
This Article  
 
Share  
Bibliographic References  
Add to:  
Digg Furl Spurl Blink Simpy Del.icio.us Y!MyWeb  
Search  
 
ASCII Text  x  
N.D. Hemkumar, J.R. Cavallaro, "Redundant and OnLine CORDIC for Unitary Transformations," IEEE Transactions on Computers, vol. 43, no. 8, pp. 941954, August, 1994.  
BibTex  x  
@article{ 10.1109/12.295856, author = {N.D. Hemkumar and J.R. Cavallaro}, title = {Redundant and OnLine CORDIC for Unitary Transformations}, journal ={IEEE Transactions on Computers}, volume = {43}, number = {8}, issn = {00189340}, year = {1994}, pages = {941954}, doi = {http://doi.ieeecomputersociety.org/10.1109/12.295856}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  JOUR JO  IEEE Transactions on Computers TI  Redundant and OnLine CORDIC for Unitary Transformations IS  8 SN  00189340 SP941 EP954 EPD  941954 A1  N.D. Hemkumar, A1  J.R. Cavallaro, PY  1994 KW  parallel algorithms; matrix algebra; eigenvalues and eigenfunctions; special purpose computers; parallel architectures; computational complexity; digital arithmetic; CORDIC; unitary transformations; twosided unitary transformation; matrices; parallel algorithms; Jacobilike methods; eigenvalue; singular value decompositions; complex matrices; specialpurpose processor array architectures; nonredundant CORDIC; online CORDIC; redundant CORDIC; Coordinate Rotation Digital Computer. VL  43 JA  IEEE Transactions on Computers ER   
Twosided unitary transformations of arbitrary 2/spl times/2 matrices are needed in parallel algorithms based on Jacobilike methods for eigenvalue and singular value decompositions of complex matrices. This paper presents a twosided unitary transformation structured to facilitate the integrated evaluation of parameters and application of the typically required transformations using only the primitives afforded by CORDIC; thus enabling significant speedup in the computation of these transformations on specialpurpose processor array architectures implementing Jacobilike algorithms. We discuss implementation in (nonredundant) CORDIC to motivate and lead up to implementation in the redundant and online enhancements to CORDIC. Both variable and constant scale factor redundant (CFR) CORDIC approaches are detailed and it is shown that the transformations may be computed in 10n+/spl delta/ time, where n is the data precision in bits and /spl delta/ is a constant accounting for accumulated online delays. A more areaintensive approach using a novel online CORDIC encoded angle summation/difference scheme reduces computation time to 6n+/spl delta/. The area/time complexities involved in the various approaches are detailed.
[1] H. M. Ahmed, J. M. Delosme, and M. Morf, "Highly concurrent computing structures for matrix arithmetic and signal processing,"IEEE Comput., vol. 15, no. 1, pp. 6582, 1982.
[2] R. P. Brent and F. T. Luk, "The solution of singularvalue and symmetric eigenvalue problems on multiprocessor arrays,"SIAM J. Sci. Stat. Comput., vol. 6, no. 1, pp. 6984, 1985.
[3] R. P. Brent, F. T, Luk and C. F. Van Loan, "Computation of the singular value decomposition using meshconnected processors,"J. VLSI Comput. Syst., vol. 1, no. 3, pp. 242270, 1985.
[4] R. L. Causey, "Computing eigenvalues of nonHermitian matrices by methods of Jacobi type,"J. Soc. Indust. Appl. Math., vol. 6, no. 2, pp. 172181, 1958.
[5] J. R. Cavallaro and A. C. Elster, "A CORDIC processor array for the SVD of a complex matrix," inSVD and Signal Processing II (Algorithms, Analysis and Applications), R. Vaccaro, Ed. New York: Elsevier, 1991, pp. 227239.
[6] J. R. Cavallaro and F. T. Luk, "CORDIC arithmetic for an SVD processor,"J. Parallel Distributed Comput., vol. 5, no. 3, pp. 271290, June 1988.
[7] D. Daggett, "Decimalbinary conversions in CORDIC,"IRE Trans. Electron. Comput., vol. EC8, no. 3, pp. 335339, 1959.
[8] J. M. Delosme, "A processor for twodimensional symmetric eigenvalue and singular value arrays," inIEEE 21th Asilomar Conf. Circuits, Syst. and Comput., 1987, pp. 217221.
