 Ornstein-Uhlenbeck power spectrum

Let $X$ be the centered stationary solution of $d{X}_t=-A{X}_{t}dt+Bd{W}_t$, where $A$ is a real $d\times d$ matrix with eigenvalues of positive real part and $W$ is an $m$-dimensional Brownian motion. Use the two-sided angular-frequency conventionThe Ornstein-Uhlenbeck power spectrum isThe stationary linear filter has transfer matrix $H(\omega )=(A+i\omega I{)}^{-1}$, so the output spectral matrix is $HB{B}^T{H}^{\ast }$, explaining the adjoint and sign of the second factor. This matrix is Hermitian positive semidefinite for every frequency.

For scalar $dX=-\lambda Xdt+\sigma dW$ with $\lambda >0$, the spectrum is ${\sigma }^2/({\lambda }^2+{\omega }^2)$, proportional to a Cauchy distribution probability density function. Its inverse-frequency integral gives variance ${\sigma }^2/(2\lambda )$, checking the normalization. With ordinary frequency $f$ and inverse transform $\int e^{2\pi ift}{S}_f(f)df$, use $S_f(f)=S(2\pi f)$. A one-sided positive-frequency spectrum doubles this density away from zero. These conventions must be matched when comparing a theoretical spectrum with measured spectral power.