Polymorphic Algorithms FFT-Implementations That Share

We denote by a polymorphic radix-n FFT an abstract algorithm scheme that is shared by all radix-n FFT algorithms, similar to the way polymorphic data types share. Given such polymorphic algorithm, particular radix-n algorithms can be obtained by specialization, thus need not be implemented separately. How to accomplish sharing between different radix-n algorithms is not obvious: for example the four major radix-2 algorithms, defining different divide-and-conquer schemes and working for different input formats, have little in common at the implementation level. At a higher level of abstraction, however, it is possible to provide a unifying framework. In this paper we introduce a polymorphic radix-n FFT, mathematically based on the Cooley-Tukey mapping, and show how to effectively realize this mapping using techniques from generic programming. Since specializations can take place entirely at compile time, their generalization does not incur any run-time overhead. We implemented the polymorphic radix-n FFT as a C++ meta-program.