Units of a Fourier Transform (FFT) when doing Spectral Analysis of a Signal

Question

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 ou

Answer 1

Here's what I've been able to come up with so far:

The y-axis seems likely to be in units of [Energy / Hz] !?

Here's how I'm deriving this (feedback welcomed!):

1. the signal v(t) is in volts

2. so after taking the Fourier integral: integral e^iwt v(t) dt , we should have units of [volts*seconds], or [volts/Hz] (e^iwt is unitless)

3. taking the magnitude squared should then give units of [volts^2 * s^2], or [v^2 * s/Hz]

4. we know Power is proportional to volts ^2, so this gets us to [power * s / Hz]

5. but Power is the time-rate of change in energy, i.e. power = energy/s, so we can also write Energy = power * s

6. this leaves us with the candidate conclusion [Energy/Hz]. (Joules/Hz ?!)

... which suggests the meaning "Energy content per Hz", and suggests as a use integrating frequency bands and seeing the energy content... which would be very nice if it were true...

Continuing... assuming the above is correct, then we are dealing with an Energy measurement, so this would suggest using 10log10 conversion to get into dB scale, instead of 20log10...

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