an extremely computationally expensive process. However, optimal structural analysis provides efficient and practical methods of structural analysis which can be employed as powerful tools to reduce significantly the computational time of structural analyses performed in the structural optimization process. In this paper, an efficient eigensolution method for free vibration analysis of rotationally repetitive structures is developed for the optimal design of cyclic symmetric structures subject to frequency constraints. Through the present eigensolution method, the initial free-vibration eigenproblem is decomposed into some sub-eigenproblems with much smaller dimensions using an efficient block-diagonalization technique. The main advantage of the present method is that it requires significantly less computational time and memory than the existing classical eigensolution method. Finally, two large-scale numerical examples are given to demonstrate the efficiency and accuracy of the present eigensolution method and compare it with the existing classical method in both terms of computational time and memory requirements. Numerical results indicate that the present eigensolution method not only guarantees the accuracy of the analysis results but also significantly reduces the computational time and memory required compared to the existing classical method. More importantly, the computational time of the optimization process is also significantly reduced. Optimization is carried out by a set-theoretical-based Jaya algorithm, named ST-JA. The proposed ST-JA aims to enhance the diversification and intensification capabilities of the original JA and strike a fine balance between them. Optimization results confirm that the proposed ST-JA outperforms the original JA and has superior or comparable performance to other state-of-the-art optimization algorithms.

A practical approach to improve product of energy efficiency and capacity rate in MU-MIMO systems

PurposeThe major challenges in the modern-day wireless communication systems are increased co-channel interference owing to large number of users and the increased energy consumption owing to high circuit and/or hardware power consumption. Hence, the purpose of this paper is to present a practical approach involving linear precoding, channel estimation, user selection (US) and transmit antenna selection (AS) for enhanced reliability in multiuser multiple-input multiple output (MU-MIMO) system.Design/methodology/approachThe proposed technique considers systematic and optimum deployment of users and transmits antennas for each selected user which enhances the sum rate or the system capacity. The comparison of algorithms, namely, norm-based and capacity-based US is presented with its implementation with precoding techniques, namely, block-diagonalization (BD) and zero-forcing with combining (ZFC) which is used to minimize co-channel interference. In this paper, a power consumption model is proposed for energy efficiency calculation in MU-MIMO system. Also, post analysis, the variant of US and AS algorithms optimizing the performance of BD and ZFC precoding technique is proposed.FindingsIt is seen that the proposed MU-MIMO system with norm-based US and norm-based AS improves over existing US-based systems by 43% in terms of sum rate and 19% in terms of energy efficiency for 100 users.Originality/valueIt is seen that the proposed MU-MIMO system with norm-based US and norm-based AS improves over existing US-based systems by 43% in terms of sum rate and 19% in terms of energy efficiency for 100 users.

BER and Capacity Improvement for Multiuser MIMO System by MRT Precoding Technique

In recent years several of endeavors have been conducted in various papers to improve capacity and bit error rate performance of multiuser MIMO system. Every effort tried to increase gain of the individual stream between base station and users’ antennas while reducing the interference among multiple users and interference between streams of single user itself. All these papers adopted precoding techniques for beam forming of individual stream. Various papers have compared precoding techniques such as DPC, BD-ZF, BD-MMSE, BD-QR-MRL, and BD-QR-LRL. Block diagonalization technique is capable to reduce multiuser interference, but not able to combat interference between antennas of single user itself. BD combines with other techniques to overcome this problem. This paper also proposes a precoding technique called modified maximum ratio transmission, which is able to deal with the problem without much complexity. Simulation is done to compare all precoding technique with MRT. The paper tests the precoding techniques on the parameters of BER and capacity, to achieve optimum receiver performance. BER curve and capacity curve are plotted to conclude the paper.

A Novel Equivalent Baseband Channel of Hybrid Beamforming in Massive Multiuser MIMO Systems

The conventional full radio frequency (RF) configuration appears to be unfeasible in massive multiple-input-multiple-output (MIMO) systems due to the still-exorbitant energy consumption of RF chains at millimeter wave frequencies. In this letter, we address this challenge by proposing a novel two step hybrid beamforming scheme for the downlink massive multiuser MIMO systems. Our purpose is to maximize the overall spectral efficiency with limited RF chains. In the first step, we aim to find an equivalent baseband channel by maximizing its capacity through the analog RF processing. In the second step, the inter-user interference is eliminated by the block diagonalization (BD) technique. Simulation results demonstrate that the proposed scheme can achieve a higher capacity than the traditional BD scheme with much fewer number of RF chains.

