From Fourier to Wavelet Analysis of Time Series

It is well known that Fourier analysis is suited to the analysis of stationary series. If {Xt,t = 0, ± 1, …} is a weakly stationary process, it can be decomposed into a linear combination of sines and cosines. Formally, 1.1$${X_t} = \int_{ - \pi }^\pi {{e^{i\lambda t}}dZ\left( \lambda \right)} ,$$ where Z(λ), ™π ≤ λ ≤ π is an orthogonal process. Moreover, 1.2$$Var\left\{ {{X_t}} \right\} = \int_{ - \pi }^\pi {dF\left( \lambda \right)} ,$$ with E|dZ(λ)|2 = dF(λ). F(λ) is the spectral distribution function of the process. In the case that dF(λ) = f(λ)dλ, f(λ) is the spectral density function or simply the (second order) spectrum of Xt. Relation (1.2) tells us that the variance of a time series is decomposed into a number of components, each one associated with a particular frequency. This is the basic idea in the Fourier analysis of stationary time series. Some references are Brillinger (1975) and Brockwell and Davis(1991).