The nature of time series and their frequency analysis. Foundations. Analytic properties of fourier transforms and complex matrices. Stochastic properties of finite fourier transforms. The estimation of power spectra. Analysis of a linear time invariant relation between a stochastic series and several deterministic series. Estimating the second-order spectra of vector-valued series. Analysis of a linear time invariant between two vector-valued stochastic series. Principal components in the frequency domain. The canonical analysis of time series. Proofs of theorems.