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
NoneExample
Program Text
* DG01ND EXAMPLE PROGRAM TEXT
* Copyright (c) 2002-2017 NICONET e.V.
*
* .. 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 -0.3116Program 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