Auto-correlation of the sum of two generic signals

Question

Be $x[n]$ and $y[n]$ two generic discrete-time signals. Given $s[n] = x[n] + y[n]$ I want to evaluate its autocorrelation $R_s[l]$.

By definition (https://en.wikipedia.org/wiki/Cross-correlation): $$R_s[l] = \sum^{+\infty}_{m = -\infty}s^*[m]s[m+l]$$ Which expands to: $$\dots = \sum^{+\infty}_{m = -\infty}(x[m]+y[m])^*\,(x[m+l]+y[m+l])$$ Since $(a + b)^* = a^* + b^*$ : $$\dots = \sum^{+\infty}_{m = -\infty}x^*[m]x[m+l]+y^*[m]y[m+l]+x^* [m]y[m+l]+y^*[m]x[m+l]$$ Using the definition again: $$\dots = R_x[l] + R_y[l] + R_{xy}[l] + R_{yx}[l]$$

Is this result correct?

Answer 1

This is correct. You can use $R_{xy}[l]=R_{yx}^{*} [l]$ to simplify a little further to

$$R_{ss}[l] = R_{xx}[l]+R_{yy}[l]+2 \cdot \Re\{ R_{xy}[l] \} $$