Error Analysis and Efficient Realization of the Multiplier-Less FFT-Like Transformation (ML-FFT) and Related Sinusoidal Transformations

Abstract

This paper studies the round-off analysis, design and implementation, and applications of the multiplier-less Fast Fourier Transform-like (ML-FFT) transformation proposed by Chan et al. [1, 2]. The ML-FFT parameterizes the twiddle factors in the conventional FFT algorithm as certain rotation-like matrices and approximates the associated parameters inside these matrices by the sum-of-power-of-two (SOPOT) or canonical signed digits representations, hence avoiding expensive multiplications. The error due to the SOPOT approximation is called the coefficient round-off error and it has been studied in [1, 2]. This paper studies the signal round-off error arising from internal rounding and develops a recursive noise model for ML-FFT. Using this model, a random search algorithm is proposed to minimize the hardware resources for realizing the ML-FFT subject to a prescribed output bit accuracy. To address the irregular structure of the ML-FFT due to the varying number of SOPOT terms used, a framework for its software implementation is also developed. The resulting algorithm has a regular implementation structure and is shown to offer a good performance similar to their floating-point counterpart. Finally, a new ML-FFT for real-valued input, called the ML-RFFT, is proposed. Because of the symmetry in the algorithm, it only requires about half the number of additions as required by the ML-FFT. Using the mappings between the DFT and the DCTs and DWTs, new ML-FFT-based transformations called ML-DCTs and ML-DWTs are derived. Design examples are given to demonstrate the usefulness of the proposed methods.