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 the spectral analysis of a signal, or, put another way, of interpreting what comes out of throwing a signal into an FFT of a math package.

Specifically:
(i) take a signal, a time-varying voltage v(t)

(ii) throw it into an FFT -- you get back a sequence of real numbers (Laurance: ok, not quite the FFT, but the abs(fft)^2 )

(iii) throw it onto a plot. You now have a frequency spectrum g(w) with frequency on the x axis, and

(?iv?) ... WHAT PHYSICAL UNITS on the y axis? (and what meaning?)

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.

But to relate them to something with meaning in the real world, it would be helpful to know the UNITS of these spectral coefficients.

The other problem is that without knowing the UNITS, I can't make the right choice for how to move to a dB scale for graphing them. (I want to use a dB scale because I have some tiny measurements and some huge measurements so dB scale is the best way to show the extreme range on a single plot.)

So I have to make a choice: do I use the 20log10 dB conversion (corresponding to a field measurement, like voltage)? Or do I use the 10log10 dB conversion (corresponding to an energy measurement, like power)? Which one to use depends on what the units are?

Any light shed on this would be greatly appreciated!

AKE (edited)

Answer 1

The y axis is complex (as opposed to real). The magnitude is the amplitude of the original signal in whatever units your original samples were in. The angle is the phase of that frequency component.

Answer 2

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...

...

Answer 3

take a signal, a time-varying voltage v(t)

units are V, values are real.

throw it into an FFT -- ok, you get back a sequence of complex numbers

units are still V, values are complex ( not V/Hz - the FFT a DC signal becomes a point at the DC level, not an dirac delta function zooming off to infinity )

now take the modulus (abs)

units are still V, values are real - magnitude of signal components

and square the result, i.e. |fft(v)|^2

units are now V², values are real - square of magnitudes of signal components

shall I call these spectral coefficients?

It's closer to an power density rather than usual use of spectral coefficient. If your sink is a perfect resistor, it will be power, but if your sink is frequency dependent it's "the square of the magnitude of the FFT of the input voltage".

AT THIS POINT, you have a frequency spectrum g(w): frequency on the x axis, and... WHAT PHYSICAL UNITS on the y axis?

Units are V²

The other reason the units matter is that the spectral coefficients can be tiny and enormous, so I want to use a dB scale to represent them. But to do that, I have to make a choice: do I use the 20log10 dB conversion (corresponding to a field measurement, like voltage)? Or do I use the 10log10 dB conversion (corresponding to an energy measurement, like power)?

You've already squared the voltage values, giving equivalent power into a perfect 1 Ohm resistor, so use 10log10.

log(x²) is 2 log(x), so 20log10 |fft(v)| = 10log10 ( |fft(v)|²), so alternatively if you did not square the values you could use 20log10.