A general framework for efficient FPGA implementation of matrix product

Original article can be found at: http://www.medjcn.com/ Copyright Softmotor LimitedHigh performance systems are required by the developers for fast processing of computationally intensive applications. Reconfigurable hardware devices in the form of Filed-Programmable Gate Arrays (FPGAs) have been proposed as viable system building blocks in the construction of high performance systems at an economical price. Given the importance and the use of matrix algorithms in scientific computing applications, they seem ideal candidates to harness and exploit the advantages offered by FPGAs. In this paper, a system for matrix algorithm cores generation is described. The system provides a catalog of efficient user-customizable cores, designed for FPGA implementation, ranging in three different matrix algorithm categories: (i) matrix operations, (ii) matrix transforms and (iii) matrix decomposition. The generated core can be either a general purpose or a specific application core. The methodology used in the design and implementation of two specific image processing application cores is presented. The first core is a fully pipelined matrix multiplier for colour space conversion based on distributed arithmetic principles while the second one is a parallel floating-point matrix multiplier designed for 3D affine transformations.Peer reviewe

Using reconfigurable computing technology to accelerate matrix decomposition and applications

Matrix decomposition plays an increasingly significant role in many scientific and engineering applications. Among numerous techniques, Singular Value Decomposition (SVD) and Eigenvalue Decomposition (EVD) are widely used as factorization tools to perform Principal Component Analysis for dimensionality reduction and pattern recognition in image processing, text mining and wireless communications, while QR Decomposition (QRD) and sparse LU Decomposition (LUD) are employed to solve the dense or sparse linear system of equations in bioinformatics, power system and computer vision. Matrix decompositions are computationally expensive and their sequential implementations often fail to meet the requirements of many time-sensitive applications. The emergence of reconfigurable computing has provided a flexible and low-cost opportunity to pursue high-performance parallel designs, and the use of FPGAs has shown promise in accelerating this class of computation. In this research, we have proposed and implemented several highly parallel FPGA-based architectures to accelerate matrix decompositions and their applications in data mining and signal processing. Specifically, in this dissertation we describe the following contributions: • We propose an efficient FPGA-based double-precision floating-point architecture for EVD, which can efficiently analyze large-scale matrices. • We implement a floating-point Hestenes-Jacobi architecture for SVD, which is capable of analyzing arbitrary sized matrices. • We introduce a novel deeply pipelined reconfigurable architecture for QRD, which can be dynamically configured to perform either Householder transformation or Givens rotation in a manner that takes advantage of the strengths of each. • We design a configurable architecture for sparse LUD that supports both symmetric and asymmetric sparse matrices with arbitrary sparsity patterns. • By further extending the proposed hardware solution for SVD, we parallelize a popular text mining tool-Latent Semantic Indexing with an FPGA-based architecture. • We present a configurable architecture to accelerate Homotopy l1-minimization, in which the modification of the proposed FPGA architecture for sparse LUD is used at its core to parallelize both Cholesky decomposition and rank-1 update. Our experimental results using an FPGA-based acceleration system indicate the efficiency of our proposed novel architectures, with application and dimension-dependent speedups over an optimized software implementation that range from 1.5ÃÂ to 43.6ÃÂ in terms of computation time

HIGH PERFORMANCE, LOW COST SUBSPACE DECOMPOSITION AND POLYNOMIAL ROOTING FOR REAL TIME DIRECTION OF ARRIVAL ESTIMATION: ANALYSIS AND IMPLEMENTATION

