Power spectrum

A power spectrum resolves the second-moment strength of a stationary process or random field by temporal frequency or spatial wavenumber. For a real, mean-zero stationary process with integrable autocovariance function $C(\tau )=\mathbb{E}[X_{t+\tau }{X}_t]$, use the angular-frequency conventionThis form of the Wiener-Khinchin theorem makes the interpretation concrete: integrating spectral density over all frequencies recovers the variance. A nonzero mean contributes a spectral atom at zero if one transforms the uncentered correlation instead of the autocovariance function. Some stationary processes have discrete spectral lines, so an ordinary integrable spectral density need not exist.

For an Ornstein-Uhlenbeck process with $C(\tau )=v{e}^{-a|\tau |}$, $a,v>0$, the Fourier transform is $S(\omega )=2av/(a^2+{\omega }^2)$. Its area is $2\pi v$ as required. Small $a$ concentrates power near zero and corresponds to slowly decaying fluctuations. Conventions using cyclic frequency rather than angular frequency move the factors of $2\pi$; stating the convention prevents a normalization error when comparing spectra.

For a statistically homogeneous vector random field, its spectral tensor is the Fourier transform of the componentwise two-point correlation $C_{ij}(\mathbf{r})=\mathbb{E}[B_i(\mathbf{x}+\mathbf{r})B_j(\mathbf{x})]$. It distributes cross-correlations between components over wavevectors. With $\hat{B}(\mathbf{k})=\int B(\mathbf{r})e^{-i\mathbf{k}\cdot \mathbf{r}}{d}^{d}r$, homogeneity gives $\mathbb{E}[{\hat{B}}_i(\mathbf{k}){\hat{B}}_j({\mathbf{k}}^{\prime })]=(2\pi {)}^d\delta (\mathbf{k}+{\mathbf{k}}^{\prime }){\hat{C}}_{ij}(\mathbf{k})$, understood as an identity of distributions for an infinite stationary field.

For a statistically homogeneous, isotropic, reflection-symmetric solenoidal vector random field in three dimensions, the nonzero-wavevector spectral tensor is a scalar multiple of the transverse projector of a vector field. Isotropy gives $A(k){\delta }_{ij}+D(k)k_i{k}_j$, while zero divergence gives $A+D{k}^2=0$. Its trace is $2A$, giving the displayed normalization. Without reflection symmetry an additional antisymmetric, helical term can occur.

A random field is statistically homogeneous when all of its joint distributions are invariant under a common spatial translation of their arguments. Its momentum-space correlation functions consequently contain a Dirac delta function imposing total-momentum conservation.

A random field is statistically isotropic when all of its joint distributions are invariant under a common spatial rotation of their arguments. A statistically isotropic scalar power spectrum depends only on the magnitude of its wavevector.

Statistical reflection symmetry means that the joint distributions of a random field are unchanged by a simultaneous spatial reflection, with each field component transformed according to its vector or pseudovector character. For an isotropic magnetic field, it excludes a parity-odd magnetic helicity contribution to the two-point spectral tensor.