Units of Fourier Transform (CTFT) vs spectral density

[amama]

I'm confused on how units work with regards to the Fourier Transform (CTFT).

I was reading the Wikipedia article on spectral density. In an example, they use Parseval's equation, along with the units calculated on the time side, to determine the units on the frequency domain side. The units of the spectral density are determined as $\frac{V^2}{Hz}$.

The reasoning in the article makes sense, but what I am struggling with is how those units make sense from the definition of the Fourier Transform. The Fourier Transform integrates the signal with respect to time $\int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt$. I would guess that the units would be volt-seconds, which would be equivalent to $\frac{V}{Hz}$. If you square that, like you do in the Parseval's equation, wouldn't the units now be $\frac{V^2}{Hz^2}$, which is different than $\frac{V^2}{Hz}$, so obviously I made a mistake somewhere.

 

[BvU]

Hello amama, !

If you square that

You do, but then you integrate over $df$ which has the dimension ${\rm s}^{-1}$ .

 

[amama]

Hello amama, !
You do, but then you integrate over $df$ which has the dimension ${\rm s}^{-1}$ .

Thanks. Based on your answer I looked back at what I was doing and realized that I was looking at energy spectral density and that I was used to looking at power spectral density.

With energy density the equation inside should be energy per Hz, which is what you get if you "flip" one of the s to the bottom as Hz, but leave the other one to turn the $v^2$ to $v^2 s$, which is indeed units of energy.

$$v^2 s^2 = \frac{v^2 s}{Hz}$$