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15
Arbitrary Precision Real Arithmetic: Design and Algorithms
, 1996
"... this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms fo ..."
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this article the second representation mentioned above. We first recall the main properties of computable real numbers. We deduce from one definition, among the three definitions of this notion, a representation of these numbers as sequence of finite Badic numbers and then we describe algorithms for rational operations and transcendental functions for this representation. Finally we describe briefly the prototype written in Caml. 2. Computable real numbers
An Efficient Jacobilike Algorithm for Parallel Eigenvalue Computation
 IEEE TRANS. ON COMPUTERS
, 1993
"... A very fast Jacobilike algorithm for the parallel solution of symmetric eigenvalue problems is proposed. It becomes possible by not focusing on the realization of the Jacobi rotation with a CORDIC processor, but by applying approximate rotations and adjusting them to single steps of the CORDIC algo ..."
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Cited by 17 (8 self)
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A very fast Jacobilike algorithm for the parallel solution of symmetric eigenvalue problems is proposed. It becomes possible by not focusing on the realization of the Jacobi rotation with a CORDIC processor, but by applying approximate rotations and adjusting them to single steps of the CORDIC algorithm, i.e., only one angle of the CORDIC angle sequence defines the Jacobi rotation in each step. This angle can be determined by some shift, add and compare operations. Although mly linear convergence is obtained for the most simple version of the new algorithm, the overall operation count (shifts and adds) decreases dramatically. A slow increase of the number of involved CORDIC angles during the runtime retains quadratic convergence.
High Performance Rotation Architectures Based On Radix4 Cordic Algorithm
, 1997
"... Traditionally, CORDIC algorithms have employed radix2 in the first n/2 microrotations (n is the precision in bits) in order to preserve a constant scale factor. In this work we will present a full radix4 CORDIC algorithm in rotation mode and circular coordinates and its corresponding selection fun ..."
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Cited by 10 (5 self)
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Traditionally, CORDIC algorithms have employed radix2 in the first n/2 microrotations (n is the precision in bits) in order to preserve a constant scale factor. In this work we will present a full radix4 CORDIC algorithm in rotation mode and circular coordinates and its corresponding selection function, and we will propose an efficient technique for the compensation of the non constant scale factor. Three radix4 CORDIC architectures are implemented: a) a word serial architecture based on the zero skipping technique; b) a pipelined architecture; and c) an application specific architecture (the angles are known beforehand). The first two are general purpose implementations in redundant arithmetic (carrysave), whereas the last one is a simplification of the first two. The proposed architectures are time and/or area efficient when compared with already existing CORDIC architectures. 1. Introduction The CORDIC (COordinate Rotation DIgital Computer) algorithm was introduced by Volder [...
A HighSpeed CORDIC Algorithm and Architecture for DSP Applications
 in Proc. of the 1999 IEEE Workshop on Signal Processing Systems (SiPS’99
, 1999
"... This paper presents a novel CORDIC algorithm and architecture for the rotation and vectoring mode in circular coordinate systems in which the directions of all microrotations are precomputed while maintaining a constant scale factor. Thus, an examination of the sign of the angle or yremainder afte ..."
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Cited by 4 (3 self)
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This paper presents a novel CORDIC algorithm and architecture for the rotation and vectoring mode in circular coordinate systems in which the directions of all microrotations are precomputed while maintaining a constant scale factor. Thus, an examination of the sign of the angle or yremainder after each iteration is no longer required. By using MostSignificant Digit (MSD) first adder/multiplier, the critical path of the entire CORDIC architecture only requires (1:5n + 2) and (1:5n + 10) fulladders (n corresponds to the wordlength of the inputs) for rotation and vectoring modes, respectively. This is a speed improvement of about 30% compared to the previously fastest reported shared rotation and vectoring mode implementations. Additionally, there is a higher degree of freedom in choosing the pipeline cutsets due to the novel independence of iteration i and i1 in the CORDIC rotation. Optional pipelining can lead for example to an online delay of three clockcycles where every cloc...
ApplicationSpecific Architecture For Fast Transforms Based On The Successive Doubling Method, Part II: Orthogonal Transforms
, 1994
"... This is the second part of a study which deals with the design of an application specific architecture for fast orthogonal transforms based on the successive doubling method. In this work we will analyze six of the most important fast transforms: Complex Valued Fourier (CFFT), Walsh (FWT), Hartley ( ..."
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Cited by 3 (3 self)
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This is the second part of a study which deals with the design of an application specific architecture for fast orthogonal transforms based on the successive doubling method. In this work we will analyze six of the most important fast transforms: Complex Valued Fourier (CFFT), Walsh (FWT), Hartley (FHT), Real Valued Fourier (RFFT), Cosine (FCT), and Haar (FHrT). Out of them, only the CFFT and the FWT posses a data flow coinciding with the one generated by the successive doubling method. The other four require some type of hardware modification to guarantee the constant geometry of the successive doubling method. The FHT and the RFFT require that the radix r butterflies (N = r n , N is the length of the input sequence and n the number of stages of the transform) be processed as pairs and the data flow must be corrected when the transformation angle is not zero. The FCT, which we compute using an indirect method (n + 1 stages) based on the FHT, requires that the two first stages be eva...
