Matrix Inversion Algorithm Research Articles

Investigations on Adaptive Beamforming for Linear and Planar Smart Antenna Arrays using Sample Matrix Inversion Algorithm

In this paper sample matrix inversion (SMI) algorithm is used to generate beam for a smart antenna. The capability of adaptive beam generation by SMI algorithm is investigated for linear and planar arrays. The accuracies of main beam toward user and null toward interferer, beam width and side lobe level with respect to the variation of block length are reported. General Terms Smart antenna, SMI algorithm

A data-driven paradigm for mapping problems

We propose data-driven paradigm for solving task mapping problems.Data-mapping paradigm is formulated for multiple-core systems.Data dependency analysis replaced task dependency analysis on mapping problems.We prototype data-mapping paradigm for Strassen and Cholesky inversion methods.Data-mapping paradigm is an efficient approach for data-intensive applications. We present a new data-driven paradigm for solving mapping problems on parallel computers. This paradigm targets at mapping data modules, instead of task modules, onto multiple processing cores. By dependency analysis of data modules, we devise a data movement matrix to reduce the need of manipulating task program modules at the expenses of handling data modules. To visualize and quantify the complex maneuver, we adopt the parallel activities trace graphs introduced earlier. To demonstrate the procedure and algorithmic values of our paradigm, we test it on the Strassen matrix multiplication and Cholesky matrix inversion algorithms. Mapping tasks has been more widely studied while mapping data is a new approach that appears to be more efficient for data-intensive applications that are becoming prevalent for today's parallel computers with millions of cores.

A Relaxed Algorithm for Online Matrix Inversion

We consider a variation of the well known problem of computing the unique solution to a nonsingular systemAx = b of n linear equations over a eld K. The variation assumes that A has generic rank prole and requires as output not only the single solution vector A 1 b2 K n 1 , but rather the solution to all leading principle subsystems. Most importantly, the rows of the augmented system A b are given one at a time from rst to last, and as soon as the next row is given the solution to the next leading principal subsystem should be produced. We call this problem OnlineSystem. The obvious iterative algorithm for OnlineSystem has a cost in terms of eld operations that is cubic in the dimension of A. In this paper we introduce a relaxed representation for the inverse and show how to obtain an algorithm for OnlineSystem that allows us to incorporate matrix multiplication. As an application we show how to introduce fast matrix multiplication into the inherently iterative algorithm for row rank prole computation presented previously by the authors.

ADAPTIVE BEAMFORMING ALGORITHMS FOR CANCELLATION OF MULTIPLE INTERFERENCE SIGNALS

This paper proposes a fast Minimum-Variance-Distortionless-Response (MVDR) beam- forming algorithm for an antenna array for cancellation of multiple interference signals. The proposed algorithm uses Sample-Average Estimate (SAE) of the data covariance matrix and reduces its compu- tational effort by applying the Matrix-Inversion-Lemma (MIL) to its covariance Matrix Inversion (MI) operation. The proposed algorithm is compared to two SAE-based algorithms: the Sample Matrix Inversion (SMI) algorithm that requires an MI operation and the Auxiliary Vector (AV) algorithm that does not need an MI operation. A non-SAE based algorithm using the Least Mean Square (LMS) method is also included for comparison. Simulation results show that the proposed algorithm converges slower than the SMI scheme but outperforms the AV and LMS schemes during the transient phase. Once convergence is achieved, the proposed algorithm converges to a better Mean Square Error than the rest of the algorithms evaluated.

Combined high-order algorithms in robust least-squares estimation with harmonic regressor and strictly diagonally dominant information matrix

This article describes new high-order algorithms in the least-squares problem with harmonic regressor and strictly diagonally dominant information matrix. Estimation accuracy and the number of steps to achieve this accuracy are controllable in these algorithms. Simplified forms of the high-order matrix inversion algorithms and the high-order algorithms of direct calculation of the parameter vector are found. The algorithms are presented as recursive procedures driven by estimation errors multiplied by the gain matrices, which can be seen as preconditioners. A simple and recursive (with respect to order) algorithm for update of the gain matrix, which is associated with Neumann series, is found. It is shown that the limiting form of the algorithm (algorithm of infinite order) provides perfect estimation. A new form of the gain matrix is also a basis for unification method of high-order algorithms. New combined and fast convergent high-order algorithms of recursive matrix inversion and algorithms of direct calculation of the parameter vector are presented. The stability of algorithms is proved and explicit transient bound on estimation error is calculated. New algorithms are simple, fast and robust with respect to round-off error accumulation.

