FFT with general radix

If the number of points, N, can be expressed as , and if the computation algorithm is carried out via a succession of r-point transforms, the resultant FFT is known as radix-r algorithm. In a radix-r FFT, an elementary computation (EC) consists of an r-point DFT followed by multiplication of r results by appropriate twiddle factor. The number of ECs required is

                                                                      

which decreases as the r increases.

Certainly, the complexity of an EC increases with the increasing r. For r = 2, the EC (the butterfly) consists of the single complex multiplication and 2 complex additions; for r = 4, the EC requires 3 complex multiplications and many complex additions. 

  Suppose that we desire an N-point DFT where N is a composite number which can be factored into product of integers

    N = N1 N2 ... Nm

If, for example, N = 64 and m = 3, we might factor N into product 64 = 4 x 4 x 4, and 64- point transform can be viewed as a 3-dimensional 4 x 4 x 4 transform. 

  If N is a prime number so that the factorization of N is not possible, the original signal can be zero-padded and resulting new composite number of points can be factored.

  We illustrate in table drawn below the situation for N = 64. As 64 = 2⁶ , we can have a radix-2 FFT; alternatively, as 64 = 4³

, we can also have a radix-4 FFT. 

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