Real-valued algorithms for the FFT

Since 1965, when Cooley and Tukey published their famous paper on the radix-2 fast Fourier transform, much effort has gone into developing even more efficient algorithms. Most algorithms, however, do not directly handle real-valued data very well, and them exist several ways to solve that problem. This paper derives a new algorithm; the decimation-in-time real-valued split-radix FFT, which can transform any length N = 2^Msequence but uses less operations than any other known real-valued FFF, which is the fastest Cooley-Tukey real-valued transform in use. Instead of breaking the transform down equally as in traditional algorithms, the even and odd indexed parts are broken down differently in the split-radix algorithm. This gives a significant savings in both additions and multiplications over any fixed radix Cooley-Tukey FFT. The paper compares the split-radix transform with several of the already existing methods such as the Hartley transform, the prime factor, Winograd, Cooley-Tukey etc, and shows in which cases a specific algorithm is faster than the rest.