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I have just started my signal and system course and I would like to know how we derive the corresponding formula for the energy of a continuous-time signal $x(t)$ over an interval $[t_{1},t_{2}]$ :

$$E_{x}=\displaystyle\int_{t_{1}}^{t_{2}}|x(t)|^{2}\;\text{d}t$$

That is to say why is the signal energy the area under the region of the square of $x(t)$. For instance, I can think of $x(t):=t$ with $t$ being geometrically the first bisector and so by squaring it we obtain a parabola so this would mean the energy is the area under this parabola but why is this true?

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The formula for the energy of a signal over an interval isn't a derivation. It's a definition. You do that integration and you call the result the signal's energy.

The reason that in the mathematical world we make that definition is because (A) it makes the math handy, and (B) for a preponderance (but by no means all) of the means of signal transmission in the real world, the actual, physical, energy of a signal over an interval really is proportional to its amplitude squared.

So if you launch a radio signal into free space with an antenna, you couple into free space where the actual physical energy of the signal is proportional to its amplitude squared. When you launch a sonar signal into the water with a transducer, the actual physical energy of the signal is proportional to its amplitude squared. When you launch an electronic signal into a transmission line, the actual energy of the forward-traveling portion of the resulting wave is proportional to the amplitude squared. Etc.

So -- just accept it because it makes the math handy, and be happy that the real world happens to fit with an expression that also makes the math handy.

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  • $\begingroup$ Thank you for explanation, I will therefore accept it the way it is $:)$ $\endgroup$
    – B E I R U T
    Commented Feb 2, 2021 at 15:31
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All of these squared energy terms come from the basic physics of simple harmonic motion (SHM), where there is a restoring force proportional to displacement. This can be mechanical, electric field, magnetic field, etc etc. When f(x) = -kx, energy = 0.5 k x^2 + c, since force is by definition the derivative of energy.

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