nature of output of the addition of white noise with colord noise I want to know that the addition of a uncorrelated time series x with a correlated time series, say y generates a time series but is that correlated or uncorrelated output? For example: x is White Gaussian noise and y is pink noise, then z = x+y Looking at the plot which is the output of the noisy time series, it is hard to say whether z is correlated or uncorrelated. Is there a rule about the nature of the output?
Are $x$ and $y$ uncorrelated? In this case the autocorrelation function of $x+y$ is just the sum of the autocorrelation functions of $x$ and $y$, which if one of them is not white will lead to its sum also not being white.
In general, for stationary processes you have $\phi_{zz}(\tau) = \phi_{xx}(\tau) + \phi_{yy}(\tau) + \phi_{xy}(\tau) + \phi_{yx}(\tau)$ where $\phi_{xy}(\tau) = \mathbb{E}\{x(t)y(t+\tau)\}$. Moreover, since the process is stationary you also have $\phi_{xy}(\tau) = \phi_{xy}(-\tau) = \phi_{yx}(\tau)$. So in general, the autocorrelation of $z$ depends not only on the autocorrelations of $x$ and $y$ but also their cross correlations.