for kl=1:64
   x(kl)=sin((kl)*pi/64.0)+sin((kl)*pi/32.0);
 end
   plot (x)
  z=fft(x);
  whos
              Name        Size       Elements     Bytes    Density   Complex

                kl       1 by 1             1         8       Full      No 
                 x       1 by 64           64       512       Full      No 
                 z       1 by 64           64      1024       Full      Yes

   bar(z)
  figure
  figure(2)
   bar(x)
   z=fft(x);
  figure(1)
  bar(real(z))

   z=fftshift(y);
  bar(z)
    WARNING: Complex parts of X and/or Y arguments ignored.

  bar(imag(y))
  bar(abs(y))
  plot(fftshift(x))

  plot(x,'.',fftshift(x),'-')

for kl=1:64
 u(kl)=(kl)*pi/64.0;
 x(kl)=sin((kl)*pi/64.0);
 x(kl)=x(kl)+1/2*sin(2*(kl)*pi/64.0)+1/3*sin(3*(kl)*pi/64.0)+1/4*sin(4*(kl)*pi/64.0);
 x(kl)=x(kl)+1/5*sin(5*(kl)*pi/64.0)+1/6*sin(6*(kl)*pi/64.0)+1/7*sin(7*(kl)*pi/64.0);
 x(kl)=x(kl)+1/8*sin(8*(kl)*pi/64.0)+1/9*sin(9*(kl)*pi/64.0)+1/10*sin(10*(kl)*pi/64.0);
end
plot(u,x,'.',u,fftshift(x),'-')
z=fft(fftshift(x));
 whos
              Name        Size       Elements     Bytes    Density   Complex

               ans       1 by 5             5        80       Full      Yes
                kl       1 by 1             1         8       Full      No 
                 u       1 by 64           64       512       Full      No 
                 x       1 by 64           64       512       Full      No 
                 y       1 by 64           64      1024       Full      Yes
                 z       1 by 64           64      1024       Full      Yes

Grand total is 262 elements using 3160 bytes

  bar(abs(z))
  bar(fftshift(abs(z)))
%EVEN
for kl=1:64
 u(kl)=(kl)*pi/64.0;
 x(kl)=sin(2*(kl)*pi/32.0);
 x(kl)=x(kl)+1/4*sin(4*(kl)*pi/32.0);
 x(kl)=x(kl)+1/6*sin(6*(kl)*pi/32.0)+1/8*sin(8*(kl)*pi/32.0)+1/10*sin(10*kl*pi/32.0);
end
plot(u,x,'-')
y=fft(x);

 plot(y,'co')                                                  
 title('Fourier Coefficients in the Complex Plane');           
 xlabel('Real Axis');                                          
 ylabel('Imaginary Axis');                                     

 plot([y(3:4) y(62:63)],'co') 
                                                                 
plot(u,abs(x),'-',u,abs(fft(fft(x)/64)),'+')
plot(u,abs(x),'-',u,abs(fft(fft(fft(fft(x))))/(64*64)),'o')

plot(u,abs(x),'-',u,abs(fft(fft(x)/64)),'+',...
          ,u,abs(fft(fft(fft(fft(x))))/(64*64)),'o')

 plot([y(3:6) y(60:63)],'co') 
 plot([y(3:4) y(62:63)],'co')   %THINK ABOUT THIS
 bar (abs([y(3:6) y(60:63)]))

gg=zeros(58);
bar (abs([y(3:6) gg(1,:) y(60:63)]))
bar (fftshift(abs([y(3:6) gg(1,:) y(60:63)]) )   )

 gg = 0*( 1:58); rot90(gg) ;

 gg = ( 1:12) ./  ( 1:12)  
 gg = 1:12 ./ 1:12   GIVES ONLY A VECTOR) 
 gg= rot90( ( 1:12) ./  ( 1:12)  )