1. ECE 734: Project Presentation Pankhuri May 8, 2013 Pankhuri May 8, 2013 64-point FFT Algorithm for OFDM Applications using 8-point DFT processor (radix-8)

2. Fast Fourier Transform Uses symmetry and periodicity properties of DFT to lower computation 64-point DFT computes a sequence X(f), Basis of FFT: DFT can be divided into smaller DFTs. e.g. radix-8 algorithm divides FFT into 8-point DFTs, radix-2: 2-point DFTs (BF) Uses symmetry and periodicity properties of DFT to lower computation 64-point DFT computes a sequence X(f), Basis of FFT: DFT can be divided into smaller DFTs. e.g. radix-8 algorithm divides FFT into 8-point DFTs, radix-2: 2-point DFTs (BF)

3. 64-point FFT Algorithm Details

4. Performance Improvement

5. 8-point FFT processor details FFT8 processor uses Winograd algorithm Minimizes multiplication (more expensive operation) at expense of increased additions and some more memory requirement. FFT8 (unit that performs base FFT operation) is pipelined One complex number is read from/written into input/output data buffer each clock cycle. (Total of 14 clock cycles) Supports clock frequency of up to 250 MHz FFT8 processor uses Winograd algorithm Minimizes multiplication (more expensive operation) at expense of increased additions and some more memory requirement. FFT8 (unit that performs base FFT operation) is pipelined One complex number is read from/written into input/output data buffer each clock cycle. (Total of 14 clock cycles) Supports clock frequency of up to 250 MHz

6. 8-point FFT processor algorithm 8 – point Winograd FFT

7. Processor Design Overview Synthesis and Simulation: Altera Quartus II Language: Verilog Target: Stratix IV FPGA Synthesis and Simulation: Altera Quartus II Language: Verilog Target: Stratix IV FPGA Buffer RAM1 Complex Input 8-point FFT unit 1 Twiddle factor multiplier Buffer RAM3 Buffer RAM2 8-point FFT unit 2 Complex Output

8. Processor Design Overview Data buffers: convert data from 8-inverse order to natural order e.g. without third buffer at the end, the output order is 0,8,16….56, 1,9,17,….. (8-inverse order). Use altsyncram, can store 2x64 complex data One bank is written to from previous stage, other can be read simultaneously. FFT Blocks: Only constant multiplications needed are 1/ √2 (bunch of shift and add operations) Data buffers: convert data from 8-inverse order to natural order e.g. without third buffer at the end, the output order is 0,8,16….56, 1,9,17,….. (8-inverse order). Use altsyncram, can store 2x64 complex data One bank is written to from previous stage, other can be read simultaneously. FFT Blocks: Only constant multiplications needed are 1/ √2 (bunch of shift and add operations)

9. Processor Design Overview Sixteen 8-point FFT units are avoided here by instead multiplexing the use of two units at expense of increased latency. Twiddle factor multiplier is a ROM having pre-calculated twiddle factors Complex multiplication is accomplished by breaking it into three multiplies and five additions. (lpm_mult mega function) (A + jB)(C + jD) = C(A-B) + B(C-D) + j(A(C-D) – C(A-B)) Sixteen 8-point FFT units are avoided here by instead multiplexing the use of two units at expense of increased latency. Twiddle factor multiplier is a ROM having pre-calculated twiddle factors Complex multiplication is accomplished by breaking it into three multiplies and five additions. (lpm_mult mega function) (A + jB)(C + jD) = C(A-B) + B(C-D) + j(A(C-D) – C(A-B))

10. Synthesis Results

11. Processor Design Overview

12. Learning Outcomes Details of various implementation issues of FFT processor design - resolving bandwidth issues when multiple stages are involved, reducing multiplier count (pipelining), total number of multiplications required (algorithm efficiency) Read about a LOT of FFT algorithms used for OFDM applications (before shortlisting this one). Various strategies to reduce computation employed in these algorithms especially popularity of radix-8 algorithms over radix-2. Details of various implementation issues of FFT processor design - resolving bandwidth issues when multiple stages are involved, reducing multiplier count (pipelining), total number of multiplications required (algorithm efficiency) Read about a LOT of FFT algorithms used for OFDM applications (before shortlisting this one). Various strategies to reduce computation employed in these algorithms especially popularity of radix-8 algorithms over radix-2.

13. Future Work & Applications Future Work: Modular design allows it to be used together with other 64-point FFTs to create larger size. (Much as this design is built using 8-point units) Structure can be configured in Xilinx, Altera, Alcatel, Lattice FPGA devices and ASIC Applications: OFDM modems, software defined radio, multichannel coding and many other high-speed real-time systems. Future Work: Modular design allows it to be used together with other 64-point FFTs to create larger size. (Much as this design is built using 8-point units) Structure can be configured in Xilinx, Altera, Alcatel, Lattice FPGA devices and ASIC Applications: OFDM modems, software defined radio, multichannel coding and many other high-speed real-time systems.