Parallel Compensation of Scale Factor for the CORDIC Algorithm
 IN PRESS IN VLSI OF SIGNAL PROCESSING
, 1998
"... The compensation of scale factor imposes significant computation overhead on the CORDIC algorithm. In this paper we present two algorithms and the corresponding architectures (one for both rotation and vectoring modes and the other only for rotation mode) to perform the scaling factor compensation i ..."
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The compensation of scale factor imposes significant computation overhead on the CORDIC algorithm. In this paper we present two algorithms and the corresponding architectures (one for both rotation and vectoring modes and the other only for rotation mode) to perform the scaling factor compensation in parallel with the classical CORDIC iterations. With these methods, the scale factor compensation overhead is reduced to a couple of iterations for any word length. The architectures presented have been optimized for conventional and redundant arithmetic.
CORDIC Algorithms and Architectures
"... Digital signal processing (DSP) algorithms exhibit an increasing need for the efficient implementation of complex arithmetic operations. The computation of trigonometric functions, coordinate transformations or rotations of complex valued phasors is almost naturally involved with modern DSP algorith ..."
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Digital signal processing (DSP) algorithms exhibit an increasing need for the efficient implementation of complex arithmetic operations. The computation of trigonometric functions, coordinate transformations or rotations of complex valued phasors is almost naturally involved with modern DSP algorithms. Popular application examples are algorithms used in digital communication technology and in adaptive signal processing. While in digital communications, the straightforward evaluation of the cited functions is important, numerous matrix based adaptive signal processing algorithms require the solution of systems of linear equations, QR factorization or the computation of eigenvalues, eigenvectors or singular values. All these tasks can be e ciently implemented using processing elements performing vector rotations. The COordinate Rotation DIgital Computer algorithm (CORDIC) o ers the opportunity to calculate all the desired functions in a rather simple and elegant way. The CORDIC algorithm was rst introduced byVolder [1] for the computation
Pseudec: Implementation Of The ComputationIntensive Partran Functionality Using A Dedicated OnLine Cordic CoProcessor
"... This paper describes PSEUDEC, a dedicated coprocessor and the rationale behind it's design. The final goal of our work is to present a single chip solution with low power consumption for an advanced digital hearing aid based on a parameterized transformation of speech (PARTRAN). Characterization of ..."
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This paper describes PSEUDEC, a dedicated coprocessor and the rationale behind it's design. The final goal of our work is to present a single chip solution with low power consumption for an advanced digital hearing aid based on a parameterized transformation of speech (PARTRAN). Characterization of the constituent parts of the PARTRAN algorithm shows, that it is well suited for implementation on a heterogeneous architecture. The design strategy used identifies a subset suited for implementation on dedicated hardware, with computational complexity roughly equivalent to the performance of a standard 10 MIPS DSP. The subset of PARTRAN implemented by PSEUDEC performs PSEUdo DEComposition of a 12th order LPC polynomial. An adapted algorithm displays improved dynamic range compared to a conventional solution suited for DSP's, calculating the amplitude spectrum rather than the power spectrum. Highly pipelined CORDICunits optimized for the application replaces complex multiplication, trigono...
A VLSI processor for computing Linear and Circular CORDIC
, 1993
"... This report consider the generalised CORDIC algorithm from a theoretical point of view, involving principles, applications and range of convergence. Further is considered the derivation of an instance of this algorithm, allowing computations in circular and linear modes, denoted LCC. Since LCC is in ..."
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This report consider the generalised CORDIC algorithm from a theoretical point of view, involving principles, applications and range of convergence. Further is considered the derivation of an instance of this algorithm, allowing computations in circular and linear modes, denoted LCC. Since LCC is intended for use as a BCU in an ASIC VLSI implementation of the DITPOS algorithm, the specifications for LCC has been derived from this application, although it may find application in other designs as well, provided the numerical needs match the one offered by LCC. Also, the derivation of a highly pipelined array architecture with high resource utilisation is treated in detail. The architecture is based upon bitserial arithmetic, and a physical implementation of a prototype for this architecture is included. The implementation is based on the SOLO 1400 standard cell tool. Table of Contents 1 Introduction 1 2 Theoretical background 2 2.1 Planar Rotations and their computation using CORDI...
Number systems and Digit Serial Arithmetic
, 1997
"... this paper. By introducing an extra termination symbol, which signals that an operand was merely terminated due to its length exceeding some bound, operands can be kept as intervals, representing an imprecise operand. Operands terminated in the ordinary way can be taken to represent exact numbers. T ..."
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this paper. By introducing an extra termination symbol, which signals that an operand was merely terminated due to its length exceeding some bound, operands can be kept as intervals, representing an imprecise operand. Operands terminated in the ordinary way can be taken to represent exact numbers. The cube modeling a function of two variables, can be generalized to a hypercube modeling a polyhomographic function of n variables. For n = 3 the function is defined as: