High Speed Recursive Realisations of Small Length Dfts

This paper develops a prime radix transform algorithm that can be used to calculate the discrete Fourier transform (DFT) of a data block of length P, a prime, using fixed coefficient second order recursive filter sections: this simple hardware structure allows compact, high performance DFT processors to be developed. Further “massaging” of the prime radix algorithm produces the zero factor transform (ZFT) implemented using first order recursive filter sections, with scaling values of -1. The prime radix and zero factor transforms are limited to data block lengths that are prime: this constraint can be removed by re-factoring the transform equation and implementing the resultant algorithm as a two-stage recursion process. This method, the composite radix Fourier transform (CRAFT), can process data blocks of any length and is particularly suitable for transforming small blocks of data of length equal to 2 All of these algorithms can be implemented using first or second order recursive filter structures with fixed coefficient scaling multipliers, and high precision transforms can be produced by exploiting parallel error cancellation techniques. The simple hardware structure that results from this development has allowed high speed transform processors to be built, using both commercially available logic families and gatearrays of moderate complexity.