A novel approach to fast discrete Fourier transform (Q1273886)

Discrete Fourier transform (DFT) is an important tool in digital signal processing. In the present paper, we propose a novel approach to performing DFT. We transform DFT into a form expressed in discrete moments via a modular mapping and truncating Taylor series expansion. From this, we extend the use of our systolic array for fast computation of moments without any multiplications to one that computes DFT with only a few multiplications and without any evaluations of exponential functions. The multiplication number used in our method is \(O(N\log 2 N/ \log 2\log 2 N)\) superior to \(O(N\log 2 N)\) in FFT. The execution time of the systolic array is only \(O(N\log 2 N/ \log 2\log 2 N)\) for 1-D DFT and \(O(Nk)\) for \(k\)-D DFT \((k\geq 2)\). The systolic implementation is a demonstration of the locality of dataflow in the algorithms and hence it implies an easy and potential hardware/VLSI realization. The approach is also applicable to DFT inverses.