The Arithmetic Cube: error analysis and simulation

This paper examines the error performance and presents simulation results of the Arithmetic Cube. The Arithmetic Cube is a special purpose architecture for computing high speed convolution and the DFT. An error analysis is performed for convolution and the DFT, as computed on the Cube. An upper bound on the number of bits lost is derived. The Cube looses at most an extra two bits (four bits), while computing convolution (DFT), more than the number of bits lost if computed by the direct, limited precision convolution (DFT). A VHDL description of the Cube was written and simulations were run. Simulation results substantiate the derived upper bounds. A comparison of the Winograd Fourier-transform-algorithm (WFTA), computed by the Cube, and a rounded FFT, shows that the Cube is at least as accurate as the rounded FFT. Contrary to previous results, it is argued that the WFTA performs better, with respect to accuracy, than the Prime Factor Algorithm (PFA), if both are computed on the Cube