# Autocorrelation of Addition of Two Independent Signals

Given a random signal $$Z \left( t \right)$$ which is addition of two independent signals $$X \left( t \right)$$ and $$Y \left( t \right)$$ with constant parameters $$a$$ and $$b$$:

$$Z (t) = aX(t) + bY (t)$$

If the auto correlation function of $$R_{XX} \left( \tau \right)$$ and $$R_{YY} \left( \tau \right)$$ is known (Assume both are Stationary Signals).
What would be the Auto correlation function of $$Z \left( t \right)$$?

Is it given by (I skipped the steps):

$${R}_{ZZ} \left( \tau \right) = a^2 R_{XX} \left( \tau \right) + b^2 R_{YY}\left( \tau \right)$$

I neglected the $${R}_{XY} \left( \tau \right)$$ contribution because it is zero as $$X \left( t \right) \perp Y \left( t \right)$$ (Independent Random Signals).

Namely, in order to have ${R}_{XY} \left( \tau \right) = 0$ having $X \left( t \right) \perp Y \left( t \right)$ isn't enough but at least of them has zero mean (Namely, $\mathbb{E} \left[ X \left( t \right) \right] = 0$, $\mathbb{E} \left[ Y \left( t \right) \right] = 0$ or both).
• just a minor point. If you're going to use $\tau$ to denote Z's correlation lag, then you should use it for x and y also. otherwise, the assumption is usually that $\tau = 0$ which can possibly confuse readers. – mark leeds May 31 '18 at 21:32