clear all; close all;  % clear all variables and figures

  L=20;   % define the computational domain [-L/2,L/2]
  p=8;  % define the number of Fourier modes 2^p
  N=2^p;  % define the number of Fourier modes 2^p

  dt=L/N;
  t= (-L/2):dt:(L/2 - dt);
  
  yfun=inline('exp( -1*t.^2 )');
  %yfun=inline('exp( -1*t.^2 )+cos(t)');
  %yfun=inline('1*ones(1,length(t))')
 
    
  y=yfun(t);
  
  FFTlist=fft(y);
  FFTlist_indices=0:(N-1) ;
  
  %utshift=fftshift(ut); % shift FFT

  figure
  subplot(131)
  set(gca,'FontSize',18)
  plot(t,y); xlabel('t'); ylabel('y');

  subplot(132)
  set(gca,'FontSize',18)
  bar(FFTlist_indices,FFTlist); xlabel('FFTlistindices'); ylabel('FFTlist');
axis tight
  


  
%%% NOW UN-TRANSFORM %%%

y_recovered = ifft(FFTlist) ;
  
  subplot(133)
  set(gca,'FontSize',18)
  plot(t,y_recovered); xlabel('t'); ylabel('yrecovered');

  
%%% NOW INTERPRET IN TERMS OF ANGULAR FREQUENCIES K %%%

k_list=2*pi/L*[0:(N/2 -1)     (-N/2):-1];

  figure
  subplot(121)
  set(gca,'FontSize',18)
  plot(t,y); xlabel('t'); ylabel('y');

  subplot(122)
  set(gca,'FontSize',18)
  bar(k_list,FFTlist); xlabel('ang. freq. k'); ylabel('FFTlist');
  axis tight
  
%%% NOW INTERPRET IN TERMS OF TEMPORAL FREQUENCIES f %%%

temporal_freq_list=k_list/(2*pi) ; 

  figure
  subplot(121)
  set(gca,'FontSize',18)
  plot(t,y); xlabel('t'); ylabel('y');

  subplot(122)
  set(gca,'FontSize',18)
  bar(temporal_freq_list,FFTlist); xlabel('temporal freq. f'); ylabel('FFTlist');
  axis tight
  
  
  %%% Add some noise to y
  
  y_noisy=y+0.2*randn(size(y));
  
  FFTlist_noisy=fft(y_noisy);
  
  figure
  subplot(121)
  set(gca,'FontSize',18)
  plot(t,y_noisy); xlabel('t'); ylabel('ynoisy');

  subplot(122)
  set(gca,'FontSize',18)
  bar(temporal_freq_list,FFTlist_noisy); xlabel('temporal freq. f'); ylabel('FFTlistnoisy');
  axis tight
  
  
  
cutoff=.3
FFTlist_denoised=FFTlist_noisy;
%denoise
for j=1:length(FFTlist_noisy)
    if abs(temporal_freq_list(j))>=cutoff
        FFTlist_denoised(j)=0;
    end
end



%recover y denoised
y_denoised=ifft(FFTlist_denoised);

  figure
  subplot(221)
  set(gca,'FontSize',18)
  plot(t,y_noisy); xlabel('t'); ylabel('ynoisy');

  subplot(222)
  set(gca,'FontSize',18)
  bar(temporal_freq_list,FFTlist_noisy); xlabel('temporal freq. f'); ylabel('FFTlistnoisy');
  axis tight


  

  subplot(223)
  set(gca,'FontSize',18)
  bar(temporal_freq_list,FFTlist_denoised); xlabel('temporal freq. f'); ylabel('FFTlistdenoised');

  subplot(224)
  set(gca,'FontSize',18)
  plot(t,y_denoised); xlabel('t'); ylabel('ydenoised');
  axis tight