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

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?

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 $$|\mathcal A\{f\}(x)|^2=f(x)^2+(\mathcal{H}\{f\}(x))^2$$
• Thank you. I think this is what I was looking for. So I could call it "the square of the envelope of the analytical representation of the function $f$"? It is a bit of a mouthful but I guess that's the best we've got. – Oliver Jan 7 at 1:39