Evaluation of Autocorrelation and Power Spectral Density of white noise through a filter
White noise is modelled as a process having spectral density $G_n(f) = k$, a constant.

After the filter, the resulting process $y$ will have spectral density $G_y(f) = G_n(f) |H(f)|^2$. And the autocorrelation will be the inverse Fourier transform of the spectral density.

This is a general result for any stable filter and finite power, wide-sense stationary input process (in this case we are pushing the boundaries because white noise has infinite power as defined above).