Transformations of matrix structures work again

Matrices with the structures of Toeplitz, Hankel, Vandermonde and Cauchy types are omnipresent in modern computations in Sciences, Engineering, and Signal and Image Processing. These four matrix classes have distinct features, but in our 1990 paper in Mathematics of Computation we showed that Vandermonde and Hankel multipliers can be applied to transform each class into the others, and then we demonstrated how by using these transforms we can readily extend any successful matrix inversion algorithm from one of these classes to all the others. The power of this approach was widely recognized later, when novel numerically stable algorithms solved nonsingular Toeplitz linear systems of equations in quadratic (versus classical cubic) arithmetic time based on transforming Toeplitz into Cauchy matrix structures. More recent papers combined the same transformation with a link of the Cauchy matrices to the Hierarchical Semiseparable matrix structure, which is a specialization of matrix representations employed by the Fast Multipole Method. This produced numerically stable algorithms that approximated the solution of a nonsingular Toeplitz linear system of equations in nearly linear arithmetic time. We first revisit the successful method of structure transformation, covering it comprehensively. Then we analyze the latter approximation algorithms for Toeplitz linear systems and extend them to approximate multiplication of Vandermonde and Cauchy matrices by a vector and to approximate solution of Vandermonde and Cauchy linear systems of equations provided that they are nonsingular and well-conditioned. We decrease the arithmetic cost of the known numerical approximation algorithms for these tasks from quadratic to nearly linear, and similarly for the computations with the matrices having structures that generalize the structures of Vandermonde and Cauchy matrices and for polynomial and rational evaluation and interpolation. We also accelerate a little further the known numerical approximation algorithms for a nonsingular Toeplitz or Toeplitz-like linear system by employing distinct transformations of matrix structures, and we briefly comment on some natural research challenges, particularly some promising applications of our techniques to high precision computations.

Высокопроизводительный алгоритм Шермана-Моррисона обращения матриц на GPU

Matrix inversion is widely used in numerical methods, such as linear solvers, preconditioning for linear system, domain decomposition, digital image processing, etc. High-performance implementation of matrix inversion requires efficient matrix storage formats and optimal distribution of computations between computing devices. In this paper, we study the performance of traditional matrix inversion algorithms, such as LU-factorization and Gauss-Jordan, as well as the conjugate gradient method and the Sherman – Morrison formula. In the last two algorithms, matrix-vector products and scalar products are efficiently executed on multicore/manycore processors. We compare the performance of the algorithms on hybrid multi-CPU multi-GPU platforms, using the matrices from well-know test suites and from the numerical simulation of wrap spring.

PERFORMANCE ANALYSIS OF DIGITAL BEAMFORMING ALGORITHMS

Interference reduction is needed for being able to effectively communicate with mobile users. One method that can be used as a solution to reduce interference is using smart antenna technology. Along with interference reduction there is increase in capacity and coverage. Adaptive beam forming techniques forms a narrow beam and directs it towards desired users and thereby forms nulls in interference directions This improves signal to noise ratio. The study documented in this paper shows the performance of variants of the Least Mean Square (LMS) Algorithm based approaches, in the realm of beam formation, used earlier in adaptive filtering, aiming at reduction in complexity, without considerable degradation in performance. The proposed algorithms namely Signum- Error Least Mean Square Algorithm along with LMS Algorithm are tested for computational complexity in weight vector updating and observed that the ones proposed perform better than the conventional Least Mean Square algorithm with respect to reduced complexity in beam formation, making them suitable for high speed digital communication systems. This paper will analyze Least Mean square algorithm and Sample Matrix Inverse algorithm for the performance. The consequence will demonstrate the Least Mean Square as a Solution to jammer cancellation when compared to Sample Matrix Inverse.

Parallel Communication-Avoiding Algorithm for Triangular Matrix Inversion on Homogeneous and Heterogeneous Platforms

We address in this paper the parallelization of a recursive algorithm for large scale triangular matrix inversion based on the `Divide and Conquer' (D&C) paradigm. A set of different versions of an original sequential algorithm are first presented. A theoretical performance study permits to establish an accurate comparison between the designed algorithms. Afterwards, we develop in the second part of the paper, an optimal parallel avoiding-communication algorithm for a given number of available homogeneous and heterogeneous processors. To reach this target, we use a so called `non equitable and incomplete' version of the D&C paradigm consisting in recursively decomposing the original problem into two sub-problems of non equal sizes, then decomposing only one sub-problem in the same previous manner. The theoretical study is validated by a series of experiments achieved on three target platforms, namely an 8-core shared memory machine, a distributed memory cluster and a heterogeneous CPU-GPU cluster. The obtained results permit to illustrate the interest of the contribution.

