FFT2D

The routine FFT2D computes the discrete complex Fourier transform of a complex two dimensional array of size (NRA = N) × (NCA = M). 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 and M are each products of small prime factors. If N and M satisfy this condition, then the computational effort is proportional to N M log N M. This considerable savings has historically led people to refer to this algorithm as the “fast Fourier transform” or FFT.

Specifically, given an N × M array a, FFT2D returns in c = COEF

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

Finally, note that an unnormalized inverse is implemented in FFT2B. The routine FFT2D 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 F2T2D/DF2T2D. The reference is:

CALL F2T2D (NRA, NCA, A, LDA, COEF, LDCOEF, WFF1, WFF2, CWK, CPY)

The additional arguments are as follows:

WFF1 — Real array of length 4 * NRA + 15 initialized by FFTCI. The initialization depends on NRA. (Input)

WFF2 — Real array of length 4 * NCA + 15 initialized by FFTCI. The initialization depends on NCA. (Input)

CWK — Complex array of length 1. (Workspace)

CPY — Real array of length 2 * MAX(NRA, NCA). (Workspace)

If the Intel® Math Kernel Library, Sun Performance Library or IBM Engineering and Scientific Subroutine Library is used, WFFT1, WFF2, CWK, and CPY are not referenced.

2. The routine FFT2D is most efficient when NRA and NCA are the product of small primes.

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

4. If FFT2D/FFT2B is used repeatedly, with the same values for NRA and NCA, then use FFTCI to fill WFF1(N = NRA) and WFF2(N = NCA). Follow this with repeated calls to F2T2D/F2T2B. This is more efficient than repeated calls to FFT2D/FFT2B.

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 compute the Fourier transform of the pure frequency input for a 5 × 4 array

for 1 ≤ n ≤ 5 and 1 ≤ m ≤ 4 using the IMSL routine FFT2D. The result

has all zeros except in the (3, 4) position.

USE FFT2D_INT

USE CONST_INT

USE WRCRN_INT

IMPLICIT NONE

INTEGER I, IR, IS, J, NCA, NRA

REAL FLOAT, TWOPI

COMPLEX A(5,4), C, CEXP, CMPLX, COEF(5,4), H

CHARACTER TITLE1*26, TITLE2*26

INTRINSIC CEXP, CMPLX, FLOAT

TITLE1 = 'The input matrix is below '

TITLE2 = 'The output matrix is below'

NRA = 5

NCA = 4

IR = 3

IS = 4

! Fill A with initial data

TWOPI = CONST('PI')

TWOPI = 2.0*TWOPI

C = CMPLX(0.0,1.0)

H = CEXP(TWOPI*C)

DO 10 I=1, NRA

DO 10 J=1, NCA

A(I,J) = CEXP(TWOPI*C*((FLOAT((I-1)*(IR-1))/FLOAT(NRA)+ &

FLOAT((J-1)*(IS-1))/FLOAT(NCA))))

10 CONTINUE

CALL WRCRN (TITLE1, A)

CALL FFT2D (A, COEF)

CALL WRCRN (TITLE2, COEF)

END

Output

The input matrix is below

1 2 3 4

1 ( 1.000, 0.000) ( 0.000,-1.000) (-1.000, 0.000) ( 0.000, 1.000)

2 (-0.809, 0.588) ( 0.588, 0.809) ( 0.809,-0.588) (-0.588,-0.809)

3 ( 0.309,-0.951) (-0.951,-0.309) (-0.309, 0.951) ( 0.951, 0.309)

4 ( 0.309, 0.951) ( 0.951,-0.309) (-0.309,-0.951) (-0.951, 0.309)

5 (-0.809,-0.588) (-0.588, 0.809) ( 0.809, 0.588) ( 0.588,-0.809)

The Output matrix is below

1 2 3 4

1 ( 0.00, 0.00) ( 0.00, 0.00) ( 0.00, 0.00) ( 0.00, 0.00)

2 ( 0.00, 0.00) ( 0.00, 0.00) ( 0.00, 0.00) ( 0.00, 0.00)

3 ( 0.00, 0.00) ( 0.00, 0.00) ( 0.00, 0.00) ( 20.00, 0.00)

4 ( 0.00, 0.00) ( 0.00, 0.00) ( 0.00, 0.00) ( 0.00, 0.00)

5 ( 0.00, 0.00) ( 0.00, 0.00) ( 0.00, 0.00) ( 0.00, 0.00)