fft importance of time sampled

I guessed that the longer one samples, the most exact result with fft. And so it seems to do with frequency, but not at all with amplitude. The amplitude decreases the longer the time sampled. Did I do something wrong ? Is this reasonable? Thanks to any help

% T=[.01 .02 .1 .2 1];
% T=[1:10]
T=[1 5 10];
n=10; % maximal number of harmonic

freq = 5;
Fs = 150e4; % Sampling frequency (frecuencia de muestreo)
nfft = 1e7; % Length of FFT % number of fft bins

tic

for m=1:length(T) 

      t = 0:1/Fs:T(m); % Time vector of 1 second
      x = cos(2*pi*t*freq); % sine wave of f Hz.
      x = x-mean(x);% restamos de la fcn 'x' su DC offset

      figure(1);
      subplot(length(T),1,m);
      plot(t,x,'.');

      % Take fft, padding with zeros so that length(X) is equal to nfft
      X = fft(x,nfft);
      % FFT is symmetric, throw away second half
      X = X(1:nfft/2);
      % Scaling is done here using the number of samples: length(x)/2
      X=X/(length(x)/2);
      % Take the magnitude of fft of x
      mx = abs(X);
      % Frequency vector
      f_fft = (0:nfft/2-1)*Fs/nfft;

      f_Axis=f_fft(1:nfft/2);

      % picos de amplitud
      pks= findpeaks(mx(1:nfft/2));
      n=min(n,length(pks));
      pks_sort=sort(pks,'descend');% vector de picos

      for k =  1:n
          locs=find(mx(1:nfft/2)==pks_sort(k));
          f(k)=f_Axis(locs);
      end

      % bar plot freq y magnitude
      figure
      uds='Amp';
      pks_sort=pks_sort(:); % to make it column array for text of bar plot
      f=f(:); % to make it column array for text of bar plot
      subplot(1,2,1)
      bar(pks_sort(1:n),'r');
      ylabel({sprintf('Amplitude (%s)'...
          ,uds)},'fontweight','bold','fontsize',16);
      text(1:n,pks_sort(1:n),num2str(pks_sort(1:n),'%.2f'),... 
          'HorizontalAlignment','center',...
          'VerticalAlignment','bottom')

      subplot(1,2,2)
      bar(f(1:n),'b');
      ylabel('Frequency (Hz)','fontweight','bold','fontsize',16);
      text(1:n,f(1:n),num2str(f(1:n),'%.2f'),...
          'HorizontalAlignment','center',...
          'VerticalAlignment','bottom')

      bar3d_pks_sort(m,:)=pks_sort(1:n);
      bar3d_f(m,:)=f(:);
  end

figure;
col(1,:)=[0 0 1]; col(2,:)=[0 .5 0];
plot(bar3d_pks_sort(:,1),'-+','Color',col(1,:),'Linewidth',2);
h1 = gca;
h2 = axes('Position',get(h1,'Position'));
plot(bar3d_f(:,1),'-o','Color',col(2,:),'Linewidth',2);
set(h1,'YColor',col(1,:))
set(h2,'YAxisLocation','right','Color','none','XTickLabel',[],'YColor',col(2,:))
set(h2,'YAxisLocation','right','Color','none','XTickLabel',[])

Answer 1

[Star Strider]

You have to normalise the Fourier transform by the length of the time-domain vector.

So this line should be:

      Fn = Fs/2;                                                          % Nyquist Frequency
      % Take fft, padding with zeros so that length(X) is equal to nfft
      X = fft(x,nfft)/nfft;

... and the following line is then:

      % FFT is symmetric, throw away second half
      Fv = linspace(0, 1, fix(X/2)+1)*Fn;                                  % Frequency Vector (For Plots, &c.)  
      Iv = 1:length(Fv);                                                   % Index Vector
      X = X(1:nfft(Iv))*2;

The current documentation is a bit confusing. Much more straightforward description is in the R2015a fft documentation.

I didn’t go through the rest of your code, but that should correct some of it.

[Juan]

magnitude scale and thus nfft variable were the key

Thanks!

