FFTCF

The routine FFTCF computes the discrete complex Fourier transform of a complex vector of size N. It uses the Intel® Math Kernel Library, Sun Performance 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, which 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. This considerable savings has historically led people to refer to this algorithm as the “fast Fourier transform” or FFT.

Specifically, given an N-vector x, FFTCF returns in c = COEF

Furthermore, a vector of Euclidean norm S is mapped into a vector of norm

Finally, note that we can invert the Fourier transform as follows:

This formula reveals the fact that, after properly normalizing the Fourier coefficients, one has the coefficients for a trigonometric interpolating polynomial to the data. An unnormalized inverse is implemented in FFTCB. FFTCF is based on the complex FFT in FFTPACK. The package FFTPACK was developed by Paul Swarztrauber at the National Center for Atmospheric Research.

Comments

1. Workspace may be explicitly provided, if desired, by use of F2TCF/DF2TCF. The reference is:

CALL F2TCF (N, SEQ, COEF, WFFTC, CPY)

The additional arguments are as follows:

WFFTC — Real array of length 4 * N + 15 initialized by FFTCI. The initialization depends on N. (Input)

CPY — Real array of length 2 * N. (Workspace)

If the Intel® Math Kernel Library, Sun Performance Library or IBM Engineering and Scientific Subroutine Library is used, WFFTC and CPY are not referenced.

2. The routine FFTCF is most efficient when N is the product of small primes.

3. The arrays COEF and SEQ may be the same.

4. If FFTCF/FFTCB is used repeatedly with the same value of N, then call FFTCI followed by repeated calls to F2TCF/F2TCB. This is more efficient than repeated calls to FFTCF/FFTCB.

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

Example

In this example, we input a pure exponential data vector and recover its Fourier series, which is a vector with all components zero except at the appropriate frequency where it has an N. Notice that the norm of the input vector is

but the norm of the output vector is N.

USE FFTCF_INT

USE CONST_INT

USE UMACH_INT

IMPLICIT NONE

INTEGER N

PARAMETER (N=7)

INTEGER I, NOUT

REAL TWOPI

COMPLEX C, CEXP, COEF(N), H, SEQ(N)

INTRINSIC CEXP

C = (0.,1.)

TWOPI = CONST('PI')

TWOPI = 2.0 * TWOPI

! Here we compute (2*pi*i/N)*3.

H = (TWOPI*C/N)*3.

! This loop fills out the data vector

! with a pure exponential signal of

! frequency 3.

DO 10 I=1, N

SEQ(I) = CEXP((I-1)*H)

10 CONTINUE

! Compute the Fourier transform of SEQ

CALL FFTCF (N, SEQ, COEF)

! Get output unit number and print

! results

CALL UMACH (2, NOUT)

WRITE (NOUT,99998)

99998 FORMAT (9X, 'INDEX', 8X, 'SEQ', 15X, 'COEF')

WRITE (NOUT,99999) (I, SEQ(I), COEF(I), I=1,N)

99999 FORMAT (1X, I11, 5X,'(',F5.2,',',F5.2,')', &

5X,'(',F5.2,',',F5.2,')')

END

Output

INDEX SEQ COEF

1 ( 1.00, 0.00) ( 0.00, 0.00)

2 (-0.90, 0.43) ( 0.00, 0.00)

3 ( 0.62,-0.78) ( 0.00, 0.00)

4 (-0.22, 0.97) ( 7.00, 0.00)

5 (-0.22,-0.97) ( 0.00, 0.00)

6 ( 0.62, 0.78) ( 0.00, 0.00)

7 (-0.90,-0.43) ( 0.00, 0.00)