What Type of Signal is x(t) in Parseval's Theorem?

[Jiho]

Homework Statement

In parseval's theorem, what is type of x(t)?? I mean.. is this voltage wave? or power wave?

Homework Equations

The Attempt at a Solution

 

[Delta2]

Mathematically, $x(t)$ can be any function $x:\mathbb{R}\rightarrow\mathbb{C}$ that is any complex valued function over the real line (or as a sub case any real valued function over the real line $x:\mathbb{R}\rightarrow\mathbb{R}$) for which its continuous Fourier transform $X(f)$ exists . So it can be a function representing the (complex) voltage between two nodes of a circuit or the power between two nodes of a circuit.

However the usual interpretation of this theorem in signal analysis is that the two sides of the equation are just two different ways of computing the total energy of a signal $x(t)$ (a voltage or a current signal).

 

[Jiho]

you mentioned it doesn't matter that signal x(t) represents voltage or power. But I can't understand if x(t) represent power,

this equation doesn't make sense as I think. In my opinion, How can power^2 be just power?? Unit of dimension is not same as I know.

 

[Delta2]

Yes I said that $x(t)$ can be any function for which the Fourier transform exists. However I also said that the usual interpretation is that of energy (the integrals in both sides represent energy) in the case $x(t)$ is a voltage or current signal. If $x(t)$ is a power signal then we can't give that usual interpretation of energy to this theorem.