The only thing to change is:

nfft=fix(length(x)/2)*2;

There is some written about scale in fft fucntion, but as Star Strider says, its a bit confusing.

I read <http://es.mathworks.com/help/matlab/ref/fft.html?searchHighlight=fft matlab fft guide > . Sumerized it says:

Datf = fft(Data,N) specifies the transformation length. In the default case, the length of the transformation is determined by the signal length. A second argument N will force FFT transformations of length N, padding with zeros if the signals in Data are shorter and truncating otherwise. Thus the number of frequencies in the real signal case will be N/2 or (N+1)/2. If Data contains several experiments, N can be a row vector of corresponding length.

I misunderstood The number of frequencies in the real signal case will be N/2. I interpreted as fft check only for frequencies as high as N (e.g. till frequency N=150 Hz). And thats false, it checks N/2 difference frequencies, no till N/2 frequency.

If anyone fancys to try the code solved complete just copy-paste this:

% T=[.01 .02 .1 .2 1];
% T=[2:2:8];
T=[1 5 10];
n=10; % maximal number of harmonic

freq = 5;
Fs = 150e4; % Sampling frequency (frecuencia de muestreo)

tic

for m=1:length(T) 

      t = 0:1/Fs:T(m); % Time vector of 1 second
      x = cos(2*pi*t*freq); % sine wave of f Hz.
      x = x-mean(x);% rest DC offset
      nfft=fix(length(x)/2)*2; % KEY OF THIS QUESTION !!!

      figure(1);
      subplot(length(T),1,m);
      plot(t,x,'.');

      % Take fft, padding with zeros so that length(X) is equal to nfft
      X = fft(x,nfft);
      % FFT is symmetric, throw away second half
      X = X(1:nfft/2);
      % Scaling is done here using the number of samples: length(x)/2
      X=X/(length(x)/2);
      % Take the magnitude of fft of x
      mx = abs(X);
      % Frequency vector
      f_fft = (0:nfft/2-1)*Fs/nfft;

      f_Axis=f_fft(1:nfft/2);

      pks= findpeaks(mx(1:nfft/2));
      n=min(n,length(pks));
      pks_sort=sort(pks,'descend');

      for k =  1:n
          locs=find(mx(1:nfft/2)==pks_sort(k));
          f(k)=f_Axis(locs);
      end

      % bar plot freq and magnitude
      figure
      uds='Amp';
      pks_sort=pks_sort(:);
      f=f(:); 
      subplot(1,2,1)
      bar(pks_sort(1:n),'r');
      ylabel({sprintf('Amplitude (%s)'...
          ,uds)},'fontweight','bold','fontsize',16);
      text(1:n,pks_sort(1:n),num2str(pks_sort(1:n),'%.2f'),...
          'HorizontalAlignment','center',...
          'VerticalAlignment','bottom')

      subplot(1,2,2)
      bar(f(1:n),'b');
      ylabel('Frequency (Hz)','fontweight','bold','fontsize',16);
      text(1:n,f(1:n),num2str(f(1:n),'%.2f'),...
          'HorizontalAlignment','center',...
          'VerticalAlignment','bottom')

      bar3d_pks_sort(m,:)=pks_sort(1:n);
      bar3d_f(m,:)=f(:);
  end

figure;
bar3(bar3d_pks_sort);
figure;
bar3(bar3d_f);

figure;
col(1,:)=[0 0 1]; col(2,:)=[0 .5 0];
plot(bar3d_pks_sort(:,1),'-+','Color',col(1,:),'Linewidth',2);
text(1:m,bar3d_pks_sort(:,1),num2str(bar3d_pks_sort(:,1)),...
    'HorizontalAlignment','right','Color','red','FontSize',14)
h1 = gca;
h2 = axes('Position',get(h1,'Position'));
plot(bar3d_f(:,1),'-o','Color',col(2,:),'Linewidth',2);
set(h1,'YColor',col(1,:))
set(h2,'YAxisLocation','right','Color','none','XTickLabel',[],'YColor',col(2,:))
set(h2,'YAxisLocation','right','Color','none','XTickLabel',[])

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[Star Strider]

As always, my pleasure!