Cfft2b Cfft2f Cfft2i Rfft2b Rfft2c Rfft2f Rfft2i

Rfft2f() Sub Rfft2f ( L As  Long, M As  Long, R() As  Double, Wsave() As  Double, Info As  Long  ) Two-dimensional real Fourier transform Purpose This routine computes the two-dimensional Fourier transform of a periodic sequence within a real array. This is referred to as the forward transform or Fourier analysis, transforming the sequence from physical to spectral space. The full complex transform is given by: c(j,k) = (1/lm)ΣΣR(l1,m1)exp(-2πi(j*l1/L+k*m1/M)) (1st Σ for l1 = 0 to L-1, 2nd Σ for m1 = 0 to M-1) (j = 0 to L-1, k = 0 to M-1) (i is imaginary unit) This transform is normalized since a call to Rfft2f followed by a call to Rfft2b (or vice-versa) reproduces the original array subject to algorithmic constraints, roundoff error, etc. Parameters [in] L Number of elements to be transformed in the first dimension of the two-dimensional array R(). (L >= 1) (The transform is most efficient when L is a product of small primes) [in] M Number of elements to be transformed in the second dimension of the two-dimensional array R(). (M >= 1) (The transform is most efficient when M is a product of small primes) [in,out] R() Array R(LR1 - 1, LR2 - 1) (LR1 >= L, LR2 >= M) [in] The two dimensional real sequence data to be transformed. [out] The Fourier forward transformed two dimensional complex sequence data c(i,j) (i = 0 to l-1, j = 0 to m-1). The complex transform of the real sequence data has conjugate symmetry. That is, c(i,j) = conj(c(l-i,m-j)), so only half the transform is computed and packed back into the original array R() as shown in the further details below. [in] Wsave() Array Wsave(LWsave - 1) (LWsave >= L + 3*M + ln(L)/ln(2) + 2*ln(M)/ln(2) + 12) Work data. Its contents must be initialized with a call to Rfft2i before the first call to Rfft2f or Rfft2b for a given transform length L and M. [out] Info = 0: Successful exit. = -1: The argument L had an illegal value. (L < 1) = -2: The argument M had an illegal value. (M < 1) = -3: The argument R() is invalid. (Array R() is not big enough) = -4: The argument Wsave() is invalid. (Array Wsave() is not big enough) Further Details Let a(i,j) = re(c(i,j)) and b(i,j) = im(c(i,j)), then the packed data of the forward transform are stored in R() as following examples: l = m = 4 a(0,0)* a(0,1) b(0,1) a(0,2)* R(i,j) = a(1,0) a(1,1) a(1,2) a(1,3) b(1,0) b(1,1) b(1,2) b(1,3) a(2,0)* a(2,1) b(2,1) a(2,2)* l = m = 5 a(0,0)* a(0,1) b(0,1) a(0,2) b(0,2) a(1,0) a(1,1) a(1,2) a(1,3) a(1,4) R(i,j) = b(1,0) b(1,1) b(1,2) b(1,3) b(1,4) a(2,0) a(2,1) a(2,2) a(2,3) a(2,4) b(2,0) b(2,1) b(2,2) b(2,3) b(2,4) *: Imaginary part = 0 The remaining c(i,j) for i = (L+1)/2 to L-1 and j = 0 to M-1 can be obtained via the conjugate symmetry. For even L, c(L/2,j) = conj(c(L/2,M-j). Reference FFTPACK Example Program Compute the two-dimensional Fourier transform and backward transform of 4 rows x 4 columns random data sequence successively, and compare with the original data sequence. Sub Ex_Rfft2() Const L = 4, M = 4 Dim Wsave() As Double, R(L - 1, M - 1) As Double, R0(L - 1, M - 1) As Double Dim LWsave As Long, Info As Long, I As Long, J As Long '-- Initialization LWsave = L + 3 * M + Log(L) / Log(2) + 2 * Log(M) / Log(2) + 12 ReDim Wsave(LWsave - 1) Call Rfft2i(L, M, Wsave, Info) If Info <> 0 Then GoTo Err '-- Generate test data For I = 0 To L - 1 For J = 0 To M - 1 R(I, J) = Rnd() R0(I, J) = R(I, J) Next Next '-- Forward transform Call Rfft2f(L, M, R(), Wsave(), Info) If Info <> 0 Then GoTo Err '-- Backward transform Call Rfft2b(L, M, R(), Wsave(), Info) If Info <> 0 Then GoTo Err '-- Print result For J = 0 To M - 1 For I = 0 To L - 1 Debug.Print R0(I, J), R(I, J), R(I, J) - R0(I, J) Next Debug.Print Next Exit Sub Err: Debug.Print "Error in Rfft2i/Rfft2f/Rfft2b: Info =", Info End Sub Example Results 0.805006325244904 0.805006325244904 0 0.204303324222565 0.204303324222565 0 0.446774303913116 0.446774303913116 0 0.939462721347809 0.939462721347809 0 0.215638399124146 0.215638399124146 0 0.995530605316162 0.995530605316162 0 0.468557119369507 0.468557119369507 0 0.22608757019043 0.22608757019043 0 0.71610301733017 0.71610301733017 0 0.69307678937912 0.69307678937912 0 0.922678887844086 0.922678887844086 0 0.193300426006317 0.193300426006317 0 0.955126643180847 0.955126643180847 0 0.324254155158997 0.324254155158997 0 0.939402937889099 0.939402937889099 0 0.747899651527405 0.747899651527405 0