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Comparison of the discrete Fourier transform, shown with red, of a self–Fourier function $A(x)=\exp(-x^2/2)$, shown with black dots, to the result of the numerical evaluation of the the Fourier operator of array $A$, shown with blue. The discrete representation is performed with number of nodes $n\!=\!16$.

The image is generated in the following way.

The lines are drawn in the EPS format by the C++ code below. The result is concerted to PDF format.

The labels are added in the latex document below.

The result is concerted to the PNG format with default reaolution.

C++ generator of curves

File fafo.cin should be loaded to the working directory for the compilation of the code below.

#include <stdlib.h>
#include <complex>
using namespace std;
#define z_type complex<double>
#define Re(x)  x.real()
#define Im(x)  x.imag()
#define RI(x)  x.real(),x.imag()
#define DB double
#define DO(x,y) for(x=0;x<y;x++)
void ado(FILE *O, int X, int Y)
{       fprintf(O,"%c!PS-Adobe-2.0 EPSF-2.0\n",'%');
       fprintf(O,"%c%cBoundingBox: 0 0 %d %d\n",'%','%',X,Y);
       fprintf(O,"/M {moveto} bind def\n");
       fprintf(O,"/L {lineto} bind def\n");
       fprintf(O,"/S {stroke} bind def\n");
       fprintf(O,"/s {show newpath} bind def\n");
       fprintf(O,"/C {closepath} bind def\n");
       fprintf(O,"/F {fill} bind def\n");
       fprintf(O,"/o {.025 0 360 arc C F} bind def\n");
       fprintf(O,"/times-Roman findfont 20 scalefont setfont\n");
       fprintf(O,"/W {setlinewidth} bind def\n");
       fprintf(O,"/RGB {setrgbcolor} bind def\n");}
// #include "ado.cin"
// DB F(DB x){DB u=x*x; return u*(-3.+u)*exp(-x*x/2.);}
DB F(DB x){ return exp(-x*x/2.);}
main(){z_type * a, *b, c; int j,m,n, N=16; FILE *o;
       double step=sqrt(2*M_PI/N),x,y,u;
       a=(z_type *) malloc((size_t)((N+1)*sizeof(z_type)));
       b=(z_type *) malloc((size_t)((N+1)*sizeof(z_type)));
//for(j=0;j<N;j++) { x=step*(j-N/2); u=x*x; a[j]=b[j]=(3.+u*(-6.+u))*exp(-x*x/2); }
for(j=0;j<N;j++) { x=step*(j-N/2); u=x*x; a[j]=b[j]=F(x); }
for(j=0;j<N;j++) printf("%2d %18.15f %18.15f  %18.15f %18.15f\n", j, RI(a[j]), RI(b[j])  );
o=fopen("FFTexample16.eps","w"); ado(o,1024,780); 
#define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
#define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
#define o(x,y) fprintf(o,"%6.4f %6.4f o\n",0.+x,0.+y);
fprintf(o,"522 340 translate 100 100 scale\n");
// M(-5,0) L(5,0) M(0,0) L(0,1) fprintf(o,".01 W S\n");
// M(-5,1) L(5,1) M(-5,-1) L(5,-1)
for(m=-5;m<6;m++) {M(m,-3) L(m,3)} fprintf(o,".004 W S\n");
for(m=-3;m<4;m++) {M(-5,m) L(5,m)} fprintf(o,".004 W S\n");
fprintf(o,"1 setlinejoin 1 setlinecap\n");
DB *X; X=(DB *) malloc((size_t)((N+1)*sizeof(DB))); DO(j,N){ x=step*(j-N/2); X[j]=x; }
DO(j,N){x=X[j]; M(x,0)L(x,.15)} fprintf(o,".01 W S\n");
DO(j,N){x=X[j];y=Re(a[j]); if(j==0)M(x,y)else L(x,y);} fprintf(o,".04 W 0 .4 1 RGB S\n");
DO(j,N){x=X[j];y=Re(b[j]); if(j==0)M(x,y)else L(x,y);} fprintf(o,".04 W 1 0 0 RGB S\n");
// DO(j,N){x=X[j];y=Im(b[j]); if(j==0)M(x,y)else L(x,y);} fprintf(o,".04 W 1 0 0 RGB S\n");
// DO(j,N){x=X[j];y=100.*(Re(b[j])-F(x)); if(j==0)M(x,y)else L(x,y);} fprintf(o,"0.007 W 0 0 .3 RGB S\n");
printf("X[0]=%9.5f step=%9.6f\n",X[0],step);
// DO(m,101){x=-5.+.1*m; y=F(x); if(m/2*2==m)M(x,y)else L(x,y);} fprintf(o,".01 W 0 0 0 RGB S\n");
fprintf(o,".01 W 0 0 0 RGB S\n");
DO(m,101){x=-5.+.1*m; y=F(x); o(x,y)}
fprintf(o,"showpage\n%cTrailer",'%'); fclose(o);
 system("epstopdf FFTexample16.eps");
 system(    "open FFTexample16.pdf"); //these 2 commands may be specific for macintosh
 getchar(); system("killall Preview");// if run at another operational system, may need to modify