[9] J. M. Delosme, "Parallel implementations of the SVD using implicit CORDIC arithmetic," inSVD and Signal Processing II (Algorithms, Analysis and Applications), R. Vaccaro, Ed. New York: Elsevier, 1991, pp. 3356.
[10] E. F. Deprettere and A.J. van der Veen, "Parallel VLSI matrix pencil algorithm for high resolution direction finding,"IEEE Trans. Signal Processing, vol. 39, pp. 383394, 1991.
[11] A. M. Despain, "Fourier transform computers using CORDIC iterations,"IEEE Trans. Comput., vol. C23, no. 10, pp. 9931001, 1974.
[12] P. J. Eberlein, "On the Schur decomposition of a matrix for parallel computation,"IEEE Trans. Comput., vol. C36, no. 2, pp. 167174, 1987.
[13] M. D. Ercegovac and T. Lang, "Redundant and online CORDIC: Application to matrix triangularization and SVD,"IEEE Trans. Comput., vol. 39, no. 6, pp. 725740, 1990.
[14] G. E. Forsythe and P. Henrici, "The cyclic Jacobi method for computing the principal values of a complex matrix,"Trans. Amer. Math. Soc., vol. 94, no. 1, pp. 123, 1960.
[15] G. H. Golub and C. F. Van Loan,Matrix Computations, second ed. Baltimore, MD: Johns Hopkins Univ. Press, 1989.
[16] G. L. Haviland and A. A. Tuszynski, "A CORDIC arithmetic processor chip,"IEEE Trans. Comput., vol. C29, no. 2, pp. 6879, 1980.
[17] N. D. Hemkumar and J. R. Cavallaro, "An efficient parallel implementation of the Jacobi SVD algorithm for arbitrary matrices," Preprint 93077, AHPCRC, Univ. Minnesota, Minneapolis, MN, Aug. 1993.SIAM J. Matrix Anal. Appl., (in second review).
[18] N. D. Hemkumar and J. R. Cavallaro, "Efficient complex matrix transformations with CORDIC," inProc. IEEE 11th Symp. Comput. Arithmetic, Windsor, Canada, June 1993, pp. 122129.
[19] S. Hitotumatu, "Complex arithmetic through CORDIC," Kodai Math. Sem. Rep., vol. 26, pp. 176186, 1975.
[20] X. Hu, R. G. Harber, and S. C. Bass, "Expanding the range of convergence of the CORDIC algorithm,"IEEE Trans. Comput., vol. 40, no. 1, pp. 1321, 1991.
[21] R. Kumaresan and D. W. Tufts, "Singular value decomposition and improved frequency estimation using linear prediction,"IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP30, no. 4, pp. 671675, 1982.
[22] J. Lee and T. Lang, "SVD by constantfactor redundant CORDIC," inProc. 10th Symp. Comput. Arithmetic, 1991, pp. 264271.
[23] J. Lee and T. Lang, "Constantfactor redundant CORDIC for angle calculation and rotation,"IEEE Trans. Comput., vol. 41, no. 8, pp. 10161025, 1992.
[24] F. T. Luk, "A triangular processor array for computing singular values,"J. Linear Algebra Appl., vol. 77, pp. 259273, 1986.
[25] J.M. Speiser, "Signal processing computational needs," inProc. SPIE Advanced Algorithms and Architectures for Signal Processing I, 1986, vol. 696, pp. 26.
[26] G. W. Stewart, "A Jacobilike algorithm for computing the Schur decomposition of a nonHermitian Matrix,"SIAM J. Sci. Stat. Comput., vol. 6, no. 4, pp. 853864, 1985.
[27] N. Takagi, T. Asada, and S. Yajima, "Redundant CORDIC methods with a constant scale factor for sine and cosine computation,"IEEE Trans. Comput., vol. 40, no. 9, pp. 989995, 1991.
[28] J. Volder, "The CORDIC trigonometric computing technique,"IRE Trans. Electronic. Comput., vol. EC8, no. 3, pp. 330334, 1959.
[29] J. S. Walther, "A unified algorithm for elementary functions,"AFIPS Spring Joint Comput. Conf., pp. 379385, 1971.
[30] B. Yang and J. F. Böhme, "Reducing the computations of the SVD array given by Brent and Luk,"SIAM J. Matrix Anal. Appl., vol. 12, no. 4, pp. 713725, Oct. 1991.