Large-Scale MIMO Detection for 3GPP LTE: Algorithms and FPGA Implementations

Large-scale (or massive) multiple-input multiple-output (MIMO) is expected to be one of the key technologies in next-generation multi-user cellular systems, based on the upcoming 3GPP LTE Release 12 standard, for example. In this work, we propose - to the best of our knowledge - the first VLSI design enabling high-throughput data detection in single-carrier frequency-division multiple access (SC-FDMA)-based large-scale MIMO systems. We propose a new approximate matrix inversion algorithm relying on a Neumann series expansion, which substantially reduces the complexity of linear data detection. We analyze the associated error, and we compare its performance and complexity to those of an exact linear detector. We present corresponding VLSI architectures, which perform exact and approximate soft-output detection for large-scale MIMO systems with various antenna/user configurations. Reference implementation results for a Xilinx Virtex-7 XC7VX980T FPGA show that our designs are able to achieve more than 600 Mb/s for a 128 antenna, 8 user 3GPP LTE-based large-scale MIMO system. We finally provide a performance/complexity trade-off comparison using the presented FPGA designs, which reveals that the detector circuit of choice is determined by the ratio between BS antennas and users, as well as the desired error-rate performance.

Algorithm and FPGA implementation of interpolation‐based soft output MMSE MIMO detector for 3GPP LTE

The number of orthogonal frequency division multiplexing (OFDM) sub-carriers in modern wireless communication systems such as Third Generation Partnership Project (3GPP) long term evolution (LTE) can be as high as 2048. Channel matrix of data sub-carriers in conventional multiple-input-multiple-output (MIMO) receivers is derived by interpolating channel matrix of pilot sub-carriers and MIMO detection is performed on each interpolated channel matrix. Symbol detection on sub-carrier by sub-carrier basis demands a very high computational power. This study presents the first very-large scale integration implementation of a novel 4 × 4 interpolation-based soft output minimum-mean-squared error (MMSE) MIMO detector which combines channel estimation and MIMO detection. The detector computes MMSE matrix on pilot sub-carriers using proposed division-free matrix inversion algorithm and obtains MMSE matrix of data sub-carriers by interpolation, reducing computational complexity by more than 66%. The throughput of the implemented detector on Virtex-7 field programmable gate array (FPGA) exceeds 1 Gbps with 2.1 μs latency.

Optimization Capabilities of LMS and SMI Algorithm for Smart Antenna Systems

In the present paper convergence characteristics of Sample Matrix Inversion (SMI) and Least Mean Square (LMS) adaptive beam-forming algorithms (ABFA) are compared for a Smart Antenna System (SAS) in a multipath environment. SAS are employed at base stations for radiating narrow beams at the desired mobile users. The ABFA are incorporated in the digital signal processors for adjusting the weights to adjust the beam on the desired user and generate null in the direction of interferer. SMI and LMS algorithms are used with SAS for improving the performance of wireless communication system by optimizing the radiation pattern according to the signal environment. This can enhance the coverage and capacity of the system in multipath environment by reducing the interference and noise. The data rate can be enhanced by mitigating fading due to cancellation of multipath components. In this paper, optimization capabilities of SMI and LMS are considered by changing the parameters. The results reveal improvement in gain, speed of convergence and reduction in side-lobe level.

A fast parallel Gauss Jordan algorithm for matrix inversion using CUDA

The ability to invert large matrices quickly and accurately determines the effectiveness of a computational tool. Current literature suggests that time complexity of matrix inversion is 2 or higher. This paper redesigns the Gauss Jordan algorithm for matrix inversion on a CUDA platform to exploit the large scale parallelization feature of a massively multithreaded GPU. The algorithm is tested for various types of matrices and the performance metrics are studied and compared with CPU based parallel methods. We show that the time complexity of matrix inversion scales as n as long as n2 threads can be supported by the GPU.

Low complexity zero-forcing beamforming for distributed massive MIMO systems in large public venues

Distributed massive MIMO systems, which have high bandwidth efficiency and can accommodate a tremendous amount of traffic using algorithms such as zero-forcing beamforming (ZFBF), may be deployed in large public venues with the antennas mounted under-floor. In this case the channel gain matrix H can be modeled as a multi-banded matrix, in which off-diagonal entries decay both exponentially due to heavy human penetration loss and polynomially due to free space propagation loss. To enable practical implementation of such systems, we present a multi-banded matrix inversion algorithm that substantially reduces the complexity of ZFBF by keeping the most significant entries in H and the preceding matrix W. We introduce a parameter p to control the sparsity of H and W and thus achieve the tradeoff between the computational complexity and the system throughput. The proposed algorithm includes dense and sparse preceding versions, providing quadratic and linear complexity, respectively, relative to the number of antennas. We present analysis and numerical evaluations to show that the signal-to-interference ratio (SIR) increases linearly with p in dense precoding. In sparse preceding, we demonstrate the necessity of using directional antennas by both analysis and simulations. When the directional antenna gain increases, the resulting SIR increment in sparse precoding increases linearly with p, while the SIR of dense precoding is much less sensitive to changes in p.

