DFT Computation Using Gauss-Eisenstein Basis: FFT Algorithms and VLSI Architectures

Abstract

A joint numerical representation based on both Gaussian and Eisenstein integers is proposed. This Gauss-Eisenstein representation maps complex numbers into four-tuples of integers with arbitrarily high precision. The representation furnishes the computation of the 3-, 6-, and 12-point discrete Fourier transform (DFT) at any desired accuracy. The associated fast algorithms based on the Gauss-Eisenstein integers are error-free up to the final reconstruction step, which can be realized in hardware as a multiplierless implementation. The introduced methods are compared with competing algorithms in terms of arithmetic complexity. We propose three FRS architectures based on the following methods: Dempster-McLeod representation, expansion factor, and addition aware quantization. The Gauss-Eisenstein 12-point DFT is physically realized on a Xilinx Virtex 6 FPGA device with maximum clock frequency of 302 MHz for the expansion factor FRS with real-time throughput of 3:62 × 109 coefficients/s. The FPGA verified digital designs were synthesized, mapped, placed and finally routed for 0:18mm CMOS technology assuming a 1.8 V DC supply employing Austria Micro Systems (AMS) standard-cell library (hitkit version 4.11). The routed ASIC is predicted to operate at a maximum frequency of 505 MHz for the expansion factor FRS with potential real-time throughput of 6:06 × 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">9</sup> coefficients/s.