Number theoretic techniques applied to algorithms and architectures for digital signal processing

Many of the techniques for the computation of a two-dimensional convolution of a small fixed window with a picture are reviewed. It is demonstrated that Winograd's cyclic convolution and Fourier Transform Algorithms, together with Nussbaumer's two-dimensional cyclic convolution algorithms, have a common general form. Many of these algorithms use the theoretical minimum number of general multiplications. A novel implementation of these algorithms is proposed which is based upon one-bit systolic arrays. These systolic arrays are networks of identical cells with each cell sharing a common control and timing function. Each cell is only connected to its nearest neighbours. These are all attractive features for implementation using Very Large Scale Integration (VLSI). The throughput rate is only limited by the time to perform a one-bit full addition. In order to assess the usefulness to these systolic arrays a 'cost function' is developed to compare them with more conventional techniques, such as the Cooley-Tukey radix-2 Fast Fourier Transform (FFT). The cost function shows that these systolic arrays offer a good way of implementing the Discrete Fourier Transform for transforms up to about 30 points in length. The cost function is a general tool and allows comparisons to be made between different implementations of the same algorithm and between dissimilar algorithms. Finally a technique is developed for the derivation of Discrete Cosine Transform (DCT) algorithms from the Winograd Fourier Transform Algorithm. These DCT algorithms may be implemented by modified versions of the systolic arrays proposed earlier, but requiring half the number of cells

Faster fourier transformation: The algorithm of S. Winograd

The new DFT algorithm of S. Winograd is developed and presented in detail. This is an algorithm which uses about 1/5 of the number of multiplications used by the Cooley-Tukey algorithm and is applicable to any order which is a product of relatively prime factors from the following list: 2,3,4,5,7,8,9,16. The algorithm is presented in terms of a series of tableaus which are convenient, compact, graphical representations of the sequence of arithmetic operations in the corresponding parts of the algorithm. Using these in conjunction with included Tables makes it relatively easy to apply the algorithm and evaluate its performance

DFT algorithms for bit-serial GaAs array processor architectures

Systems and Processes Engineering Corporation (SPEC) has developed an innovative array processor architecture for computing Fourier transforms and other commonly used signal processing algorithms. This architecture is designed to extract the highest possible array performance from state-of-the-art GaAs technology. SPEC's architectural design includes a high performance RISC processor implemented in GaAs, along with a Floating Point Coprocessor and a unique Array Communications Coprocessor, also implemented in GaAs technology. Together, these data processors represent the latest in technology, both from an architectural and implementation viewpoint. SPEC has examined numerous algorithms and parallel processing architectures to determine the optimum array processor architecture. SPEC has developed an array processor architecture with integral communications ability to provide maximum node connectivity. The Array Communications Coprocessor embeds communications operations directly in the core of the processor architecture. A Floating Point Coprocessor architecture has been defined that utilizes Bit-Serial arithmetic units, operating at very high frequency, to perform floating point operations. These Bit-Serial devices reduce the device integration level and complexity to a level compatible with state-of-the-art GaAs device technology

Design of microprocessor-based hardware for number theoretic transform implementation

Number Theoretic Transforms (NTTs) are defined in a finite ring of integers Z (_M), where M is the modulus. All the arithmetic operations are carried out modulo M. NTTs are similar in structure to DFTs, hence fast FFT type algorithms may be used to compute NTTs efficiently. A major advantage of the NTT is that it can be used to compute error free convolutions, unlike the FFT it is not subject to round off and truncation errors. In 1976 Winograd proposed a set of short length DFT algorithms using a fewer number of multiplications and approximately the same number of additions as the Cooley-Tukey FFT algorithm. This saving is accomplished at the expense of increased algorithm complexity. These short length DFT algorithms may be combined to perform longer transforms. The Winograd Fourier Transform Algorithm (WFTA) was implemented on a TMS9900 microprocessor to compute NTTs. Since multiplication conducted modulo M is very time consuming a special purpose external hardware modular multiplier was designed, constructed and interfaced with the TMS9900 microprocessor. This external hardware modular multiplier allowed an improvement in the transform execution time. Computation time may further be reduced by employing several microprocessors. Taking advantage of the inherent parallelism of the WFTA, a dedicated parallel microprocessor system was designed and constructed to implement a 15-point WFTA in parallel. Benchmark programs were written to choose a suitable microprocessor for the parallel microprocessor system. A master or a host microprocessor is used to control the parallel microprocessor system and provides an interface to the outside world. An analogue to digital (A/D) and a digital to analogue (D/A) converter allows real time digital signal processing

Fast Fourier transform on a 3D FPGA

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2005.Includes bibliographical references (p. 53-55).Fast Fourier Transforms perform a vital role in many applications from astronomy to cellphones. The complexity of these algorithms results from the many computational steps, including multiplications, they require and, as such, many researchers focus on implementing better FFT systems. However, all research to date focuses on the algorithm within a 2-Dimensional architecture ignoring the opportunities available in recently proposed 3-Dimensional implementation technologies. This project examines FFTs in a 3D context, developing an architecture on a Field Programmable Gate Array system, to demonstrate the advantage of a 3D system.by Elizabeth Basha.S.M

Orthogonal transforms and their application to image coding

Imperial Users onl

Generic low power reconfigurable distributed arithmetic processor

Higher performance, lower cost, increasingly minimizing integrated circuit components, and higher packaging density of chips are ongoing goals of the microelectronic and computer industry. As these goals are being achieved, however, power consumption and flexibility are increasingly becoming bottlenecks that need to be addressed with the new technology in Very Large-Scale Integrated (VLSI) design. For modern systems, more energy is required to support the powerful computational capability which accords with the increasing requirements, and these requirements cause the change of standards not only in audio and video broadcasting but also in communication such as wireless connection and network protocols. Powerful flexibility and low consumption are repellent, but their combination in one system is the ultimate goal of designers. A generic domain-specific low-power reconfigurable processor for the distributed arithmetic algorithm is presented in this dissertation. This domain reconfigurable processor features high efficiency in terms of area, power and delay, which approaches the performance of an ASIC design, while retaining the flexibility of programmable platforms. The architecture not only supports typical distributed arithmetic algorithms which can be found in most still picture compression standards and video conferencing standards, but also offers implementation ability for other distributed arithmetic algorithms found in digital signal processing, telecommunication protocols and automatic control. In this processor, a simple reconfigurable low power control unit is implemented with good performance in area, power and timing. The generic characteristic of the architecture makes it applicable for any small and medium size finite state machines which can be used as control units to implement complex system behaviour and can be found in almost all engineering disciplines. Furthermore, to map target applications efficiently onto the proposed architecture, a new algorithm is introduced for searching for the best common sharing terms set and it keeps the area and power consumption of the implementation at low level. The software implementation of this algorithm is presented, which can be used not only for the proposed architecture in this dissertation but also for all the implementations with adder-based distributed arithmetic algorithms. In addition, some low power design techniques are applied in the architecture, such as unsymmetrical design style including unsymmetrical interconnection arranging, unsymmetrical PTBs selection and unsymmetrical mapping basic computing units. All these design techniques achieve extraordinary power consumption saving. It is believed that they can be extended to more low power designs and architectures. The processor presented in this dissertation can be used to implement complex, high performance distributed arithmetic algorithms for communication and image processing applications with low cost in area and power compared with the traditional methods