calculate frequency band using Parseval

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?

[Sk Group]

Parseval’s Theorem in MATLAB Compete (Code + Output)

Answer 1

[Wayne King]

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.

[biomed]

thank you so much!!

Answer 2

[Wayne King]

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)

[Wayne King]

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]

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?