FFT of a complex signal

I try to compute the FFT of the signal : x(t)*exp(I*angle(t)) where I = sqrt(-1) . So far I rewrite this signal as :

realPart = x(t)*cos(angle(t)) ;
imagPart = x(t)*sin(angle(t)) .

I use the following structure for my code:

• reading the file in which I have time, realPart, imagPart;

• save the total integration time and the timestep:

  deltat = time(2)-time(1) TotTIME = time(n)-time(1)

• call the FFT, where AR is the realPart and AI is the ImagPart, N is the number of points to use, M is the exponent (in power of two: 2**M) for the data

  CALL FFT(AR,AI,N,M)

• Write the output frequencies (O) and amplitude (AR):

  DO I = 1, N/2 O(I) = (I-1)*(1.d0/deltat - 1.d0/TotTIME)/(N-1) + 1.d0/TotTIME AR(I) = dsqrt( AR(I)**2 + AI(I)**2 ) / N END DO

The FFT proceeding is as follow:

`SUBROUTINE FFT (AR,AI,N,M)
!
! An example of the fast Fourier transform subroutine with N = 2**M.
! AR and AI are the real and imaginary part of data in the input and
! corresponding Fourier coefficients in the output.
! Copyright (c) Tao Pang 1997.
!
  IMPLICIT NONE
  INTEGER, INTENT (IN) :: N,M
  INTEGER :: N1,N2,I,J,K,L,L1,L2
  REAL(8) :: PI,A1,A2,Q,U,V
  REAL(8), INTENT (INOUT), DIMENSION (N) :: AR,AI
!
  PI = 4.D0*ATAN(1.D0)
  N2 = N/2
!
  N1 = 2**M
  IF(N1.NE.N) STOP 'Indices do not match'
!
! Rearrange the data to the bit reversed order
!
  L = 1
  DO K = 1, N-1
    IF (K.LT.L) THEN
      A1    = AR(L)
      A2    = AI(L)
      AR(L) = AR(K)
      AR(K) = A1
      AI(L) = AI(K)
      AI(K) = A2
    END IF
    J   = N2
    DO WHILE (J.LT.L)
      L = L-J
      J = J/2
    END DO
    L = L+J
  END DO
!
! Perform additions at all levels with reordered data
!
  L2 = 1
  DO L = 1, M
    Q  =  0.D0
    L1 =  L2
    L2 =  2*L1
    DO K = 1, L1
      U   =  DCOS(Q)
      V   = -DSIN(Q)
      Q   =  Q + PI/L1
      DO J = K, N, L2
        I     =  J + L1
        A1    =  AR(I)*U-AI(I)*V
        A2    =  AR(I)*V+AI(I)*U
        AR(I) =  AR(J)-A1
        AR(J) =  AR(J)+A1
        AI(I) =  AI(J)-A2
        AI(J) =  AI(J)+A2
      END DO
    END DO
  END DO
END SUBROUTINE FFT`

In a very simple example of real function ( AR = cos(t) ; AI = 0. ) I was be able to compute the correct position of the peak (in freq. or in period: 1/f ) as you can see from the figure:

The problem(s) are that I can not be able to reproduce,with this code the correct signal, in particular the position of the peaks in the frequencies and that I don't know if dividing the signal using Euler's formula (cos,sin) and passing those two vector to the AR and AI is the correct way.

Many thanks