Power and Energy Computations in the Frequency Domain

Question

How do you calculate the power and energy of a signal given only the frequency domain form of the signal function? For the purposes of this question, please do not assume that it is possible to find a closed form representation of the inverse fourier transform time domain function. Or, in other words, assuming you cannot look at the time domain at all, how can you derive power and energy from the frequency domain representation of a function?

To add some clarification:

Rayleigh's Property:

$$\int_{-\infty}^{\infty}|x(t)|^2dt = \int_{-\infty}^{\infty}|X(f)|^2df$$

Definitions of Power and Energy:

$$E_x = \lim_{T\to\infty}\int_{-T}^{T}|x(t)|^2dt$$

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

Since the limit approximates Rayleigh's property, it seems it should be possible to find Energy and maybe power even if you cannot access the time domain function.

Answer 1

How do you calculate the power and energy of a signal given only the frequency domain form of the signal function?

For power you do it in exactly the same way that you would in the time domain;

Time domain: \$ P(t)=V(t)I(t) \$

S-Domain: \$ \tilde{P}(\omega)=\tilde{V}(\omega)\tilde{I}(\omega) \$

If you don't know \$ \tilde{V}(\omega) \$ and \$ \tilde{I}(\omega) \$ Then the situation is the same as if you were to try and calculate the power in the time domain without knowing \$ V(t) \$ and \$ I(t) \$, it is not possible.

For energy you pick any number you like..

The s-domain does not care about time. If we assume that you have a signal at 1kHz with a power of 1W. if you observe this signal for 1s in the time domain then the energy is 1J, is you observe it for 1000s in the time domain then the energy is 1kJ.