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I want to know how much energy is at a specific frequency in a signal. I am using FFT to get the spectrum, and the frequency step is determined by the length of my signal.

My spectrum looks, for example, like this : Spectrum, the red line is where I want a value, but falls between two frequencies

I want to get the spectrum peak at a specific frequency, -0.08. However, the discretization of the spectrum only give me a peaks at -0.0729 and -0.0833.

Is there a way to shift the spectrum to make sure there is a data point at the frequency I want? Or a way to get the value without necessarily using fft?

Thank you very much!

FunkySheep
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2 Answers2

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What you're actually doing when you take a DFT (or any Fourier Transform) is measuring how much of your signal "intersects" with sines of certain frequencies. This is done by summing product of your signal with the complex conjugate of the wave of whatever frequency. Technically, this is called an inner product, which is a generalization of the dot product, and measures how "close" a signal is to another. So if you're only interested in one frequency, don't take the whole DFT, just look at one you want.

I'm not sure what your units are, so I'll assume you want the peak at f0 = -0.08 Hz (If your units are something else, like normalized to the sampling frequency, then you'll need to account for that). This corresponds to the complex exponential exp(2*pi*j*f0*t). Because you're sampling, your t is discrete, so t = n/fs, where fs is the sampling frequency (in Hz). 

# assuming you're using numpy arrays
w = exp(-2*pi*1j*f0*arange(len(signal))/fs)
peak = abs(sum(signal*w))

There are different definitions of the DFT; I'm pretty sure numpy's corresponds to what I have above. The extra minus in the exponential is because it's the complex conjugate.

Notice that it's unlikely that w is actually periodic. If the number of samples is large enough this doesn't really matter. A good heuristic is at least 10 periods.

jpkotta
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  • Thank you so much, this is exactly what I was looking for! You are right, no need to take the whole spectrum here. – FunkySheep Mar 16 '16 at 15:16
  • *measuring how much of your signal "intersects" with sines of certain frequencies* - only now I finally get it, 5 years removed from school – Pavol Hanzel Mar 31 '18 at 22:23
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If you have discrete data but need an output for a continuous variable you'll necessarily need some kind of interpolation function. For a value by request style I would advise Scipy interp1d (example of the use of a interp1d function). I believe it's the fastest way to achieve your intended results.

armatita
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