Spectral Analysis

Spectral analysis treats a time series as a sum of sinusoids and asks how its variance is distributed across frequencies. The lag-domain language of autocovariance and autocorrelation reorganizes into the frequency-domain language of spectral density, with the two perspectives connected by the Fourier transform. Periodic structure that is hard to read off an autocorrelation plot becomes obvious as a peak in a periodogram; a filter that is awkward to write as a convolution becomes a multiplication in the frequency domain. The toolkit covered here is the standard one used in audio, neuroscience, geophysics, and any setting where the data carry meaningful oscillations.

Prerequisites

• Time series and weak stationarity
• Autocovariance function
• Complex exponentials and Euler’s formula
• Sinusoidal regression (motivates the Fourier basis)

Contents

• 2.3.1 Discrete Fourier Transform: the finite Fourier basis and the complex coefficients $d({\omega }_j)$ that decompose a length-$n$ record
• 2.3.2 Periodogram: the squared modulus $P({\omega }_j)=(4/n)|d({\omega }_j){|}^2$, a noisy but unbiased estimate of the spectral density
• 2.3.3 Spectral Density: the Fourier pair of the autocovariance, $f(\omega )={\sum }_h\gamma (h)e^{-2\pi ih\omega }$
• 2.3.4 Linear Filters and Frequency Response: convolution in time as multiplication in frequency, with $f_y=|A{|}^2{f}_x$ as the central identity
• 2.3.5 Time-Frequency Analysis: the short-time Fourier transform and spectrogram for non-stationary signals

Recommended path

A reader new to the frequency domain should read 2.3.1 Discrete Fourier Transform first to anchor the basis, then 2.3.3 Spectral Density to see what the population quantity is, and finally 2.3.2 Periodogram to connect data to estimation. 2.3.4 Linear Filters and Frequency Response and 2.3.5 Time-Frequency Analysis are extensions that build on those three.