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
NoneExample
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 -0.3116Program 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