Copyright by Jason Todd Arbaugh 2004 The Dissertation Committee for Jason Todd Arbaugh certifies that this is the approved version of the following dissertation: Table Look -up CORDIC: Effective Rotations Through Angle Partitioning Committee: ______________________________ Earl E. Swartzlander, Jr., Supervisor ______________________________ Anthony P. Ambler ______________________________ John H. Davis ______________________________ Chang Yong Kang ______________________________ Nur A. To uba Table Look -up CORDIC: Effective Rotations Through Angle Partitioning by Jason Todd Arbaugh, B.S., M.S. Dissertation Presented to the Faculty of the Graduate School of the University of Texas at Austin in Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy The University of Texas at Austin December 2004 This dissertation is dedicated to all of the teachers in my life: those who have taught me theory in my classrooms, application in my jobs in industry, hard wo rk in my activities, integrity in my interactions, and balance in my life. Acknowledgements There are many people that deserve acknowledgement for their help in completing this dissertation. The help they provided, while not always technical in nature, was sorely needed at various times during my research, implementation, writing, and editing. I would like to thank Dr. Earl E. Swartzlander, Jr. for all of his teaching, help, and encouragement in graduate school. From his class on High Speed Arithmetic to his advice as my supervisor, I have enjoyed his humor, imagination, knowledge, and support. I have, and will continue to appreciate his mentorship. And even when he insists that I call him Earl, I will always regard him as Dr. Swartzlander. I am also thankful to the members of my dissertation committee Dr. Tony Ambler, Dr. John Davis, Dr. Chang Yong Kang, and Dr. Nur Touba. In addition to their time, which is so very valuable in today’s hectic world, they have provided valuable insights and timely sug gestions that have widened the scope and improved the content and quality of my dissertation. Brian P. Klawinski, Esquire, provided seemingly instantaneous assistance in the development of several of the C++ programs used in the calculation, evaluation, an d verification of the Table Look-up CORDIC algorithm. Mr. Paul T. Muehr helped with the setup of the backend tools. Without his help with the environment, scripts for the synthesis, auto place and route, and static timing analysis of my Verilog, I would not have been able to meet my deadlines. In addition to my committee and supervisor, Mr. Charles Pummill and Mrs. Sherri Staley assisted in reading, critiquing, and improving this dissertation. Their help on my dissertation has been greatly appreciated. My close friend, Mr. Chris W. Mobley, helped keep me motivated and healthy through the many long days of sitting in front of the computer typing. The v games of racquetball, the workouts, the pep talks, the reality checks, and the social interaction have kept me from going completely insane. I would like to thank Dr. Dhananjay Phatak, Associate Professor at the University of Maryland, Baltimore County, for providing my a copy of some of the C code used in simulating his Double Step Branching CORDIC algorithm. The C code was very helpful for figuring out some of the nuances of the algorithm that were not covered in the published paper. And finally, I would like to thank my entire family for helping make me who I am today: my mother and father, Sherri E. Staley and Wayne L. Arbaugh, my stepparents, Dr. Leo G. Staley and Kathy Arbaugh, my brothers and step -sister, “Huck” Fenn J. Arbaugh, Dr. Jesse L. Arbaugh, and Kristi Bogans. vi Table Look -up CORDIC: Effective Rotations Through Angle Partitioning Publication No . ___________________ Jason Todd Arbaugh, Ph.D. The University of Texas at Austin, 2004 Supervisor: Earl E. Swartzlander, Jr. This dissertation documents the development, derivation, verification, implementation, and evaluation of an improved versio n of the COordinate Rotation DIgital Computer (CORDIC) algorithm for calculating sine and cosine values. The CORDIC algorithm was originally developed to calculate trigonometric relationships in navigation systems using a family of linearly converging ite ration equations. The CORDIC algorithm computes numerous elementary functions including powers, exponentials, logarithms, trigonometric and hyperbolic functions. Many different versions of the classic CORDIC algorithm have been developed to enhance the performance of calculating these elementary functions. These alternative algorithms utilize methods that vary from using different number systems, to increasing the number of rotations performed in each iteration, to calculating the rotations using differen t Arc Tangent Radices. Even though each of vii these methods improves the performance of the CORDIC calculations, they still require a significant number of iterations through the CORDIC equations to obtain the final answer. The new CORDIC algorithm utilizes look-up tables and standard microprocessor arithmetic functional units to perform the calculations. The look-up tables employ either the traditional CORDIC or the new Parallel Arc Tangent Radix (ATR). The traditional CORDIC ATR combines multiple CORDIC iterations into a single effective rotation. The parallel ATR uses the exact angle value to perform the rotation rather than a summation of CORDIC ATR angles. Utilizing exact angles reduces the complexity of the decoders and permits parallel access to th e ROMs. The Table Look-up CORDIC (TLC) algorithm is shown to be correct through the development of a mathematical proof utilizing the polar form of the CORDIC iteration equations. The TLC algorithm and other versions of the CORDIC algorithm are implemente d in MatLab and simulated. The results of these simulations are compared with the bit correct value calculated with MatLab’s built in trigonometric functions to verify the correct operation. The same CORDIC algorithms are then modeled in Verilog. The Ver ilog models are synthesized to gates, placed and routed, and statically timed. The auto place and route of these circuits allows area estimates to be obtained for the different algorithms. The static timing analysis allows the worst -case path to be timed for frequency and latency comparisons. viii Table of Contents List of Tables ............................................................................................................. xii List of Figures ........................................................................................................... xiv List of Supplemental Files ........................................................................................ xvi Chapter 1 Introduction .................................................................................................1 1.1 Elementary Functions .................................................................................. 1 1.2 Previous Research ........................................................................................ 3 1.3 Table Look-up CORDIC .............................................................................. 6 Chapter 2 Algorithm Classes ....................................................................................... 8 2.1 Polynomial Approximation.......................................................................... 8 2.2 Rational Approximation............................................................................. 10 2.3 Linear Convergence ................................................................................... 13 2.4 Quadratic Convergence .............................................................................. 14 2.5 Research Opportunities .............................................................................. 16 Chapter 3 Classic CORDIC Algorithm...................................................................... 18 3.1 The Unit Circle .......................................................................................... 19 3.2 Calculation by Rotation ............................................................................. 21 3.3 Angle Selection.......................................................................................... 26 3.4 Rotation Direction...................................................................................... 28 3.5 Scale Factor................................................................................................ 28 3.6 Angle Criteria ............................................................................................. 29 3.7 Iteration Equations..................................................................................... 30 3.8 Pseudo Rotations........................................................................................ 31 Chapter 4 Previous Work ........................................................................................... 32 4.1 Unified CORDIC ....................................................................................... 33 4.2 Step -Branching CORDIC .......................................................................... 34 4.3 Double Step -Branching CORDIC .............................................................. 37 ix 4.4 Hybrid CORDIC ........................................................................................ 40 Chapter 5 Table Look-up CORDIC Algorit hm ......................................................... 42 5.1 Effective Rotations....................................................................................