Analytically determine a PSD from a transient function This question is related to a series of questions I have asked about the units of PSD and ESDs. I include it as a separate question as it may have worth in isolation.

As I understand it to compute the power spectrum density from some transient signal one takes the approach of computing the truncated Fourier transform
$$\mathcal{F}_T(\omega) = \int_{-T/2}^{+T/2} e^{-i \omega t} f(t) \ \ dt$$
Taking the absolute value square and then taking the limit as $T\rightarrow \infty$
$$\text{PSD} = S_{x}(\omega) = \lim_{T\rightarrow\infty}\frac{1}{T}|\mathcal{F}_T(\omega)|^{2}$$
When ignoring the normal dimensioning of units as described here by a very nice answer by Matt L., and also descibed here and in the Wikipedia article here, we get the familiar units of a PSD of $\rm{[V^{2}/Hz]}$.

My question



So that is the background and I believe I follow it (please say if there is a mistake or something overtly wrong!)

But If I want to analytically derive a function for a PSD or $S_{x}(\omega)$, I always end up with zero as the $\lim_{T\rightarrow\infty}$ kills the expression because of the factor of $1/T$. Is this the correct approach to analytically deriving a function or lineshape for a PSD with correct units? The end goal is to use it to fit a function to spectra acquired with an FFT spectrum analyser.
Energy signals, i.e., signals with finite energy

$$\int_{-\infty}^{\infty}|f(t)|^2dt&lt;\infty$$

have zero power and, consequently, a power spectrum that is equal to zero. They do have an energy density spectrum, which is the squared magnitude of their Fourier transform.

You will only get a non-zero power spectrum, according to the definition in your question, for power signals, which have a finite non-zero power defined by

$$\overline{f^2(t)}=\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}|f(t)|^2dt$$

I.e., power signals satisfy

$$0&lt;\overline{f^2(t)}&lt;\infty$$