4 Spectral analysis and filtering

The following section is adapted from Percival and Walden Spectral Analysis for physical applications, CUP (1993). I will try to extract the most relevant notions so you can understand in greater details nonparametric spectral estimation, the periodogram, tapering and smoothing, and testing procedures.

As Percival and Walden say: “Spectral analysis is an analysis of variance”.

Roughly speaking, the aim of spectral analysis is to break the variability of the series into different contributions corresponding to a frequency.

We shall analyze stationary sequences \(\boldsymbol{X} \in \mathbb{R}^n\) thought to originate from a stationary process \(\{\boldsymbol{X}_t\}, t \in \mathbb{Z}\). All the information is contained in the spectral density function assuming the latter exists.