Latex generator of lables

FIle FFTexample16.pdf shouild be generated with the code above for the compiation of the latex document below.

\documentclass[12pt]{article} %<br> \usepackage{geometry} %<br> \paperwidth 1012pt %<br> \paperheight 740pt %<br> \textheight 800pt \topmargin -104pt %<br> \oddsidemargin -92pt %<br> \parindent 0pt %<br> \pagestyle{empty} %<br> \usepackage{graphicx} %<br> \newcommand \sx \scalebox %<br> \begin{document} %<br> \begin{picture}(1024,742) %<br> \put(4,0){\includegraphics{FFTexample16}} %<br> %\put(510,732){\sx{2.4}{$y$}} %<br> \put(510,634){\sx{2.5}{$y$}} %<br> \put(510,532){\sx{2.4}{$2$}} %<br> \put(510,432){\sx{2.4}{$1$}} %<br> %\put(510,320){\sx{2.4}{$0$}} %<br> \put(496,232){\sx{2.4}{$-\!1$}} %<br> \put(496,132){\sx{2.4}{$-\!2$}} %<br> \put(496, 32){\sx{2.4}{$-\!3$}} %<br> %<br> \put(104,310){\sx{2.4}{$-4$}} %<br> \put(204,310){\sx{2.4}{$-3$}} %<br> \put(304,310){\sx{2.4}{$-2$}} %<br> \put(404,310){\sx{2.4}{$-1$}} %<br> \put(522,310){\sx{2.4}{$0$}} %<br> \put(622,310){\sx{2.4}{$1$}} %<br> \put(722,310){\sx{2.4}{$2$}} %<br> \put(822,310){\sx{2.4}{$3$}} %<br> \put(923,310){\sx{2.4}{$4$}} %<br> \put(1016,311){\sx{2.5}{$x$}} %<br> \put(19,360){\sx{2.6}{$x_0$}} %<br> \put(79,360){\sx{2.6}{$x_1$}} %<br> \put(140,360){\sx{2.6}{$x_2$}} %<br> \put(199,360){\sx{2.6}{$x_3$}} %<br> \put(261,360){\sx{2.6}{$x_4$}} %<br> \put(327,360){\sx{2.6}{$x_5$}} %<br> \put(388,360){\sx{2.6}{$x_6$}} %<br> \put(450,360){\sx{2.6}{$x_7$}} %<br> \put(518,360){\sx{2.6}{$x_8$}} %<br> \put(578,360){\sx{2.6}{$x_9$}} %<br> \put(640,360){\sx{2.6}{$x_{10}$}} %<br> \put(704,360){\sx{2.6}{$x_{11}$}} %<br> \put(767,360){\sx{2.6}{$x_{12}$}} %<br> \put(829,360){\sx{2.6}{$x_{13}$}} %<br> \put(890,360){\sx{2.6}{$x_{14}$}} %<br> \put(950,360){\sx{2.6}{$x_{15}$}} %<br> \end{picture} %<br> \end{document} %


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