# Graphical nature of signal energy

In signals, we learn that the energy of a signal is the integral of its square.

E.g.

$$\E=\int_{-\infty}^\infty x^2(t) dt\$$

We know that integrals are the areas under the function in a graph. But what is the graphical representation of the energy of a signal? Or is the integral of a square just a purely algebraic thing?

Thanks

• Have you read the wiki article? It is a definition (and not derivation) and is analogous but not the same as energy in physics. You might also want to read this Jan 22 '19 at 21:38

Rotate the graph about the 't' axis, and you end up with a volume enclosed that is proportional to the energy.

• Where does the square come in here? Could you elaborate more? Jan 22 '19 at 21:00
• In numerical integration, the area under the graph is the sum of the slices of area x.dt. Consider that for each 'dt' the 'x' value rotates to define a disc, of area π.x^2, volume π.x^2.dt, so the total volume is the sum of those discs is equal to the integral of the square of x multiplied by π. Jan 22 '19 at 21:43

Square of what? power? no just the integral.

$$\E=\int_{-\infty}^\infty P(t) dt\$$

For electrical: Yes it is signal squared only for V,I but not P all of which are time-variant signals.

$$\\frac{V_{(t)}^2}{R}\$$ = $$\I_{(t)}^2R=P_{(t)}\$$ yes then the square law applies only to "certain signals except power signals" ( mechanical, acoustic pressure, thermal, electrical, nuclear, gravitational)

Integrals from Calculus 101 is just the area under the curve.

• Energy of signal is not the same as energy in physics. Jan 22 '19 at 20:57
• It is exactly the same, It must be a voltage or current signal. Yet we know x(t) can be defined as a power signal. So "Physics" teaching is making assumptions not given in the question. i.e. anytime assumptions are not given, it is vague. V*I=P is a rectangle ∫V(t)*I(t)dt could be a rectangle if constant, otherwise just the area. Jan 22 '19 at 21:29
• Signal is not necessarily voltage or current. It can be acoustic or optical signal. Or an abstract mathematical function. Yes, the energy is defined in analogy with physics, but it is not the same. Jan 22 '19 at 21:41
• True it could be , but it can also be a power signal in uW in which case the terms are vague. It could be a pressure signal in Pascal or a momentum signal in mv Jan 22 '19 at 21:44
• Tony, take the time to click on the link @EugeneSh. provided in his comment to the original question. The term "energy" means something different in the field of signals & systems than it means in physics (or dc circuits). Jan 22 '19 at 23:37