What is Spectral density, periodogram?

The spectral density of a weakly stationary process with summable autocovariances is $f(\omega) = \frac1{2\pi}\sum_h\gamma(h)e^{-ih\omega}$, $\omega \in [-\pi, \pi]$: the decomposition of its variance by frequency, $\gamma(0) = \int_{-\pi}^\pi f$. The periodogram $I(\omega_j) = \frac1{2\pi n}\bigl\lvert\sum_tX_te^{-it\omega_j}\bigr\rvert^2$ at the Fourier frequencies $\omega_j = 2\pi j/n$ is its sample counterpart.