title: Units of a Fourier Transform (FFT) when doing Spectral Analysis of a Signal
body: <p>My question has to do with the physical meaning of the results of doing a spectral analysis of a signal, or of throwing the signal into an FFT and interpreting what comes out using a suitable numerical package, </p>
  
  <p>Specifically:</p>
  
  <ul>
  <li>take a signal, say a time-varying voltage  v(t)</li>
  <li>throw it into an FFT    (you get back a sequence of complex numbers)</li>
  <li>now take the modulus (abs) and square the result, i.e. |fft(v)|^2.  </li>
  </ul>
  
  <p>So you now have real numbers on the y axis -- shall I call these spectral coefficients?</p>
  
  <ul>
  <li>using the sampling resolution, you follow a cookbook recipe and associate the spectral coefficients to frequencies.  </li>
  <li>AT THIS POINT, you have a frequency spectrum g(w) with frequency on the x axis, <strong>but WHAT PHYSICAL UNITS on the y axis?</strong></li>
  </ul>
  
  <p>My understanding is that this frequency spectrum shows how much of the various frequencies are present in the voltage signal -- they are spectral coefficients in the sense that they are the coefficients of the sines and cosines of the various frequencies required to reconstitute the original signal.</p>
  
  <p>So the first question is, <strong>what are the UNITS of these spectral coefficients?</strong></p>
  
  <p>The reason this matters is that spectral coefficients can be tiny and enormous, so I want to use a dB scale to represent them.  </p>
  
  <p>But to do that, I have to make a choice: </p>
  
  <ul>
  <li>Either I use the 20log10 dB conversion, corresponding to a field measurement, like voltage.  </li>
  <li>Or I use the 10log10 dB conversion, corresponding to an energy measurement, like power.  </li>
  </ul>
  
  <p><strong>Which scaling I use depends on what the units are.</strong></p>
  
  <p>Any light shed on this would be greatly appreciated!</p>