◆ Rfftmf()

One-dimensional real Fourier transform (multiple sequences)

Purpose: This routine computes the one-dimensional Fourier transform of multiple periodic sequences within a real array. This is referred to as the forward transform or Fourier analysis, transforming the sequence from physical to spectral space.
R(l*Jump) = (1/N)ΣR(l*Jump+j) (Σ for j = 0 to N-1) (l = 0 to Lot-1)

R(l*Jump+2k-1) = (2/N)ΣR(l*Jump+j)cos(2πjk/N) (Σ for j = 0 to N-1) (l = 0 to Lot-1, k = 1 to Nh)

R(l*Jump+2k) = (2/N)ΣR(l*Jump+j)sin(2πjk/N) (Σ for j = 0 to N-1) (l = 0 to Lot-1, k = 1 to Nh)

R(l*Jump+N-1) = (1/N)Σ(-1)^j R(l*Jump+j) (only if N is even) (Σ for j = 0 to N-1) (l = 0 to Lot-1)

(If N is even, Nh = N/2-1. If N is odd, Nh = (N-1)/2)

This transform is normalized since a call to Rfftmf followed by a call to Rfftmb (or vice-versa) reproduces the original array subject to algorithmic constraints, roundoff error, etc.

Parameters

[in]	Lot	Number of sequences to be transformed. (Lot >= 1)
[in]	Jump	Increment between the locations, in array R(), of the first elements of two consecutive sequences to be transformed. (Jump >= 1)
[in]	N	Length of the sequence to be transformed. (N >= 1) (The transform is most efficient when N is a product of small primes)
[in,out]	R()	Array R(LR - 1) (LR >= (Lot - 1)Jump + Inc(N - 1) + 1) [in] The sequences to be transformed. [out] The Fourier forward transformed sequences of data.
[in]	Wsave()	Array Wsave(LWsave - 1) (LWsave >= N + ln(N)/ln(2) + 4) Work data. Its contents must be initialized with a call to Rfftmi before the first call to Rfftmf or Rfftmb for a given transform length N.
[out]	Info	= 0: Successful exit. = -1: The argument Lot had an illegal value. (Lot < 1, or, Lot, Jump, N and Inc are inconsistent) = -2: The argument Jump had an illegal value. (Jump < 1) = -3: The argument N had an illegal value. (N < 1) = -4: The argument R() had an illegal value. (Array R() is not big enough) = -5: The argument Wsave() had an illegal value. (Array Wsave() is not big enough) = -7: The argument Inc had an illegal value. (Inc < 1)
[in]	Inc	(Optional) Integer increment between the locations, in array R(), of two consecutive elements within the sequence. (Inc >= 1) (default = 1)

Example Program: Compute the Fourier transform and backward transform of 2 sequences of 5 random data successively, and compare with the original data sequences.
Sub Ex_Rfftm()

Const N = 5, Lot = 2, Jump = N

Dim Wsave() As Double, R(Lot * N - 1) As Double, R0(Lot * N - 1) As Double

Dim LWsave As Long, Info As Long, I As Long, J As Long, K As Long

'-- Initialization

LWsave = N + Log(N) / Log(2) + 4

ReDim Wsave(LWsave - 1)

Call Rfftmi(N, Wsave, Info)

If Info <> 0 Then GoTo Err

'-- Generate test data

For I = 0 To Lot * N - 1

R(I) = Rnd()

R0(I) = R(I)

Next

'-- Forward transform

Call Rfftmf(Lot, Jump, N, R(), Wsave(), Info)

If Info <> 0 Then GoTo Err

'-- Backward transform

Call Rfftmb(Lot, Jump, N, R(), Wsave(), Info)

If Info <> 0 Then GoTo Err

'-- Print result

For J = 0 To Lot - 1

For I = 0 To N - 1

K = J * Jump + I

Debug.Print R0(K), R(K), R(K) - R0(K)

Next

Debug.Print

Next

Exit Sub

Err:

Debug.Print "Error in Rfftmi/Rfftmf/Rfftmb: Info =", Info

End Sub

Example Results: 0.774740099906921 0.774740099906921 1.11022302462516E-16

0.014017641544342 0.014017641544342 -5.55111512312578E-17

0.76072359085083 0.76072359085083 0

0.814490020275116 0.814490020275116 0

0.709037899971008 0.709037899971008 0

4.53527569770813E-02 4.53527569770813E-02 2.08166817117217E-17

0.414032697677612 0.414032697677612 0

0.862619340419769 0.862619340419769 1.11022302462516E-16

0.790480017662048 0.790480017662048 1.11022302462516E-16

0.373536169528961 0.373536169528961 0