## DG01MD

### Discrete Fourier transform, or inverse transform, of a complex signal

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

      To compute the discrete Fourier transform, or inverse transform,
      of a complex signal.

Specification

          SUBROUTINE DG01MD( INDI, N, XR, XI, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         INDI
          INTEGER           INFO, N
    C     .. Array Arguments ..
          DOUBLE PRECISION  XI(*), XR(*)

Arguments

Mode Parameters

      INDI    CHARACTER*1
              Indicates whether a Fourier transform or inverse Fourier
              transform is to be performed as follows:
              = 'D':  (Direct) Fourier transform;
              = 'I':  Inverse Fourier transform.

Input/Output Parameters

      N       (input) INTEGER
              The number of complex samples.  N must be a power of 2.
              N >= 2.

      XR      (input/output) DOUBLE PRECISION array, dimension (N)
              On entry, this array must contain the real part of either
              the complex signal z if INDI = 'D', or f(z) if INDI = 'I'.
              On exit, this array contains either the real part of the
              computed Fourier transform f(z) if INDI = 'D', or the
              inverse Fourier transform z of f(z) if INDI = 'I'.

      XI      (input/output) DOUBLE PRECISION array, dimension (N)
              On entry, this array must contain the imaginary part of
              either z if INDI = 'D', or f(z) if INDI = 'I'.
              On exit, this array contains either the imaginary part of
              f(z) if INDI = 'D', or z if INDI = 'I'.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value.

Method

      If INDI = 'D', then the routine performs a discrete Fourier
      transform on the complex signal Z(i), i = 1,2,...,N. If the result
      is denoted by FZ(k), k = 1,2,...,N, then the relationship between
      Z and FZ is given by the formula:

                      N            ((k-1)*(i-1))
             FZ(k) = SUM ( Z(i) * V              ),
                     i=1
                                      2
      where V = exp( -2*pi*j/N ) and j  = -1.

      If INDI = 'I', then the routine performs an inverse discrete
      Fourier transform on the complex signal FZ(k), k = 1,2,...,N. If
      the result is denoted by Z(i), i = 1,2,...,N, then the
      relationship between Z and FZ is given by the formula:

                     N             ((k-1)*(i-1))
             Z(i) = SUM ( FZ(k) * W              ),
                    k=1

      where W = exp( 2*pi*j/N ).

      Note that a discrete Fourier transform, followed by an inverse
      discrete Fourier transform, will result in a signal which is a
      factor N larger than the original input signal.

References

      [1] Rabiner, L.R. and Rader, C.M.
          Digital Signal Processing.
          IEEE Press, 1972.

Numerical Aspects

      The algorithm requires 0( N*log(N) ) operations.

Further Comments

      None

Example

Program Text

    *     DG01MD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX
          PARAMETER        ( NMAX = 128 )
    *     .. Local Scalars ..
          INTEGER          I, INFO, N
          CHARACTER*1      INDI
    *     .. Local Arrays ..
          DOUBLE PRECISION XI(NMAX), XR(NMAX)
    *     .. External Subroutines ..
          EXTERNAL         DG01MD
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) N, INDI
          IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99995 ) N
          ELSE
             READ ( NIN, FMT = * ) ( XR(I), XI(I), I = 1,N )
    *        Find the Fourier transform of the given complex signal.
             CALL DG01MD( INDI, N, XR, XI, INFO )
    *
             IF ( INFO.NE.0 ) THEN
                WRITE ( NOUT, FMT = 99998 ) INFO
             ELSE
                WRITE ( NOUT, FMT = 99997 )
                DO 20 I = 1, N
                   WRITE ( NOUT, FMT = 99996 ) I, XR(I), XI(I)
       20       CONTINUE
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' DG01MD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from DG01MD = ',I2)
    99997 FORMAT (' Components of Fourier transform are',//'   i',6X,
         $       'XR(i)',6X,'XI(i)',/)
    99996 FORMAT (I4,3X,F8.4,3X,F8.4)
    99995 FORMAT (/' N is out of range.',/' N = ',I5)
          END

Program Data

     DG01MD EXAMPLE PROGRAM DATA
       8     D
      -0.1862   0.1288
       0.3948   0.0671
       0.6788  -0.2417
       0.1861   0.8875
       0.7254   0.9380
       0.5815  -0.2682
       0.4904   0.9312
      -0.9599  -0.3116

Program Results

     DG01MD EXAMPLE PROGRAM RESULTS

     Components of Fourier transform are

       i      XR(i)      XI(i)

       1     1.9109     2.1311
       2    -1.9419    -2.2867
       3    -1.4070    -1.3728
       4     2.2886    -0.6883
       5     1.5059     1.3815
       6    -2.2271     0.2915
       7     0.1470     2.1274
       8    -1.7660    -0.5533

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