Why does "Amplitude Scale Factor of FFT" depend on data?

% November 14, 2013 - 
% Here, 2 functions were sampled using same sampling frequncy and record length. 
% Next, data were fed to fft.
% However, their absolute value of fft output (i.e., amplitude) needs to be scaled differently.
% This suggest fft applies different numerical algorithms to data even they share same sampling frequency and record length.
%
% This file plots FFT amplitude for 2 functions:
%
% 1. Sine Function:   Y(t)=sin(2*pi*f0*t)
% 2. Sinc Function:   Y(t)=sin(2*pi*f0*t)./(pi*t) if t~=0,
%                         =2*f0                   if t==0
%
% Both functions are sampled at same frequncy and share same record length.
% However, amplitude needs to be scaled differently:
%
% 1. Amplitude of FFT(sine) needed to be multiplied by 2/(Record Length -1)
%    Furthermore, the amolitude at zero frequncy must also be divided by 2.
%
% 2. Amplitude of FFT(sinc) needed to be multiplied by sampling time
%

clear

close all

% Setting frequency of sine and sinc
f0=2;

    % Defining sampling rate and time vector in seconds
    t_samp=0.05;
    t_length=1000;
    t=-t_length:t_samp:t_length;

    % Frequency Vector
    f=-1/t_samp/2:1/t_samp/max(size(t)-1):1/t_samp/2;

    % Sampling sine and sinc function for f0=1Hz.
    y_sine=sin(2*pi*f0*t);
    y_sinc=sin(2*pi*f0*t)./(pi*t);
        % Sinc needs to be set to 2*f0 at t=0.
            loc=find(t==0);
            y_sinc(loc)=2*f0;

    % FFT
    FFT_sine=fft(y_sine);
    FFT_sinc=fft(y_sinc);

    % Calculating Amplitude
    Amp_FFT_sine=abs(FFT_sine);
    Amp_FFT_sinc=abs(FFT_sinc);

    %%%%%Beginning of Amplitude Scaling:

            % For sine function, zero frequnecy amplitude must be divided by 2
            % Sinc function does not need that
            Amp_FFT_sine(1)=Amp_FFT_sine(1)/2;

            % Sine and sinc need to be scaled differently
            Amp_FFT_sine=Amp_FFT_sine*(2/max(size(t)-1));
            Amp_FFT_sinc=abs(Amp_FFT_sinc)*t_samp;

    %%%%%End of Scaling

    % Shifting zero frequncy to center
    Amp_FFT_sine=fftshift(Amp_FFT_sine);
    Amp_FFT_sinc=fftshift(Amp_FFT_sinc);

    % Plotting
    figure(1)
      h0=subplot(2,1,1);
      hold
      set(h0,'FontSize',14)
      set(h0,'FontWeight','b')
      h1=plot(f,Amp_FFT_sine);
      set(h1,'LineWidth',2)
      set(h1,'color','r')
      set(h0,'XTick',[min(f),-f0,0,f0,max(f)]);
      set(h0,'YTick',[0,1,2])
      axis([min(f),max(f),0,2])
      grid on
      legend('Fourier Transform of {sin(2*pi*f_0*t)}')
      title(strcat('f_0=',num2str(f0),'Hz'))
      ylabel('Amplitude')

      h00=subplot(2,1,2);
      hold
      set(h00,'FontSize',14)
      set(h00,'FontWeight','b')
      h2=plot(f,Amp_FFT_sinc);
      set(h2,'LineWidth',2)
      set(h2,'color','k')
      set(h00,'XTick',[min(f),-f0,0,f0,max(f)])
      set(h00,'YTick',[0,1,2])
      axis([min(f),max(f),0,2])
      grid on
      legend('Fourier Transform of {sin(2*pi*f_0*t)/(pi*t)}')
      ylabel('Amplitude')
      xlabel('Frequency  [Hz]')

[Muthu Annamalai]

You mean scale factor and how it depends on the number of points? Well thats what it takes for FFT/IFFT to be a pair of unitary transformations.

[Ali Razavi]

 No. 
 Actually, number of samples are the same.

Answer 1

[shirish pataki]

You can check following answer: We scale the amplitude in FFT domain just to satisfy the parsevel's relation (the law of conservation of energy). (I am unable to write the equation here however you can find that anywhere on google). Parsevel's relation says that in FFT domain the Energy/power is scaled by N to maintain same energy as time domain. In sinc i.e Sin x/ (Pi. x) treat it as two signal division. Therefore as per Parsevels relation, in FFT domain it is scaling by N/N thus scaling by 1. Thus in sinc you need not divide/scale the amplitude of FFT by FFT length but for FFT of sin you need to divide/scale.