Clemson University TigerPrints All Dissertations Dissertations 8-2008 Fast Fourier Transform Algorithms with Applications Todd Mateer Clemson University, [email protected] Follow this and additional works at: https://tigerprints.clemson.edu/all_dissertations Part of the Applied Mathematics Commons Recommended Citation Mateer, Todd, "Fast Fourier Transform Algorithms with Applications" (2008). All Dissertations. 231. https://tigerprints.clemson.edu/all_dissertations/231 This Dissertation is brought to you for free and open access by the Dissertations at TigerPrints. It has been accepted for inclusion in All Dissertations by an authorized administrator of TigerPrints. For more information, please contact [email protected]. FAST FOURIER TRANSFORM ALGORITHMS WITH APPLICATIONS A Dissertation Presented to the Graduate School of Clemson University In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Mathematical Sciences by Todd Mateer August 2008 Accepted by: Dr. Shuhong Gao, Committee Chair Dr. Joel Brawley Dr. Neil Calkin Dr. Kevin James ABSTRACT This manuscript describes a number of algorithms that can be used to quickly evaluate a polynomial over a collection of points and interpolate these evaluations back into a polynomial. Engineers define the “Fast Fourier Transform” as a method of solving the interpolation problem where the coefficient ring used to construct the polynomials has a special multiplicative structure. Mathematicians define the “Fast Fourier Transform” as a method of solving the multipoint evaluation problem. One purpose of the document is to provide a mathematical treatment of the topic of the “Fast Fourier Transform” that can also be understood by someone who has an understanding of the topic from the engineering perspective. The manuscript will also introduce several new algorithms that efficiently solve the multipoint evaluation problem over certain finite fields and require fewer finite field operations than existing techniques. The document will also demonstrate that these new algorithms can be used to multiply polynomials with finite field coefficients with fewer operations than Sch¨onhage’s algorithm in most circumstances. A third objective of this document is to provide a mathematical perspective of several algorithms which can be used to multiply polynomials whose size is not a power of two. Several improvements to these algorithms will also be discussed. Finally, the document will describe several applications of the “Fast Fourier Transform” algorithms presented and will introduce improvements in several of these applications. In addition to polynomial multiplication, the applications of polynomial division with remainder, the greatest common divisor, decoding of Reed-Solomon error-correcting codes, and the computation of the coefficients of a discrete Fourier series will be addressed. ii DEDICATION I dedicate this work to my wife Jennifer and my children Nathan, Laura, Jonathan, and Daniel. In terms of our family, the progress of my graduate research program has been measured through a collection of fifty quarters produced by the United States mint over roughly the same ten year period while I completed my graduate studies. I look forward to placing the final quarter on our “doctor school” map at the end of this year (2008) when I anticipate being finished with several publications related to the research presented in this manuscript. I have really treasured this time to be at home with my family while my children were young and for their “company” and “support” while completing this project. Much of this dissertation was written with little children in my lap or by my side as my wife and I worked together to get through these first few years of parenthood. I consider this time to be more valuable than the degree for which this dissertation was written. iii ACKNOWLEDGMENTS There are many people who deserve recognition for their role in my education which has led me to this point in my academic career. I will attempt to be as complete as possible here, knowing that are likely several people that I have left out. Obviously, my advisor Shuhong Gao and committee members deserve mention for being willing to mentor me through this process. This was an especially more challenging task given the fact that we were separated geographically during the entire writing of this dissertation. In fact, my advisor and I only saw each other two times in the three years it took to complete the research program. I would also like to thank Shuhong Gao for teaching a computer algebra class in 2001 which got me interested in this topic for my doctoral studies. I would also like to thank several anonymous reviewers who read over this entire manuscript several times. There are likely more places where the