= Power spectrum
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= Power spectra
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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 convention
$$
S(\omega)=\int_{\mathbb R}C(\tau)e^{-i\omega\tau}\,d\tau,\qquad C(0)=\frac1{2\pi}\int_{\mathbb R}S(\omega)\,d\omega.
$$
This 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 process>[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.
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