Why does signal energy use the magnitude squared in the integral?

[cybernoodles]

Hi All,

I am confused about what is meant in the mathematical definition for the energy of a signal. Why is it the integral of the magnitude of the signal squared?

∫(|f(t)|^2)

How do I deal with the absolute value sign? I do not have much experience with absolute value signs in integrals. I did do a search but am still confused. How do I even know if a function will need to be readjusted according to the absolute value sign and, if so, how do I rewrite the function so that it obeys the absolute value condition?

 

[AlephZero]

If $f(t)$ is a real-valued function the absolute values just "go away", because $$|f\,|^2 = |f\,| \times |f\,| = f\,{}^2$$
is always true - it doesn't matter whether $f$ is positive, negative, or zero.

If $f$ is a complex-valued function and $f(t) = p(t) + i\,q(t)$, then
$$|f\,|^2 = p^2 + q^2$$

 

[cybernoodles]

Wow that was so simple. Thanks!

 

[AlephZero]

Wow that was so simple.

Somebody once said, "There are only two types of math problem: the trivial ones that you know how to solve, and the impossible ones that you don't"