GPS 항재밍을 위한 적응 배열 안테나의 성능 분석

In anti-jamming GPS receiver, adaptive signal processing techniques in which the radiation pattern of adaptive array antenna of elements may be adaptively changed used to reject interference, clutter, and jamming signals. In this paper, I describes adaptive signal processing technique using the sample matrix inversion(SMI) algorithm. This adaptive signal processing technique can be applied effectively to wideband/narrowband anti-jamming GPS receiver because it does not consider the satellite signal directions and GPS signal power level exists below the thermal noise. I also analyzed the effects of covariance matrix sample size and diagonal loading technique on the system performance of five-element circular array antenna. To attain near optimum performance, more samples required for calculation covariance matrix. Diagonal loading technique reduces the system nulling capability against low-power jamming signals, but this technique improves robustness of adaptive array antenna.Keywords : Adaptive Array Antenna(적응 배열 안테나), Anti-Jamming(항재밍), Adaptive Signal Processing(적응 신호처리), Sample Matrix Inversion(SMI), Diagonal Loading(대각 로드)

MIMO radar clutter mitigation based on joint beamforming and joint domain localized processing

In this article, we propose a space–time adaptive processing scheme via a generalized sidelobe canceler (GSC) architecture for airborne multiple-input multiple-output (MIMO) radar. This scheme employs the waveforms extracted by the matched filter bank that is cascaded at the receive end and utilizes digital beamforming technique to synthesize a certain number of transmit–receive beams, therefore, the operation of target detection in clutter environment can be conducted in all the directions of the formed beams in parallel. The GSC architecture is derived to implement adaptive reduced-rank (RR) clutter mitigation in a localized angle-Doppler space based on a novel RR multistage Wiener filter algorithm. The number of iterative stages in this algorithm is automatically selected in terms of a rank decision methodology. Meanwhile, beamforming and beam selecting methods are provided for this scheme, aiming at adaptively suppressing the clutter in localized domain. This scheme reverses the unavailability of the PA-efficient joint domain localized algorithm for MIMO radar. Moreover, it adapts to MIMO radar with arbitrary transmit–receive array space ratio. Even better, the proposed scheme has lower computation complexity than the traditional sample matrix inversion algorithm. The simulation results show that the proposed algorithm provide a signal-to-interference-plus-noise ratio improvement than traditional algorithms.

Clutter Nulling Performance of SMI in Amplitude Heterogeneous Clutter Environments

In order to investigate the clutter nulling performance of the sample matrix inversion (SMI) algorithm in amplitude heterogeneous environments, we derive an analytical expression for the average signal-to-interference plus noise ratio (SINR) loss based on random matrix theory. The results show that comparing with the case where the training data utilizes the homogeneous secondary samples (which are independent and identically distributed (IID) to the snapshot in the cell under test (CUT)), formulating the adaptive weight with the secondary samples which have stronger clutter power than that in CUT will increase the output SINR. Nevertheless, the achievable performance improvement might be quite limited. Conversely, selecting the secondary samples with weaker clutter power will degrade the performance.

Taylor 가중치와 LSMI 알고리즘을 이용한 광대역 적응형 빔형성 연구

This paper represents an adaptive beamforming technique to suppress interference or jamming signals in wideband. In order to maintain low side lobe level(SLL) at an antenna element level, Taylor-weighting was used. Also, to make a nulling beam pattern toward jammer's directions in wideband, we used the modified Loaded Sample Matrix Inversion(LSMI) algorithm and Tapped Delay Line(TDL). To verify the proposed algorithm, we applied it to a rectangular array antenna. Finally, the results show beam pattern with low SLL and jammers suppression.

Facilitating analysis of Monte Carlo dense matrix inversion algorithm scaling behaviour through simulation

With the latest developments in the area of advanced computer architectures, we are already seeing large-scale machines at petascale level and are faced with the exascale computing challenge. All these require scalability at system, algorithmic and mathematical model levels. In particular, efficient scalable algorithms are required to bridge the performance gap. Being able to predict application demeanour, performance and scalability of currently used software on new supercomputers of different architectures, varying sizes, and utilising distinct ways of intercommunication, can be of great benefit for researchers as well as application developers. This paper is concerned with scaling characteristics of Monte Carlo based algorithms for matrix inversion. The algorithmic behaviour on both, a shared memory and a large-scale cluster system will be predicted with the help of an extreme-scale high-performance computing (HPC) simulator.

Block recursive computation of generalized inverses

C � Abstract. A fully block recursive method for computing outer generalized inverses of given square matrix is introduced. The method is applicable even in the case when some of main diagonal minors of A are singular or A is singular. Computational complexity of the method is not harder than the matrix multiplication, under the assumption that the Strassen matrix inversion algorithm is used. A partially recursive algorithm for computing various classes of generalized inverses is also developed. This method can be efficiently used for the acceleration of the known methods for computing generalized inverses.