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What is the units of FFT, when doing Spectral Analysis of a Signal?

  1. For above question, the answer could be V or V/HZ for voltage signal. Which one is right? I would expect the result to be V.t or V/Hz because of dt.

  2. I used the pspectrum function in MATLAB to create a spectrogram image with power spectrum and dB magnitude.

In general, the spectrogram is obtained as the square of the absolute value of the Short Time Fourier Transform. Also, the power spectral density of a normal signal is studied as |FFT|^2. Then since it is a density function, does the integral value by applying the window function to the frequency domain of |FFT|^2 express the power of the signal in dB? I have seen different interpretations of power spectrum and power spectral density. Parseval's theorem also means that the energy in the frequency domain and the time domain could be the same. If the dB scale power spectrum is integrated by multiplying |fft|^2 by the window function, is this also a power spectrum?

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    $\begingroup$ Does this answer your question? stackoverflow.com/q/1523814/16637112 $\endgroup$
    – Ryan
    Sep 10, 2021 at 11:46
  • $\begingroup$ Yes It is similar. I already saw that question. Thank you :) $\endgroup$
    – Daeil Noh
    Sep 14, 2021 at 9:23

2 Answers 2

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A few things to note here

  1. There are four different types of Fourier Transforms and they work all somewhat differently
  2. The FFT is an implementation of the Discrete Fourier Transform (DFT), not the Continuous Fourier Transform FT. The DFT uses sums, the FT uses integrals
  3. If your signal is in Volts, the units of the DFT will also be Volts. The units of the FT would be $V/Hz$.
  4. In order for Perceval's theorem to hold for the DFT, you need to adopt a scaling of $1/\sqrt{N}$ for both forward and backward transform.
  5. The spectral bin power of of the FFT is given by the magnitude squared and has the units of $V^2$
  6. The spectral power density is given by the spectral bin power divided by the bin bandwidth and has units of $V^2/Hz$
  7. The physical interpretation is always somewhat complicated. None of the units discussed so far represent actual physical power, intensity or energy (in $W$, $W/m^2$, or $J$). In any real physical situation there is always a second quantity and/or and impedance in play that determines the actual power, etc.
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  • $\begingroup$ Thank you! I will study more :) $\endgroup$
    – Daeil Noh
    Sep 14, 2021 at 9:23
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Note that an FFT is an implementation of a DFT, and various different but common implementations come with different scalings (1 forward + 1/N reverse, 1/N forward + 1 reverse, or 1/sqrt(N) for both). So the answer depends on your choice of FFT implementation.

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  • $\begingroup$ Thank you! I will study more about FFT :) $\endgroup$
    – Daeil Noh
    Sep 14, 2021 at 9:24

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