16.5 Overview of real data FFTs

The functions for real data are similar to those for complex data. However, there is an important difference between forward and inverse transforms. The Fourier transform of a real sequence is not real. It is a complex sequence with a special symmetry:

     z_k = z_{n-k}^*

A sequence with this symmetry is called conjugate-complex or half-complex. This different structure requires different storage layouts for the forward transform (from real to half-complex) and inverse transform (from half-complex back to real). As a consequence the routines are divided into two sets: functions in gsl_fft_real which operate on real sequences and functions in gsl_fft_halfcomplex which operate on half-complex sequences.

Functions in gsl_fft_real compute the frequency coefficients of a real sequence. The half-complex coefficients c of a real sequence x are given by Fourier analysis,

     c_k = \sum_{j=0}^{n-1} x_j \exp(-2 \pi i j k /n)

Functions in gsl_fft_halfcomplex compute inverse or backwards transforms. They reconstruct real sequences by Fourier synthesis from their half-complex frequency coefficients, c,

     x_j = {1 \over n} \sum_{k=0}^{n-1} c_k \exp(2 \pi i j k /n)

The symmetry of the half-complex sequence implies that only half of the complex numbers in the output need to be stored. The remaining half can be reconstructed using the half-complex symmetry condition. This works for all lengths, even and odd—when the length is even the middle value where k=n/2 is also real. Thus only n real numbers are required to store the half-complex sequence, and the transform of a real sequence can be stored in the same size array as the original data.

The precise storage arrangements depend on the algorithm, and are different for radix-2 and mixed-radix routines. The radix-2 function operates in-place, which constrains the locations where each element can be stored. The restriction forces real and imaginary parts to be stored far apart. The mixed-radix algorithm does not have this restriction, and it stores the real and imaginary parts of a given term in neighboring locations (which is desirable for better locality of memory accesses).