Is there a name for the square of a function plus the square of its Hilbert transform?

Question

Given a real-valued analytic function $f$ defined on the whole real line, and its Hilbert transform ${\cal H}f$, it seems that the quantity $f(x)^2+{\cal H}f(x)^2$ should have some kind of importance as an energy measure. It is the square of the complex modulus of the analytic extension of $f$ to the complex plane: $f(x)+i{\cal H}f(x)$. What are keywords (or textbooks, or even academic papers) that focus on this quantity?

Answer 1

Engineers often call real valued functions "signals". Denoting Hilbert transform $\mathcal H\{\cdot\}$, there exists a concept called Analytic signal denoted $\mathcal A\{\cdot\}$, where for a real valued signal $f(x)$:

$$\mathcal{A}\{f\}(x) = f(x) + i\mathcal{H}\{f\}(x)$$

Then, since $|a+ib|^2=a^2+b^2$ for any pair of reals $a,b$ we can see that $$|\mathcal A\{f\}(x)|^2=f(x)^2+(\mathcal{H}\{f\}(x))^2$$

As you suspect it is a kind of energy measure. In electrical engineering and signal processing the complex absolute value above is often called envelope of a signal. And if we square it (which is just application of a monotonic growing nonlinear function) we get the function you investigated.