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For an random signal $X$ (does not matter if it is correlated or uncorrelated), given lag $τ$. If white noise is added to this signal, then can the summation of them be a correlated signal? Also, even if these two are individually uncorrelated. Could somehow be related mutually?

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Can white noise be correlated to a random singal

Something can be correlated to something else but still be white. For example, let $X(t)$ be a white noise process, and let $Y(t) = 2X(t)$, then clearly $Y$ is still a white noise process, but just as clearly very correlated to $X$.

If white noise is added to this signal, then can the summation of them be a correlated signal?

Say $X_1(t)$ is now some arbitrary process that's not 0, $Y_1(t)$ is white noise, stochastically independent from and hence uncorrelated with $X_1$, and let $Z_1(t)=X_1(t)+Y_1(t)$. Then, clearly $Z$ is correlated with $X$. The same is trivial to prove: Test for correlation:

\begin{align} E\{X_1(t)Z_1(t)\} &= E\{X_1(t)(X_1(t)+Y_1(t))\}\\ &=E\{X^2_1(t)\}+E\{X_1(t)Y_1(t)\}&\text{linearity of expectation}\\ &= E\{X_1^2(t)\}+E\{X_1(t)\}\cdot E\{Y_1(t)\} &\text{$X_1,Y_1$ independent}\\ &=E\{X_1^2(t)\}+E\{X_1(t)\}\cdot 0 &\text{$Y_1$ is white, $\rightarrow$ zero-mean}\\ &=E\{X_1^2(t)\}\\ &\ne 0 \end{align}

Hence, the sum of a random variable and independent white noise is always correlated to the random variable.

Whether the sum is white: that depends fully on $X$.

I'm not sure you're intending to ask this, just in case:

Can I add two white processes $V,W$ and get a non-white processes $U= V+W$?

Sure; let $V(t)$ be a non-zero power white noise process and $W(t)=V(t-1)$,

\begin{align} E\{U(t)U(t+\tau)\}&=E\{(V(t)+V(t-1))(V(t+\tau)+V(t-1+\tau)\}\\ &=E\left\{V(t)V(t+\tau)+V(t)V(t-1+\tau)+\right.\\ &\phantom{E\left\{V(\right.}\left.V(t-1)V(t+\tau)+V(t-1)V(t-1+\tau)\right\}\\ &\ne0 \text{ for } \tau\in\{-1,0,+1\}. \end{align}

That's actually a super common case in the signal processing: your $W$ is the superposition of the transmit signal with a delayed version of itself; the multipath channel. The fact that the signals in these paths correlate is the basis for the multipath channel needing a different treatment than the AWGN channel; the fact is also the foundation of diversity gain methods in MIMO, many if –not even most channel– equalization methods, passive radar, lightning location, necessary for ultrasound imaging to work…

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