Crosscorrelation

Given two sequences as an input signal, can you find the output signal after being subjected to noise and time delay?

General formula:

$$y[n]=\alpha x[n-D]+w[n]$$

Where $\alpha$ is the degradation of the signal, $D$ is the time delay and $w$ is noise

Properties

Since

r_{xy}[l] = \sum_{n=-\infty}^{\infty}x[n]y[n-l] \\ r_{yx}[l] = \sum_{n=-\infty}^{\infty}x[n+l]y[n] \end{align}

then

$$r_{xy}[l]=r_{yx}[-l]$$

and the lag is

$$0-(-l)=l$$