This thesis develops high performance real-time signal processing modules for direction of arrival (DOA) estimation for localization systems. It proposes highly parallel algorithms for performing subspace decomposition and polynomial rooting, which are otherwise traditionally implemented using sequential algorithms. The proposed algorithms address the emerging need for real-time localization for a wide range of applications. As the antenna array size increases, the complexity of signal processing algorithms increases, making it increasingly difficult to satisfy the real-time constraints. This thesis addresses real-time implementation by proposing parallel algorithms, that maintain considerable improvement over traditional algorithms, especially for systems with larger number of antenna array elements. Singular value decomposition (SVD) and polynomial rooting are two computationally complex steps and act as the bottleneck to achieving real-time performance. The proposed algorithms are suitable for implementation on field programmable gated arrays (FPGAs), single instruction multiple data (SIMD) hardware or application specific integrated chips (ASICs), which offer large number of processing elements that can be exploited for parallel processing. The designs proposed in this thesis are modular, easily expandable and easy to implement. Firstly, this thesis proposes a fast converging SVD algorithm. The proposed method reduces the number of iterations it takes to converge to correct singular values, thus achieving closer to real-time performance. A general algorithm and a modular system design are provided making it easy for designers to replicate and extend the design to larger matrix sizes. Moreover, the method is highly parallel, which can be exploited in various hardware platforms mentioned earlier. A fixed point implementation of proposed SVD algorithm is presented. The FPGA design is pipelined to the maximum extent to increase the maximum achievable frequency of operation. The system was developed with the objective of achieving high throughput. Various modern cores available in FPGAs were used to maximize the performance and details of these modules are presented in detail. Finally, a parallel polynomial rooting technique based on Newton’s method applicable exclusively to root-MUSIC polynomials is proposed. Unique characteristics of root-MUSIC polynomial’s complex dynamics were exploited to derive this polynomial rooting method. The technique exhibits parallelism and converges to the desired root within fixed number of iterations, making this suitable for polynomial rooting of large degree polynomials. We believe this is the first time that complex dynamics of root-MUSIC polynomial were analyzed to propose an algorithm. In all, the thesis addresses two major bottlenecks in a direction of arrival estimation system, by providing simple, high throughput, parallel algorithms

Singular value decomposition based pipeline architecture for MIMO communication systems

This thesis presents a design, implementation and performance benchmark of custom hardware for computing Singular Value Decomposition (SVD) of the radio communication channel characteristic matrix. Software Defined Radio (SDR) is a concept in which the radio transceiver is implemented by software programs running on a processor. SVD of the channel characteristic matrix is used in pre-coding, equalization and beamforming for Multiple Input Multiple Output (MIMO) and Orthogonal Frequency Division Modulation (OFDM) communication systems (e.g., IEEE 802.11n). Since SVD is computationally intensive, it may require custom hardware to reduce the computing time. The pipeline processor developed in this thesis is suitable for computing the SVD of a sequence of 2 × 2 matrices. A stream of 2×2 matrices is sent to the custom hardware, which returns the corresponding streams of singular values and unitary matrices. The architecture is based on the two sided Jacobi method utilizing Coordinate Rotation Digital Computer (CORDIC) algorithms. A 2×2 SVD prototype was implemented on Field-Programmable Gate Array (FPGA) for SDR applications. The 2×2 SVD prototype design can output the singular values and the corresponding unitary matrices in pipeline while operating at a data rate of 324 MHz on a Virtex 6 (xc6vlx240t-lff1156) FPGA. The prototype design consists of fifty-five CORDIC cores which takes 32 percent of available logic on the FPGA. It achieves the optimal pipeline rate equaled to the maximum hardware clock rate. The depth of the pipeline (latency) is 173 clock-cycles for 16-bit data hardware. The proposed architecture provides performance gains over standard software libraries, such as the ZGESVD function of Linear Algebra PACKage (LAPACK) library, which is based on Golub-Kahan-Reinsch SVD algorithm, when running on standard processors. The ZGESVD function of LAPACK implemented in Intel’s Math Kernel Library (MKL) will achieve a projected data rate of 40 MHz on a 2.50 GHz Intel Quad (Q9300) CPU. The pipeline SVD hardware ban width equals the clock frequency and the data rate can reach 324 MHz on the ML605 board (Virtex 6 xc6vlx240t). The proposed architecture also has the potential to be easily extended to solve 4×4 SVD problems used in pre-coding and equalization schemes. The proposed algorithm and design have better performance for small matrices, even though the general timing complexity is n2 when compared to nlog(n) complexity of Brent-Luk-Van Loan (BLV) systolic array using non-pipeline 2×2 processors. The performance gain of the proposed design is at the cost of increased circuit area.M.S., Computer Engineering -- Drexel University, 201

Rapid Frequency Estimation

Frequency estimation plays an important role in many digital signal processing applications. Many areas have benefited from the discovery of the Fast Fourier Transform (FFT) decades ago and from the relatively recent advances in modern spectral estimation techniques within the last few decades. As processor and programmable logic technologies advance, unconventional methods for rapid frequency estimation in white Gaussian noise should be considered for real time applications. In this thesis, a practical hardware implementation that combines two known frequency estimation techniques is presented, implemented, and characterized. The combined implementation, using the well known FFT and a less well known modern spectral analysis method known as the Direct State Space (DSS) algorithm, is used to demonstrate and promote application of modern spectral methods in various real time applications, including Electronic Counter Measure (ECM) techniques