## DG01ND

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

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

Purpose

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

Specification

          SUBROUTINE DG01ND( 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
              Half the number of real samples.  N must be a power of 2.
              N >= 2.

      XR      (input/output) DOUBLE PRECISION array, dimension (N+1)
              On entry with INDI = 'D', the first N elements of this
              array must contain the odd part of the input signal; for
              example, XR(I) = A(2*I-1) for I = 1,2,...,N.
              On entry with INDI = 'I', the first N+1 elements of this
              array must contain the the real part of the input discrete
              Fourier transform (computed, for instance, by a previous
              call of the routine).
              On exit with INDI = 'D', the first N+1 elements of this
              array contain the real part of the output signal, that is
              of the computed discrete Fourier transform.
              On exit with INDI = 'I', the first N elements of this
              array contain the odd part of the output signal, that is
              of the computed inverse discrete Fourier transform.

      XI      (input/output) DOUBLE PRECISION array, dimension (N+1)
              On entry with INDI = 'D', the first N elements of this
              array must contain the even part of the input signal; for
              example, XI(I) = A(2*I) for I = 1,2,...,N.
              On entry with INDI = 'I', the first N+1 elements of this
              array must contain the the imaginary part of the input
              discrete Fourier transform (computed, for instance, by a
              previous call of the routine).
              On exit with INDI = 'D', the first N+1 elements of this
              array contain the imaginary part of the output signal,
              that is of the computed discrete Fourier transform.
              On exit with INDI = 'I', the first N elements of this
              array contain the even part of the output signal, that is
              of the computed inverse discrete Fourier transform.

Error Indicator

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

Method

      Let A(1),....,A(2*N) be a real signal of 2*N samples. Then the
      first N+1 samples of the discrete Fourier transform of this signal
      are given by the formula:

                   2*N           ((m-1)*(i-1))
           FA(m) = SUM ( A(i) * W              ),
                   i=1
                                                   2
      where m = 1,2,...,N+1, W = exp(-pi*j/N) and j = -1.

      This transform can be computed as follows. First, transform A(i),
      i = 1,2,...,2*N, into the complex signal Z(i) = (X(i),Y(i)),
      i = 1,2,...,N. That is, X(i) = A(2*i-1) and Y(i) = A(2*i). Next,
      perform a discrete Fourier transform on Z(i) by calling SLICOT
      Library routine DG01MD. This gives a new complex signal FZ(k),
      such that

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

      where k = 1,2,...,N, V = exp(-2*pi*j/N).  Using the values of
      FZ(k), the components of the discrete Fourier transform FA can be
      computed by simple linear relations, implemented in the DG01NY
      subroutine.

      Finally, let

           XR(k) = Re(FZ(k)), XI(k) = Im(FZ(k)),   k = 1,2,...,N,

      be the contents of the arrays XR and XI on entry to DG01NY with
      INDI = 'D', then on exit XR and XI contain the real and imaginary
      parts of the Fourier transform of the original real signal A.
      That is,

           XR(m) = Re(FA(m)),  XI(m) = Im(FA(m)),

      where m = 1,2,...,N+1.

      If INDI = 'I', then the routine evaluates the inverse Fourier
      transform of a complex signal which may itself be the discrete
      Fourier transform of a real signal.

      Let FA(m), m = 1,2,...,2*N, denote the full discrete Fourier
      transform of a real signal A(i), i=1,2,...,2*N. The relationship
      between FA and A is given by the formula:

                  2*N            ((m-1)*(i-1))
           A(i) = SUM ( FA(m) * W              ),
                  m=1

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

      Let

           XR(m) = Re(FA(m)) and XI(m) = Im(FA(m)) for m = 1,2,...,N+1,

      be the contents of the arrays XR and XI on entry to the routine
      DG01NY with INDI = 'I', then on exit the first N samples of the
      complex signal FZ are returned in XR and XI such that

           XR(k) = Re(FZ(k)), XI(k) = Im(FZ(k)) and k = 1,2,...,N.

      Next, an inverse Fourier transform is performed on FZ (e.g. by
      calling SLICOT Library routine DG01MD), to give the complex signal
      Z, whose i-th component is given by the formula:

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

      where i = 1,2,...,N and V = exp(2*pi*j/N).

      Finally, the 2*N samples of the real signal A can then be obtained
      directly from Z. That is,

           A(2*i-1) = Re(Z(i)) and A(2*i) = Im(Z(i)), for i = 1,2,...N.

      Note that a discrete Fourier transform, followed by an inverse
      transform will result in a signal which is a factor 2*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

    *     DG01ND EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX
          PARAMETER        ( NMAX = 128 )
    *     .. Local Scalars ..
          INTEGER          I, IEND, INFO, N
          CHARACTER*1      INDI
    *     .. Local Arrays ..
          DOUBLE PRECISION A(2*NMAX), XI(NMAX+1), XR(NMAX+1)
    *     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    *     .. External Subroutines ..
          EXTERNAL         DG01ND
    *     .. 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 = * ) ( A(I), I = 1,2*N )
    *        Copy the odd and even parts of A into XR and XI respectively.
             DO 20 I = 1, N
                XR(I) = A(2*I-1)
                XI(I) = A(2*I)
       20    CONTINUE
    *        Find the Fourier transform of the given real signal.
             CALL DG01ND( INDI, N, XR, XI, INFO )
             IF ( INFO.NE.0 ) THEN
                WRITE ( NOUT, FMT = 99998 ) INFO
             ELSE
                WRITE ( NOUT, FMT = 99997 )
                IEND = N
                IF ( LSAME( INDI, 'D' ) ) IEND = N + 1
                DO 40 I = 1, IEND
                   WRITE ( NOUT, FMT = 99996 ) I, XR(I), XI(I)
       40       CONTINUE
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' DG01ND EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from DG01ND = ',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

     DG01ND 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

     DG01ND EXAMPLE PROGRAM RESULTS

     Components of Fourier transform are

       i      XR(i)      XI(i)

       1     4.0420     0.0000
       2    -3.1322    -0.2421
       3     0.1862    -1.4675
       4    -2.1312    -1.1707
       5     1.5059    -1.3815
       6     2.1927    -0.1908
       7    -1.4462     2.0327
       8    -0.5757     1.4914
       9    -0.2202     0.0000

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