Positive-frequency spectral normalization ($g(\omega )=2f(\omega )$)

For a real weakly stationary process, restricting its even angular-frequency density $f$ to $[0,\pi ]$ gives $\gamma (k)=2{\int }_0^{\pi }f(\omega )\cos (k\omega )d\omega$. Folding both frequency halves into one density gives $g=2f$ and removes that factor two. Consequently white noise of variance $v$ has restricted density $v/(2\pi )$ and folded density $v/\pi$.