Ionization and Single and Double Electron Capture in Proton-Ar Collisions.

Total cross sections for formation of H and H-, and electron production, in H+ + Ar collisions have been calculated at energies between 100 eV and 200 keV by employing two methods: for E < 10 keV, a semiclassical treatment with an expansion in a basis of electronic wave functions of the ArH+ quasimolecule and, for E > 10 keV, the switching-classical-trajectory-Monte Carlo method (s-CTMC). The semiclassical calculation involves transitions to molecular autoionizing states, calculated by applying a block-diagonalization technique. The s-CTMC method is adept to treat two-electron processes and yields total cross sections for H- formation in reasonably good agreement with the experimental data. Cross sections for electron- and H-production processes, which are dominated by one-electron transitions, are in good agreement with the experimental data.

Optimal design of cyclically symmetric trusses with frequency constraints using cyclical parthenogenesis algorithm

Special kinds of structures exhibit repetitive patterns that can be used in the process of structural analysis in order to decrease the required computational time. This is especially helpful when a structure should be analyzed numerous times as in the structural optimization problems. In this article, weight optimization of cyclically symmetric spatial trusses subjected to frequency constraints is performed utilizing such repetitive patterns. Large initial eigenproblems are first decomposed into smaller and less time-consuming problems using an efficient block diagonalization technique. A recently developed meta-heuristic method, cyclical parthenogenesis algorithm, is then utilized to perform the optimization task. Three numerical examples are optimized to illustrate the viability and efficiency of the proposed method.

Precoding Design with Joint Dynamic Channel Assignment and Band Selection Techniques for Downlink CoMP OFDMA Systems

Multi cell transmission can prevent the multi-cell interference by localizing the signals at different sub-band for the downlink (DL) localized orthogonal frequency division multiple access (OFDMA) systems. However, the synchronized errors, such as carrier frequency offset (CFO), will induce the challenging problems of interference due to the non-orthogonality between the subcarriers and cells. In this paper, we propose the novel precoding design with joint dynamic channel assignment and band selection techniques for DL coordinated multipoint (CoMP) OFDMA systems with CFO effect. Further, a two-stage pre-processing technique is proposed to eliminate the above interference problems and reduce the computational complexity of the cell-edge users for DL CoMP OFDMA systems. Moreover, the central unit can assign the resource dynamically to enhance the link quality. Therefore, the dynamic channel on/off assignment techniques and the dynamic band selection techniques are proposed to enhance the bit error rate (BER) performance. Simulation results confirm that the proposed precoding transceiver provides excellent BER and lower computational complexity of the cell-edge communication than the conventional block diagonalization technique.

Symbolic algorithms for the inverses of general [formula omitted]-tridiagonal matrices

Two symbolic algorithms for inverting k-tridiagonal matrices have been recently found by El-Mikkawy and Atlan (2014, 2015). These two algorithms are mainly based on the Doolittle LU factorization of the k-tridiagonal matrix. In the current paper, we present a new explicit analytic expression for the inverses of general tridiagonal matrices at first. By using a block diagonalization technique, we then relate k-tridiagonal matrix inversion to tridiagonal matrix inversion. Meanwhile, an efficient algorithm is derived for computing the inverses of nonsingular k-tridiagonal matrices with the help of any algorithm for computing the inverses of tridiagonal matrices. Three examples are given in order to illustrate the performance and efficiency of the proposed algorithms.

Solving intertemporal CGE models in parallel using a singly bordered block diagonal ordering technique

The paper introduces a direct ordering method that employs a special feature of an intertemporal Computable General Equilibrium (CGE) model to reorder its first-order partial derivative matrix into a Singly Bordered Block Diagonal (SBBD) form. The matrix can then be decomposed into LU form and solved in parallel. With this method, the numerical results from the paper show a substantial advantage in computational time and memory use for parallel solutions of intertemporal CGE models in comparison to current serial solution methods. A solution for an intertemporal and regional model of the Vietnamese economy is provided as an example and comparator for the different methods.

