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calculate frequency band using Parseval

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biomed
biomed 2011-9-28
评论： Sk Group 2021-2-8
I have the fft of my signal, but I don't know its frequency band and there are no lobe to extimate it.
Can I use Parseval's Theorem to calculate the band?
Could anyone post the code?

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Wayne King
Wayne King 2011-9-28
Hi, You can determine what percentage of power lies within a certain band. Here I determine the frequency with maximum power and then find the percentage of power lying in a band that is the maximum frequency plus or minus 2 DFT bins
Fs = 1e3;
t = 0:1/Fs:1-(1/Fs);
x = cos(2*pi*100*t)+randn(size(t));
psdest = psd(spectrum.periodogram,x,'Fs',1e3,'NFFT',length(x));
[mx,I] = max(psdest.Data);
relperc = ...
100*avgpower(psdest,[psdest.Frequencies(I-2) psdest.Frequencies(I+2)])/avgpower(psdest,[0 Fs/2])
Or you could use cumsum() to get the cumulative percent power and figure out the power in a band with the appropriate subtraction.
xdft = fft(x);
xdft = xdft(1:length(x)/2+1);
relpower = 100*cumsum(abs(xdft).^2)/norm(xdft,2)^2;
plot(relpower);
The following gives you the percentage power at 100 Hz plus or minus two DFT bins.
relpower(103)-relpower(99)
Note it's very close to the answer returned by the first method.

更多回答（1 个）

Wayne King
Wayne King 2011-9-28
Hi, If you have the DFT (discrete Fourier transform) as implemented by fft(), then you have the frequencies. The frequencies are of the form (Fs*k)/N where Fs is the sampling frequency, N is the length of the signal and k runs from 0,1,...N/2 (for N even)
Parseval's theorem just demonstrates that energy is conserved.
x = randn(1e3,1);
xdft = fft(x);
norm(x,2)
(1/sqrt(length(x)))*norm(xdft,2)
  2 个评论
Wayne King
Wayne King 2011-9-28
I should note that the DFT is periodic with period N so the other half of the frequencies can be thought of as the negative version of the above, or you can just let k run from 0,..... N-1
biomed
biomed 2011-9-28
Hi, I read that using Pareseval's theorem it'ss possible to calculate the width of the band when the signal has, for example, 90% of its energy.
So if the band is [-W,W], the integral between -W and W of (|X(f)|^2)df is equal to 0.9*total_energy of the signal
What should I write in the code to find the width band 2W?

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