## DG01MD

### Discrete Fourier transform, or inverse transform, of a complex signal

[Specification] [Arguments] [Method] [References] [Comments] [Example]

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
```   Rabiner, L.R. and Rader, C.M.
Digital Signal Processing.
IEEE Press, 1972.

```
Numerical Aspects
```  The algorithm requires 0( N*log(N) ) operations.

```
```  None
```
Example

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
```
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  -0.3116
```
Program 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
```