Spectral measure of a stationary random field ($C(\zeta )={\int }_{\mathbb{R}}{e}^{i\nu \zeta }F(d\nu )$)

A second-order stationary complex random field on the line has uncentered correlation $C(\zeta )=\mathbb{E}[E(z+\zeta )\overline{E(z)}]$. When this correlation is continuous, its Fourier transform is a finite positive measure $F$ characterized by $C(\zeta )=\int e^{i\nu \zeta }F(d\nu )$. Its total mass is $\mathbb{E}|E(z){|}^2$. If it has a density, $S(\nu )=(2\pi {)}^{-1}\int C(\zeta )e^{-i\nu \zeta }d\zeta$. A constant mean $m$ contributes $|m{|}^2{\delta }_0$; the remaining measure is the covariance spectrum. This uncentered convention is useful for coherent and diffuse wave fields.

For the Fourier transform ${\hat{E}}_L(\nu )=(2\pi {)}^{-1}{\int }_{-L/2}^{L/2}E(z)e^{-i\nu z}dz$ of a stationary random field, the periodogram measure $(2\pi /L)\mathbb{E}|{\hat{E}}_L(\nu ){|}^{2}d\nu$ converges weakly to the spectral measure of a stationary random field. The expected squared modulus is $(4{\pi }^2{)}^{-1}{\int }_{-L}^L(L-|\zeta |)C(\zeta )e^{-i\nu \zeta }d\zeta$. The factor $2\pi /L$ follows from Parseval identity, and preserves the total power per unit length. It also avoids meaningless squares of Dirac delta distributions for coherent plane-wave components.