File:FFTexample16T.png

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.

using namespace std;
 * 1) include
 * 2) include
 * 3) include 
 * 4) include
 * 1) define z_type complex
 * 2) define Re(x) x.real
 * 3) define Im(x) x.imag
 * 4) define RI(x) x.real,x.imag
 * 5) define DB double
 * 6) 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"


 * 1) include"fafo.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); } fft(b,N,1); 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); 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 free(a); free(b); free(X); }
 * 1) define M(x,y) fprintf(o,"%6.4f %6.4f M\n",0.+x,0.+y);
 * 2) define L(x,y) fprintf(o,"%6.4f %6.4f L\n",0.+x,0.+y);
 * 3) define o(x,y) fprintf(o,"%6.4f %6.4f o\n",0.+x,0.+y);

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