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<H2><A Name="DG01MD">DG01MD</A></H2>
<H3>
Discrete Fourier transform, or inverse transform, of a complex 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 discrete Fourier transform, or inverse transform,
of a complex signal.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE DG01MD( INDI, N, XR, XI, INFO )
C .. Scalar Arguments ..
CHARACTER INDI
INTEGER INFO, N
C .. Array Arguments ..
DOUBLE PRECISION XI(*), XR(*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
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.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
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'.
</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>
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.
</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.
</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>
* 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
</PRE>
<B>Program Data</B>
<PRE>
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
</PRE>
<B>Program Results</B>
<PRE>
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
</PRE>
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