FFT Algorithm Details

IDL's implementation of the fast Fourier transform is based on the Cooley-Tukey algorithm. The algorithm takes advantage of the fact that the discrete Fourier transform (DFT) of a discrete time series with an even number of points is equal to the sum of two DFTs, each half the length of the original. For data lengths that are a power of 2, this algorithm is used recursively, each iteration subdividing the data into smaller sets to be transformed. In the IDL FFT, this method is also extended to powers of 3 and 5. If the number of points in the original time series does not contain powers of 2, 3, or 5, the original data are still subdivided into data sets with lengths equal to the prime factors of N. The resulting subdivisions with lengths equal to prime numbers other than 2, 3, or 5 must be transformed using a slow DFT. The slow DFT is mathematically equivalent to the FFT, but requires N² operations instead of Nlog2(N).

This implementation means that the FFT function is fastest when the number of points is rich in powers of 2, 3, or 5. The slowest case is when the number of samples is a large prime number. In this case, a significant improvement in efficiency can be gained by padding the data set with zeros to increase the number of data points to a power of 2, 3, or 5.

For real input data of even lengths, the FFT algorithm also takes advantage of the fact that the real array can be packed into a complex array of half the length, and unpacked at the end, thus cutting the running time in half.