Low complexity multiuser detection with recursively successive zero-forcing and SIC based on nullspace for multiuser MIMO-OFDM system

To improve the spectrum efficiency, this paper considers the multiuser detection with the MU-MIMO technology for multiuser MIMO-OFDM system uplink with the same subcarrier shared by multiple users. A low complexity multiuser detection algorithm with recursively successive zero-forcing and successive interference cancellation (RSZF-SIC) based on nullspace is proposed. The RSZF process based on the block diagonalization (BD) technique eliminates the co-channel interference (CCI) by a recursive method based on the nullspace orthogonal theorem. The SIC process detects the user signals respectively with the reasonable user detection sequence based on the results of the RSZF process. The computational complexity of the proposed algorithm is effectively reduced by reducing the total number of singular value decomposition (SVD) operations and the dimension of the SVD matrix in the recursive procedure. The performance of the proposed algorithm is improved in terms of bit error rate and sum capacity of the system, especially in the high-SNR regime.

Green Resource Allocation Schemes for Relay-Enhanced MIMO-OFDM Networks

This paper studies the problem of joint allocation of subchannel, transmission power, and phase duration in relay-enhanced bidirectional multiple-input-multiple-output orthogonal frequency-division multiplexing (MIMO-OFDM) networks. The goal of resource allocation is to minimize transmission energy consumption in networks with multiple decode-and-forward relay stations (RSs) under the data rate constraints of user equipment (UE). The challenges of this resource allocation problem arise from the complication of multiple-phase assignments within a subchannel since the RS can provide an additional transmission path from the base station to the UE. Existing research does not fully take into account all of the influential factors to achieve feasible resource allocation for relay-enhanced MIMO-OFDM networks. Green resource allocation (GRA) schemes with reduced computational complexity are proposed in this paper to develop joint allocation of subchannel, power, and phase duration for the UE with the consideration of direct and two-hop communications. Both the separate-downlink (DL)-and-uplink (UL) and mixed-DL-and-UL relaying assignments and the linear block diagonalization (LBD) technique are adopted to obtain the solutions for the proposed GRA schemes. Simulation results show that the proposed GRA schemes can provide comparably better energy conservation with the consideration of quality-of-service (QoS) support.

An efficient user scheduling scheme for downlink Multiuser MIMO-OFDM systems with Block Diagonalization

The combination of multiuser multiple-input multiple-output (MU-MIMO) technology with orthogonal frequency division multiplexing (OFDM) is an attractive solution for next generation of wireless local area networks (WLANs), currently standardized within IEEE 802.11ac, and the fourth-generation (4G) mobile cellular wireless systems to achieve a very high system throughput while satisfying quality of service (QoS) constraints. In particular, Block Diagonalization (BD) scheme is a low-complexity precoding technique for MU-MIMO downlink channels, which completely pre-cancels the multiuser interference. The major issue of the BD scheme is that the number of users that can be simultaneously supported is limited by the ratio of the number of base station transmit antennas to the number of user receive antennas. When the number of users is large, a subset of users must be selected, and selection algorithms should be designed to maximize the total system throughput. In this paper, the BD technique is extended to MU-MIMO-OFDM systems and a low complexity user scheduling algorithm is proposed to find the optimal subset of users that should transmit simultaneously, in light of the instantaneous channel state information (CSI), such that the total system sum-rate capacity is maximized. Simulation results show that the proposed scheduling algorithm achieves a good trade-off between sum-rate capacity performance and computational complexity.

Application of block diagonal technique to a Hamiltonian matrix in performing spin-splitting calculations for GaN wurtzite materials

The bulk inversion asymmetry (Dresselhaus) terms (i.e., B2, B1, and B′1 terms) of wurtzite materials are determined. The 2 × 2 conduction band, 2 × 2 heavy-hole band, 2 × 2 light-hole band, and 2 × 2 crystal-field split-off hole band matrices of wurtzite semiconductors are developed and decoupled by using a block diagonal technique. Importantly, those 2 × 2 block diagonal matrices incorporate not only the interband coupling effect but also the bulk inversion asymmetry effect. Analytical expressions for the conduction and the valence band spin-splitting parameters and energies of GaN wurtzite materials are formulated by solving the block diagonal matrices. The presence of these terms is shown to include the spin-splitting phenomenon.

