A NOVEL HIGH-THROUGHPUT FFT ARCHITECTURE FOR WIRELESS COMMUNICATION SYSTEMS A Thesis Presented to the Faculty of San Diego State University In Partial Fulfillment of the Requirements for the Degree Master of Science in Electrical Engineering by Nikhilesh Vinayak Bhagat Spring 2016 iii Copyright c 2016 by Nikhilesh Vinayak Bhagat iv DEDICATION To Aai and Pappa. v ABSTRACT OF THE THESIS A Novel High-Throughput FFT Architecture for Wireless Communication Systems by Nikhilesh Vinayak Bhagat Master of Science in Electrical Engineering San Diego State University, 2016 The design of the physical layer (PHY) of Long Term Evolution (LTE) standard is heavily influenced by the requirements for higher data transmission rate, greater spectral efficiency, and higher channel bandwidths. To fulfill these requirements, orthogonal frequency division multiplex (OFDM) was selected as the modulation scheme at the PHY layer. The discrete Fourier transform (DFT) and the inverse discrete Fourier transform (IDFT) are fundamental building blocks of an OFDM system. Fast Fourier transform (FFT) is an efficient implementation of DFT. This thesis focuses on a novel high-throughput hardware architecture for FFT computation utilized in wireless communication systems, particularly in the LTE standard. We implement a fully-pipelined FFT architecture that requires fewer number of computations. Particularly, we discuss a novel approach to implement FFT using the combined Good-Thomas and Winograd algorithms. It is found that the combined Good-Thomas and Winograd FFT algorithms provides a significantly more efficient FFT solution for a wide range of applications. A detailed analysis and comparison between different FFT algorithms and potential architectures suitable for the requirements of the LTE standard is presented. Theoretical results have been validated by the implementation of the proposed approach on a field-programmable gate array (FPGA). As demonstrated by the mathematical analysis, a significant reduction has been achieved in all the design parameters, such as computational delay and the number of arithmetic operations as compared to conventional FFT architectures currently used in various wireless communication standards. It is concluded that the proposed algorithm and its hardware architecture can be efficiently used as an enhanced alternative in the LTE wireless communication systems. vi TABLE OF CONTENTS PAGE ABSTRACT .................................................................................... v LIST OF TABLES.............................................................................. viii LIST OF FIGURES ............................................................................ ix ACKNOWLEDGMENTS ..................................................................... xi CHAPTER 1 INTRODUCTION ..................................................................... 1 1.1 Review of the FFT Algorithms ................................................. 1 1.2 Motivation ....................................................................... 2 1.3 Contribution of Thesis .......................................................... 3 1.4 Organization of Thesis .......................................................... 3 2 FAST FOURIER TRANSFORM ALGORITHMS................................... 4 2.1 Mapping to Two Dimensions ................................................... 4 2.2 The Cooley-Tukey FFT Algorithm ............................................. 5 2.2.1 Workload Computation.................................................... 7 2.3 Radix-2 Cooley-Tukey FFT .................................................... 9 2.3.1 Architecture of the Radix-2 FFT.......................................... 11 2.3.2 Workload Computation.................................................... 13 2.4 Radix-4 Cooley-Tukey FFT .................................................... 14 2.4.1 Architecture of Radix-4 FFT .............................................. 16 2.4.2 Workload Computation.................................................... 18 2.5 The Good-Thomas Prime-Factor Algorithm ................................... 19 2.5.1 Workload Computation.................................................... 22 2.5.2 Comparison and Summary of the FFT Algorithms ...................... 24 3 FAST FOURIER TRANSFORMS via CONVOLUTION ........................... 26 3.1 Rader’s Algorithm............................................................... 26 3.1.1 Workload Computation.................................................... 30 3.2 Winograd Short Convolution ................................................... 31 3.3 Winograd Fourier Transform Algorithm ....................................... 33 vii 3.4 Summary ........................................................................ 41 4 GOOD-THOMAS AND WINOGRAD PRIME-FACTOR FFT ALGORITHMS .. 44 4.1 Introduction...................................................................... 44 4.2 Data Format ..................................................................... 45 4.3 The Winograd FFT modules .................................................... 46 4.4 The Prime-Factor FFT Algorithm .............................................. 46 4.5 Architecture ..................................................................... 47 4.6 Hardware Design ................................................................ 47 4.6.1 Design Considerations..................................................... 51 4.6.2 Parallel Processing Architecture .......................................... 52 4.6.3 Matlab Design ............................................................. 53 4.6.4 Verilog Design ............................................................. 55 4.7 Testing ........................................................................... 57 4.8 Hardware Cost and Implementation Results ................................... 58 4.8.1 Latency and Throughput .................................................. 60 4.9 Analysis and Comparison....................................................... 61 5 APPLICATION OF FFT IN WIRELESS COMMUNICATION SYSTEMS ....... 63 5.1 Overview ........................................................................ 63 5.2 OFDM Technique ............................................................... 63 5.3 LTE Physical Layer ............................................................. 65 5.3.1 Generic Frame Structure .................................................. 65 5.3.2 LTE Parameters ............................................................ 66 6 CONCLUSION ........................................................................ 69 6.1 Future Work ..................................................................... 69 6.1.1 VLSI layout ................................................................ 70 BIBLIOGRAPHY .............................................................................. 71 viii LIST OF TABLES PAGE Table 2.1. Cooley-Tukey multiplications compared to one-dimensional DFT calculation . 9 Table 2.2. Comparison of the radix-2 and radix-4 algorithms ............................... 19 Table 2.3. Good-Thomas products savings ratio with respect to direct DFT calculation. .. 22 Table 2.4. Comparison of the Cooley-Tukey, Radix-2, Radix-4, and Good- Thomas FFT algorithms ............................................................... 25 Table 3.1. Determination of N k(x)............................................................ 32 Table 3.2. Multiplier coefficients for the 3-point WFT. ...................................... 34 Table 3.3. Multiplier coefficients for the 5-point WFT. ...................................... 35 Table 3.4. Multiplier coefficients for the 7-point WFT. ...................................... 36 Table 3.5. Multiplier coefficients for the 8-point WFT. ...................................... 36 Table 3.6. Multiplier coefficients for the 9-point WFT. ...................................... 38 Table 3.7. Multiplier coefficients for the 16-point WFT. ..................................... 39 Table 3.8. Computational requirements of Radix-2 and Winograd FFT algorithms ........ 41 Table 4.1. Resource utilization of the Winograd FFT algorithms ........................... 60 Table 4.2. Resource utilization of the combined Good-Thomas and Winograd FFT algorithm ......................................................................... 60 Table 4.3. Performance of the Winograd FFT algorithms ................................... 60 Table 4.4. Performance of the combined Good-Thomas and Winograd FFT algorithms .............................................................................. 61 Table 4.5. Resource utilization comparison with other fixed-point FFT proces- sors ..................................................................................... 61 Table 4.6. Performance comparison with other fixed-point FFT processors ................ 62 Table 5.1. FFT sizes and other physical parameters used in the current LTE standard ..... 66 Table 5.2. Available resource blocks and occupied sub-carriers ............................. 67 Table 5.3. Minimum required FFT lengths and other physical parameters for LTE ........ 67 Table 5.4. Possible FFT lengths using Good-Thomas algorithm and other phys- ical parameters for LTE ................................................................ 67 Table 5.5. Maximum possible FFT lengths using Good-Thomas algorithm and other physical parameters for LTE .................................................... 68 ix LIST OF FIGURES PAGE Figure 2.1. The two-dimensional