 Autocovariance of an AR(1) process observed with white noise

For a causal AR(1) plus uncorrelated observation white noise, cross covariances vanish by L2 convergence of the autoregressive noise expansion. Its covariance is ${\sigma }_z^2{\varphi }^{|h|}/(1-{\varphi }^2)+{\sigma }_w^2{1}_{\{h=0\}}$. The additional noise changes only lag zero, attenuating the normalized positive-lag correlations.