A MMSE Vector Precoding with Block Diagonalization for Multiuser MIMO Downlink

Block diagonalization (BD) algorithm is a generalization of the channel inversion that converts multiuser multi-input multi-output (MIMO) broadcast channel into single-user MIMO channel without inter-user interference. In this paper, we combine the BD technique with a minimum mean square error vector precoding (MMSE-VP) for achieving further gain in performance with minimal computational overhead. Two key ingredients to make our approach effective are the QR decomposition based block diagonalization and joint optimization of transmitter and receiver parameters in the MMSE sense. In fact, by optimizing precoded signal vector and perturbation vector in the transmitter and receiver jointly, we pursue an optimal balance between the residual interference mitigation and the noise enhancement suppression. From the sum rate analysis as well as the bit error rate simulations (both uncoded and coded cases) in realistic multiuser MIMO downlink, we show that the proposed BD-MVP brings substantial performance gain over existing multiuser MIMO algorithms.

Successive User Scheduling Schemes for a Multiuser MIMO System Employing Generalized Channel Inversion

Generalized channel inversion (GCI) is a precoding technique for multiuser multiple-input multiple-output system. While producing each user's precoding matrix, GCI takes into account noise and thus it is more robust compared with alternative techniques such as block diagonalization technique in terms of sum rate capacity and frame error rate. In this paper, two suboptimal multiuser scheduling schemes for GCI are proposed that by scheduling a subset of mobile users nearly maximize the sum rate capacity. They employ an iterative approach involving a number of search steps. At each step, unselected mobile users are evaluated one by one, and only one of them is chosen according to given criteria. It is shown via computer simulations that the proposed schemes are capable of achieving a large portion of the sum rate capacity that is offered by an exhaustive search. The performance of the proposed multiuser scheduling schemes is evaluated when the antenna mutual coupling effects are taken into account at the mobile users' sides. Numerical results reveal that the presence of antenna mutual coupling can result in an increased sum rate capacity when the array inter-element spacing is in the range of 0.3---0.4 wavelength.

Block diagonalization coordinated transmission with zero-forcing in MIMO broadcast

This work has investigated coordinated multiple-input multiple-output (MIMO) transmission schemes in an interference-limited cellular downlink. It has proposed a novel block diagonalization (BD) coordinated transmission scheme, which combines with zero-forcing (ZF) criterion. In the scheme, the BD technique has advantages in suppressing multi-user interference while the ZF technique enables to mitigate interference among spatial data streams for a user. Based on the proposed coordinated scheme, an efficient power allocation is also put forward. The analyses show that the ergodic capacity of the proposed coordinated scheme is that of the MIMO channel with the maximum transmit power at each transmitter. Computer simulations demonstrate the effectiveness of the proposed coordinated scheme and its corresponding power allocation.

Simplified Block Diagonalization for Multiuser MIMO Systems with Gram-Schmidt Orthogonalization

In multiuser multiple-input multiple-output (MU-MIMO) wireless downlink systems, block diagonalization (BD) is a technique, where the transmit precoding matrix of each user is designed such that its subspace lies in the null space of all the other remaining users, so that multiuser interference (MUI) is completely canceled. In low signal to noise power ratio (SNR) or low signal to interference plus noise power ratio (SINR) environments, regularized BD, that lets some MUI remain and maximizes the sum rate capacity of the BD MIMO channel, was also proposed. One of the problems of both the approaches is high complexity of computation due to a lot of singular value decomposition (SVD) processes. In this paper we propose new BD techniques utilizing QR decomposition (QRD) which can be practically achieved by Gram-Schmidt orthogonalization (GSO) with lower complexity compared to the conventional method employing SVD. We can show that the performance of the proposed approaches is close to the conventional approaches, while the proposed approaches have much lower complexity.