**Purpose**

To compute the discrete Fourier transform, or inverse transform, of a complex signal.

SUBROUTINE DG01MD( INDI, N, XR, XI, INFO ) C .. Scalar Arguments .. CHARACTER INDI INTEGER INFO, N C .. Array Arguments .. DOUBLE PRECISION XI(*), XR(*)

**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.

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'.

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value.

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.

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

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

None

**Program Text**

* DG01MD 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, 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

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

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