Spectral density of a stationary process ($f_X(\lambda )$)

When its autocovariance is absolutely summable, a weakly stationary process has spectral density $f(\lambda )=(2\pi {)}^{-1}{\sum }_h\gamma (h)e^{-ih\lambda }$. The inverse relation is $\gamma (h)={\int }_{-\pi }^{\pi }{e}^{ih\lambda }f(\lambda )d\lambda$. A linear filter multiplies this density by the squared modulus of its frequency response.

In cycles per observation, frequency lies in $[-1/2,1/2]$ and $\gamma (h)=\int e^{2\pi ih\omega }{f}_c(\omega )d\omega$. Compared with angular-frequency density $f_a$, one has $f_c(\omega )=2\pi f_a(2\pi \omega )$. In particular, white noise of variance ${\sigma }^2$ has $f_c={\sigma }^2$.

A centered weakly stationary process has a representation $X_t={\int }_{-1/2}^{1/2}{e}^{2\pi it\omega }dZ(\omega )$ with orthogonal random increments. Their variance measure is the spectral measure. When it has density $f$, the autocovariance is $\gamma (h)=\int e^{2\pi ih\omega }f(\omega )d\omega$.

For a centered record of length $T$, the periodogram is the squared modulus of its discrete Fourier transform, with normalizationUnder suitable short-memory assumptions its expectation approaches the spectral density of a stationary process, but its variance generally does not vanish. Smoothing nearby frequencies can provide an estimate with statistical consistency.