I normalized the spectrum of a time series and windowed the spectrum, then something strange happened. The code below can run in MATLAB or Octave.
figure;
n = 2e3;
t = 1:n;
m = n/2+1;
f = linspace(0,1,m);
x = rand(1,n) - 0.5;
y = fft(x);
subplot(2,2,1); plot(t,x,'k');
axis tight; xlabel('t/s'); title('original signal x');
subplot(2,2,2); plot(f,abs(y(1:m)),'k');
axis tight; xlabel('f/hz'); title('original spectrum');
y1 = y ./ abs(y);
k = n/10;
w = sin(linspace(0,pi/2,k));
y1(1:k) = y1(1:k) .* w;
y1(m:-1:m-k+1) = y1(m:-1:m-k+1) .* w;
y1(n-m+3:n) = y1(m-1:-1:2);
x1 = real(ifft(y1));
y2 = fft(x1);
x2 = real(ifft(y2));
subplot(2,2,3); plot(t,x1,'k',t,x2-x1,'r');
axis tight; xlabel('t/s'); title('whitened signal x1(black) and x2-x1(red)');
subplot(2,2,4); plot(f,abs(y1(1:m)),'k',f,abs(y2(1:m)),'r');
axis tight; xlabel('f/hz'); title('whitened spectrum y1(black) and y2(red)');
As shown in subplot 3, the red line is the difference between x1
and x2
, meaning that they are exactly the same, but as shown in subplot 4, their spectrum (black and red, respectively) is different.
So, It's weird FFT of IFFT of a spectrum is different from the spectrum itself.
fft()
andifft()
. If I remember correctly, running through afft()
/ifft()
round trip in MATLAB gives you a net gain ofN
, the FFT size. $\endgroup$y2 = fft(real(ifft(y1));
, so he expectsy1 == y2
, which is the crux of his question from my reading of it. $\endgroup$