## DF01MD

### Sine transform or cosine transform of a real signal

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

Purpose

      To compute the sine transform or cosine transform of a real
      signal.

Specification

          SUBROUTINE DF01MD( SICO, N, DT, A, DWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         SICO
          INTEGER           INFO, N
          DOUBLE PRECISION  DT
    C     .. Array Arguments ..
          DOUBLE PRECISION  A(*), DWORK(*)

Arguments

Mode Parameters

      SICO    CHARACTER*1
              Indicates whether the sine transform or cosine transform
              is to be computed as follows:
              = 'S':  The sine transform is computed;
              = 'C':  The cosine transform is computed.

Input/Output Parameters

      N       (input) INTEGER
              The number of samples.  N must be a power of 2 plus 1.
              N >= 5.

      DT      (input) DOUBLE PRECISION
              The sampling time of the signal.

      A       (input/output) DOUBLE PRECISION array, dimension (N)
              On entry, this array must contain the signal to be
              processed.
              On exit, this array contains either the sine transform, if
              SICO = 'S', or the cosine transform, if SICO = 'C', of the
              given signal.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (N+1)

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),..., A(N) be a real signal of N samples.

      If SICO = 'S', the routine computes the sine transform of A as
      follows. First, transform A(i), i = 1,2,...,N, into the complex
      signal B(i), i = 1,2,...,(N+1)/2, where

         B(1) = -2*A(2),
         B(i) = {A(2i-2) - A(2i)} - j*A(2i-1) for i = 2,3,...,(N-1)/2,
         B((N+1)/2) = 2*A(N-1) and j**2 = -1.

      Next, perform a discrete inverse Fourier transform on B(i) by
      calling SLICOT Library Routine DG01ND, to give the complex signal
      Z(i), i = 1,2,...,(N-1)/2, from which the real signal C(i) may be
      obtained as follows:

         C(2i-1) = Re(Z(i)),  C(2i) = Im(Z(i)) for i = 1,2,...,(N-1)/2.

      Finally, compute the sine transform coefficients S ,S ,...,S
                                                        1  2      N
      given by

         S  = 0,
          1
                 {                     [C(k) + C(N+1-k)]     }
         S  = DT*{[C(k) - C(N+1-k)] - -----------------------},
          k      {                    [2*sin(pi*(k-1)/(N-1))]}

            for k = 2,3,...,N-1, and

         S = 0.
          N

      If SICO = 'C', the routine computes the cosine transform of A as
      follows. First, transform A(i), i = 1,2,...,N, into the complex
      signal B(i), i = 1,2,...,(N+1)/2, where

         B(1) = 2*A(1),
         B(i) = 2*A(2i-1) + 2*j*{[A(2i-2) - A(2i)]}
         for i = 2,3,...,(N-1)/2 and B((N+1)/2) = 2*A(N).

      Next, perform a discrete inverse Fourier transform on B(i) by
      calling SLICOT Library Routine DG01ND, to give the complex signal
      Z(i), i = 1,2,...,(N-1)/2, from which the real signal D(i) may be
      obtained as follows:

         D(2i-1) = Re(Z(i)),  D(2i) = Im(Z(i)) for i = 1,2,...,(N-1)/2.

      Finally, compute the cosine transform coefficients S ,S ,...,S
                                                          1  2      N
      given by

         S  = 2*DT*[D(1) + A0],
          1
                 {                     [D(k) - D(N+1-k)]     }
         S  = DT*{[D(k) + D(N+1-k)] - -----------------------},
          k      {                    [2*sin(pi*(k-1)/(N-1))]}

            for k = 2,3,...,N-1, and

         S  = 2*DT*[D(1) - A0],
          N
                  (N-1)/2
      where A0 = 2*SUM   A(2i).
                   i=1

References

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

      [2] Oppenheim, A.V. and Schafer, R.W.
          Discrete-Time Signal Processing.
          Prentice-Hall Signal Processing Series, 1989.

Numerical Aspects

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

Further Comments

      None

Example

Program Text

    *     DF01MD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX
          PARAMETER        ( NMAX = 129 )
    *     .. Local Scalars ..
          DOUBLE PRECISION DT
          INTEGER          I, INFO, N
          CHARACTER*1      SICO
    *     .. Local Arrays ..
          DOUBLE PRECISION A(NMAX), DWORK(NMAX+1)
    *     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    *     .. External Subroutines ..
          EXTERNAL         DF01MD
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) N, DT, SICO
          IF ( N.LE.1 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99994 ) N
          ELSE
             READ ( NIN, FMT = * ) ( A(I), I = 1,N )
    *        Compute the sine/cosine transform of the given real signal.
             CALL DF01MD( SICO, N, DT, A, DWORK, INFO )
    *
             IF ( INFO.NE.0 ) THEN
                WRITE ( NOUT, FMT = 99998 ) INFO
             ELSE
                IF ( LSAME( SICO, 'S' ) ) THEN
                   WRITE ( NOUT, FMT = 99997 )
                   DO 20 I = 1, N
                      WRITE ( NOUT, FMT = 99995 ) I, A(I)
       20          CONTINUE
                ELSE
                   WRITE ( NOUT, FMT = 99996 )
                   DO 40 I = 1, N
                      WRITE ( NOUT, FMT = 99995 ) I, A(I)
       40          CONTINUE
                END IF
             END IF
          END IF
    *
          STOP
    *
    99999 FORMAT (' DF01MD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from DF01MD = ',I2)
    99997 FORMAT (' Components of sine transform are',//'   i',6X,'A(i)',/)
    99996 FORMAT (' Components of cosine transform are',//'   i',6X,'A(i)',
         $       /)
    99995 FORMAT (I4,3X,F8.4)
    99994 FORMAT (/' N is out of range.',/' N = ',I5)
          END

Program Data

     DF01MD EXAMPLE PROGRAM DATA
      17     1.0     C
      -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
       0.8743

Program Results

     DF01MD EXAMPLE PROGRAM RESULTS

     Components of cosine transform are

       i      A(i)

       1    28.0536
       2     3.3726
       3   -20.8158
       4     6.0566
       5     5.7317
       6    -3.9347
       7   -12.8074
       8    -6.8780
       9    16.2892
      10   -17.0788
      11    21.7836
      12   -20.8203
      13    -7.3277
      14    -2.5325
      15    -0.3636
      16     7.8792
      17    11.0048

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