Computes the Fourier coefficients of a real periodic sequence.

The routine FFTRF computes the discrete Fourier transform of a real vector of size N. It uses the Intel® Math Kernel Library or IBM Engineering and Scientific Subroutine Library for the computation, if available. Otherwise, the method used is a variant of the Cooley-Tukey algorithm that is most efficient when N is a product of small prime factors. If N satisfies this condition, then the computational effort is proportional to N log N.

Specifically, given an N-vector s = SEQ, FFTRF returns in c = COEF, if N is even:

If N is odd, cm is defined as above for m from 2 to (N + 1)/2.

We now describe a fairly common usage of this routine. Let f be a real valued function of time. Suppose we sample f at N equally spaced time intervals of length Δ seconds starting at time t0. That is, we have

The routine FFTRF treats this sequence as if it were periodic of period N. In particular, it assumes that f (t0) = f (t0 + NΔ). Hence, the period of the function is assumed to be T = NΔ.

Now, FFTRF accepts as input SEQ and returns as output coefficients c = COEF that satisfy the following relation when N is odd (N even is similar):

This formula is very revealing. It can be interpreted in the following manner. The coefficients produced by FFTRF produce an interpolating trigonometric polynomial to the data. That is, if we define

then, we have

Now, suppose we want to discover the dominant frequencies. One forms the vector P of length N/2 as follows:

These numbers correspond to the energy in the spectrum of the signal. In particular, Pk corresponds to the energy level at frequency

Furthermore, note that there are only (N + 1)/2 ≈ T/(2Δ) resolvable frequencies when N observations are taken. This is related to the Nyquist phenomenon, which is induced by discrete sampling of a continuous signal.

Similar relations hold for the case when N is even.

Finally, note that the Fourier transform has an (unnormalized) inverse that is implemented in FFTRB. The routine FFTRF is based on the real FFT in FFTPACK. The package FFTPACK was developed by Paul Swarztrauber at the National Center for Atmospheric Research.

If the Intel® Math Kernel Library or IBM Engineering and Scientific Subroutine Library is used, parameters computed by FFTRI are not used. In this case, there is no need to call FFTRI.

In this example, a pure cosine wave is used as a data vector, and its Fourier series is recovered. The Fourier series is a vector with all components zero except at the appropriate frequency where it has an N.