document can be improved and if there is an interest, I will make revisions to the manuscript as these places are pointed out to me. I would also like this opportunity to thank the teachers which taught me the subjects related to this research program. In particular, I would like to thank Rebecca Nowakowski, James Payne, Dale McIntyre, Yao-Huan Xu, Jenny Key, and Joel Brawley for equipping me with the algebra and complex variables background to take on this assignment. Also, Timothy Mohr, Frank Duda and Robert Mueller taught me the signals analysis which introduced me to the “engineering perspective” of the FFT. In classes taught by Joe Churm and Robert Jamison, I learned how the material of this dissertation is closely related to basic music theory. Through the instruction of Curt Frank, Jim Kendall, Fred Jenny, Michelle Claus, and James Peterson I learned iv the computer science and programming skills needed for this research project. Finally, I would like to thank my father Robert Mateer who taught me the trigonometry that is so fundamental to the topic of this dissertation. While I turned out to be an algebraist instead of following in his footsteps and making analysis (i.e. Calculus) my specialty, I still hope to become as good of a teacher as my father someday. ...but there is a God in heaven who reveals mysteries. Daniel 2:28a v TABLE OF CONTENTS Page TITLEPAGE.................................... i ABSTRACT .................................... ii DEDICATION ................................... iii ACKNOWLEDGMENTS ............................. iv LISTOFTABLES................................. x LISTOFFIGURES ................................ xi CHAPTER 1. INTRODUCTION............................. 1 1.1 The mathematician’s perspective . 3 1.2 Prerequisitemathematics . 4 1.3 Operationcountsofthealgorithms. 5 1.4 Multipoint polynomial evaluation . 7 1.5 Fast multipoint evaluation . 9 1.6 Lagrangian interpolation . 15 1.7 Fastinterpolation .......................... 17 1.8 Concludingremarks ......................... 23 2. MULTIPLICATIVE FAST FOURIER TRANSFORM ALGORITHMS 24 2.1 Thebitreversalfunction . 25 2.2 Classicalradix-2FFT ........................ 26 2.3 Twistedradix-2FFT ........................ 30 2.4 Hybridradix-2FFTs ........................ 35 2.5 Classicalradix-4FFT ........................ 36 2.6 Twistedradix-4FFT ........................ 41 2.7 Radix-8FFT............................. 43 2.8 Split-radixFFT ........................... 47 2.9 Modifiedsplit-radixFFT . 54 2.10 Theternaryreversalfunction . 59 2.11 Classicalradix-3FFT . 60 2.12 Twistedradix-3FFT ........................ 66 2.13 Hybridradix-3FFTs ........................ 73 vi Table of Contents (Continued) Page 2.14 Radix-3 FFT for symbolic roots of unity . 75 2.15 Concludingremarks . 77 3. ADDITIVE FAST FOURIER TRANSFORM ALGORITHMS . 78 3.1 VonzurGathen-GerhardadditiveFFT . 78 3.2 Wang-Zhu-CantoradditiveFFT . 87 3.3 ShiftedadditiveFFT ........................ 94 3.4 Gao’sadditiveFFT ......................... 99 3.5 AnewadditiveFFT......................... 108 3.6 Concludingremarks . 115 4. INVERSE FAST FOURIER TRANSFORM ALGORITHMS . 116 4.1 Classicalradix-2IFFT. 116 4.2 Twistedradix-2IFFT . 120 4.3 OthermultiplicativeIFFTs . 122 4.4 Wang-Zhu-Cantor additive IFFT . 129 4.5 AnewadditiveIFFT ........................ 132 4.6 Concludingremarks . 139 5. POLYNOMIAL MULTIPLICATION ALGORITHMS . 141 5.1 Karatsuba multiplication . 141 5.2 Karatsuba’s algorithm for other sizes . 145 5.3 FFT-based multiplication . 146 5.4 Sch¨onhage’salgorithm. 148 5.5 FFT-based multiplication using the new additive FFT algorithm 156 5.6 Comparison of the multiplication algorithms . 158 5.7 Concludingremarks . 160 6. TRUNCATED FAST FOURIER TRANSFORM ALGORITHMS . 162 6.1 AtruncatedFFTalgorithm. 164 6.2 An inverse truncated FFT algorithm . 167 6.3 Illustration of truncated FFT algorithms . 175 6.4 Truncated algorithms based on roots of xN 1 .......... 177 6.5 Truncated algorithms based on roots of xN − x .......... 179 6.6 Concludingremarks . .− . 180 7. POLYNOMIAL DIVISION WITH REMAINDER . 181 7.1 Classical division . 182 vii Table of Contents (Continued) Page 7.2 Newtondivision ........................... 183 7.3 Newton divison using the multiplicative FFT . 188 7.4 Newton divison for finite fields of characteristic 2 . 192 7.5 Concludingremarks . 194 8. THE EUCLIDEAN ALGORITHM . 195 8.1 TheEuclideanAlgorithm . 195 8.2 The Extended Euclidean Algorithm . 199 8.3 Normalized Extended Euclidean Algorithm . 209 8.4 The Fast Euclidean Algorithm . 210 8.5 Algorithm improvements due to the Fast Fourier Transform . 226 8.6 Concludingremarks . 231 9. REED-SOLOMON ERROR-CORRECTING CODES . 234 9.1 Systematic encoding of Reed-Solomon codewords . 235 9.2 A transform of the Reed-Solomon