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<HTML>
<HEAD><TITLE>DF01MD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="DF01MD">DF01MD</A></H2>
<H3>
Sine transform or cosine transform of a real signal
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
To compute the sine transform or cosine transform of a real
signal.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
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(*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
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.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
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.
</PRE>
<B>Workspace</B>
<PRE>
DWORK DOUBLE PRECISION array, dimension (N+1)
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
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
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[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.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The algorithm requires 0( N*log(N) ) operations.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
* 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
</PRE>
<B>Program Data</B>
<PRE>
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
</PRE>
<B>Program Results</B>
<PRE>
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
</PRE>
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