Efficient methods for FFT calculations using prime factor algorithm

Abstract

FFT algorithms are mainly used for the calculation of DFT because they are preferred due to their increased speed and higher efficiency, which arises due to the fact that for the calculation of an N-pt DFT, the sequence is broken into several segments and the DFT for each segment is calculated. However, for this many redundant memory spaces and the Butterfly structures are required for DFT calculations. Grouping the identical twiddle factors of different stages together reduces the number of memory references and the storage space due to twiddle factors, therein reducing the number of clock cycles needed for the complete implementation of the algorithm. For further reduction combine the DIT at the initial stage and DIF at the final stage. This technique is referred as DITF. It combines both the benefits of DIT and DIF and thus reducing the memory reference. The DITF algorithm reduces the number of the complex additions and multiplications in the calculation and hence in turn reduces the computational time. Prime Factor Algorithm provides an unified derivation of the Cooley-Tukey FFT and the Winograd Fourier Transform Algorithm (WFTA) FFT. In this method of reduction, arithmetic is used as an index mapping (a change of variables) to change the one-dimensional DFT into a two or higher dimensional DFTs. This is done in order to change a large problem into several easier smaller ones which makes its computing quicker and simpler and thereby reducing the computational time and does not require any butterfly structures. This algorithm is implemented in C language to verify their reduction in computation time. The same can also be verified using Code composer studio (CCS) and implemented in DSP Processors for the verification of reduced clock cycles. The reduction in memory spaces and memory references of the twiddle factors is evident in all the methods implicitly from the nature of the methods. Thus this can be an efficient method for the calculation of FFT algorithms.