# Hilbert transform LTI

I'm studying the Hilbert transform and its properties. I found out that Hilbert transform can be considered as a linear time-invariant system. How can I show this? I guess this should be trivial as my book does not show it.

A linear operator $\Theta$ is an operator acting on functions $F$ and $G$, with coefficients $a$ and $b$, such that the following equality holds:
$$\Theta(aF+bG) = a \Theta(F) + b \Theta(G)$$
To show that the Hilbert Transform is a linear operator, apply the principle given above with the Hilbert Transform in place of $\Theta$.
• Try comparing $\Theta(t)$ with $\Theta(t+p)$. – wwarriner Apr 13 '16 at 15:23