Shift in inverse Fourier Transform

Question

I have written a little script (MATLAB) using fast Fourier transform in order to filter messy time series.

Given a signal S(t) my function filter the n biggest amplitude components from the FFT transformation of S(t) and return's the filtered signal by inverse FFT.

The problem is that when I use a test signal the filtered signal is somehow shifted by approx. -1 in amplitude. I have tried to change the windowing but it doesn't appear to help (but the shift is moved toward +1).

Here is a sample code of my script:

% test Signal

t=0:0.1:1000;
f1=1/50; f2=1/300; f3=1/20;
Signal=100*sin(f1*2*pi*t)6x1+8*sin(f2*2*pi*t)+x3+x4+65*sin(f3*2*pi*t);

[FiltredSignal]=Filter(data,5,0); % call Filter function

figure
plot(t,real(a)*2,t,data) 

figure
plot(t,real(a)*2-data) % ???? shift of -1 in ifft ????


function [FiltredSignal]=Filter(Signal,n)   
% n= nbr of "most significative spectra components"

timelen=length(Signal);

FT=fft(Signal); % fast fourrier transform

FTcopy=FT/timelen; % Copy for spectral analysis
FTcopy(floor(timelen/2+1):end)=[]; % cut at nyquist point
FTcopy(1,1)=0; 


% sort FT2 in descending amplitude order
sortedFT=sort(abs(FTcopy),'descend');

% spectral selection --> n biggest amplitudes
FiltredFT=zeros(1,timelen); 
for i=1:n
  freq=find(abs(FTcopy)==sortedFT(i));      
  FiltredFT(freq)=FT(1,freq);
  %FiltredFT(freq-1)=FT(1,freq-1); % windowing
  %FiltredFT(freq+1)=FT(1,freq+1); % windowing
end

FiltredData=ifft(FiltredFT); % return filtred signal time serie

% END

Could someone explain what is happening?

I surely miss some theoretical facts of FFT (I am self learning so everything is not that clear).

It's hard to follow your code, but it looks like you are zeroing the DC (0 Hz) component in the frequency domain (FTcopy(1,1)=0; ?), which will of course result in an overall shift in the Y axis in the time domain. — Paul R
The fact is that when I leave the DC component the shift in time domaine goes from -1 to +1. — user3272910
It also looks like you may not be maintaining complex conjugate symmetry in the frequency domain - you're throwing away the complex conjugate half of the spectrum, it seems, and never actually restoring it after your frequency domain processing ? — Paul R
You are right, everything work's fine when leaving DC components AND reconstruct complex conjugate. I thought that the conjugate part was irrelevant when taking twice the real part of inverse FFT but apparently it is not... thanks for your help. I will shortly post a corrected version of the code. — user3272910

user3272910 user3272910 · Accepted Answer · 2014-03-18T17:27:14

Here is a revised version of the function above that is working, thanks to Paul R remarks.

function [FiltredSignal]=Filter(Signal,n)   
"% n= nbr of spectral components to keep "

timelen=length(Signal);

FT=fft(Signal); "% fast fourrier transform"

FTcopy=FT/timelen; "% Copy for spectral analysis"
FTcopy(floor(timelen/2+1):end)=[]; "% cut at nyquist point"


"% sort FT2 in descending amplitude order"
sortedFT=sort(abs(FTcopy),'descend');

"% spectral selection --> n biggest amplitudes"

FiltredFT=zeros(1,timelen); 

for i=1:n
  freq=find(abs(FTcopy)==sortedFT(i));      
  FiltredFT(freq)=FT(1,freq);
end

"% return filtred signal time serie with complex conjugate reconstruction"
FiltredData=ifft(FiltredFT,'symmetric');

Shift in inverse Fourier Transform

1 Answers