About FFT of Exponential decay - two-sided

Dear Forum

I just do the FFT of a Exponential decay - two-sided

 Fs=512; 
 N=2.^12;
 t=[-N/Fs/2:1/Fs:N/Fs/2-1/Fs]; 
 S=exp(-2*abs(t));

The FFT is:

 Y = fft(S,N); 
 Ayy = (abs(Y));
 Ayy(1)=Ayy(1)/N;% to get the amplitude of zero frequency component
 Ayy(2:end)=Ayy(2:end)/(N/2); %to get the amplitude of other frequencies
 Ayy=fftshift(Ayy);
 F=([-N/2:N/2-1])*Fs/N; 
 plot(F,Ayy);

My question is the FFT of the Exponential decay - two-sided is not 4/(4+F.^2)(F is the frequency vector) shown as follows:

 F=([-N/2:N/2-1])*Fs/N; 
 Ground_Truth=4./(4+F.^2);
 figure;
 plot(Ground_Truth);

The Equations of the Exponential decay-two-sided and its corresponding Fourier transform can be found in the website of "Fourier transform of typical signals" as follows: http://fourier.eng.hmc.edu/e101/lectures/handout3/node3.html

Would you please tell me why "Ayy" is different from "Ground_Truth" according to my Matlab code?

Many Thanks Erick

Answer 1

[Miriam]

Hi Erick,

I think the problem lies in your definition of "Ground_Truth". Based on the website you linked to, it should be defined using angular frequency:

Ground_Truth=4./(4+(2*pi*F).^2);

[Erick Zhou]

Thank you, Miriam. Your comment about angular frequency is right. According to your suggestion, I can get a Exponential decay - two-sided form the inverse Fourier transform of the Ground_Truth as follows:

 F=([-N/2:N/2-1])*Fs/N; 
 Ground_Truth=4./(4+(2*pi*F).^2);
 figure;
 plot(Ground_Truth);

 z=ifft(fftshift(Ground_Truth));
 z2=fftshift(z);
 figure;
 plot(z2);

But the amplitude of "Ayy" is still different from the modified "Ground_Truth". Moreover, if I use following codes to modify the amplitude of "Ayy" according to suggestions from internet

 Ayy(1)=Ayy(1)/N;% to get the amplitude of zero frequency component
 Ayy(2:end)=Ayy(2:end)/(N/2); %to get the amplitude of other frequencies

the curve shape of Ayy is distorted.

Would you please give me some suggestions? Thank you.

[Miriam]

Hi Erick,

I would suggest normalizing Ayy as follows (after taking the absolute value):

Ayy = Ayy/max(Ayy);

and removing the other two lines modifying amplitude.

Answer 2

[Erick Zhou]

Thank you for your suggestion,Miriam.

I am still have the question about how to get the real amplitude of the signal in FFT. From the internet, I found some suggestions as follows: "Some FFTs require dividing by 1/N to represent magnitude "naturally" (which is non-energy preserving). " The above suggestion can be found in the website of "https://dsp.stackexchange.com/questions/14636/how-to-get-the-fft-of-a-sine-wave".

I made a example to identify the suggestion as follows

close all; 
Adc=2;  % DC signal(zeros frequency signal)
A1=3;   %
A2=1.5; %
F1=50;  %
F2=75;  %
Fs=256; %
P1=-30; 
P2=90;  
N=256; 
t=[0:1/Fs:N/Fs]; %采样时刻
%signal
S=Adc+A1*cos(2*pi*F1*t+pi*P1/180)+A2*cos(2*pi*F2*t+pi*P2/180);
%show signal
plot(S);
title('orignal signal');

figure;
Y = fft(S,N); 
Ayy = (abs(Y)); 
plot(Ayy(1:N)); 
title('FFT result');

figure;
Ayy=Ayy/(N/2);   %to get the real amplitude
Ayy(1)=Ayy(1)/2;
F=([1:N]-1)*Fs/N; %to get the real amplitude
plot(F(1:N/2),Ayy(1:N/2));  
title('amplitude-Frequency');

The result demonstrated that the above suggestion is right. But I don't know how to modify the amplitude of my FFT result. Would you please give me some suggestion? Thank you.

Answer 3

[Erick Zhou]

For Periodic signal, it should be Ayy=Ayy/(N/2); %to get the real amplitude Ayy(1)=Ayy(1)/2;

But for Non-Periodic signal, it should be Ayy=Ayy/Fs;

I don't know the reason, but the result is right according to above code