Play a chord on a piano and a microphone records a single wobbling voltage — one number per instant, a textbook time series. Yet your ear does not hear "a wobbling voltage": it hears three notes at once. Somewhere between the air and the auditory cortex, the signal gets re-sorted from a story about time ("what is the value now?") into a story about frequency ("how much of each pitch is in the mix?"). That change of viewpoint — same data, rotated ninety degrees — is the whole of spectral analysis, and the object that carries it is the spectral density.

Everything you already know about a stationary series lives in its autocovariance function \gamma(h) — how a value co-varies with its own past h steps back. The spectral density is that same information written in the language of frequencies. Nothing is added and nothing is lost; we merely choose a basis in which certain questions become easy.

From autocovariance to spectrum

Take a zero-mean stationary process with autocovariances \gamma(h). Its spectral density is the Fourier transform of that sequence:

The frequency \omega is in radians per time step: a slow drift lives near \omega = 0, the fastest wiggle a sampled series can even represent lives at \omega = \pi (one full flip every two observations). By Euler's formula, e^{-i\omega h} = \cos(\omega h) - i\sin(\omega h); because \gamma(h)=\gamma(-h) is symmetric, the imaginary parts cancel in pairs and f(\omega) comes out real and non-negative — a genuine density over frequency.

That last line is the punchline. The spectral density is a recipe for the series' variance: the area under f over a band of frequencies is exactly the variance contributed by oscillations in that band. Fast structure and slow structure each get their share, and the shares sum to the total.

Why noise is called "white"

Take the simplest series of all, white noise: mean zero, variance \sigma^2, and no serial dependence, so \gamma(0)=\sigma^2 and \gamma(h)=0 for h\neq 0. Only one term survives the sum:

Every frequency carries the same power — exactly as white light contains every colour in equal measure. That is where the name comes from, and it runs the other way too: a process with a flat spectrum is white noise. Structure in a series always shows up as a spectrum that is not flat — humped, tilted, or spiked.

Red, blue, and a peak: the AR(1) spectrum

A first-order autoregression, AR(1) x_t = \phi x_{t-1} + \varepsilon_t, has autocovariance \gamma(h) = \sigma^2 \phi^{|h|}/(1-\phi^2). Feed that geometric sequence into the sum and it collapses to a clean closed form:

Drag \phi and watch where the power piles up (here \sigma^2 = 1):

• \phi > 0 — a "red" spectrum. Power concentrates at low frequencies (near \omega = 0): the series is smooth and drifts in long, slow swings, like red light sitting at the low-frequency end.
• \phi < 0 — a "blue" spectrum. Power piles up at high frequencies (near \omega = \pi): the series zig-zags, flipping sign almost every step.

The peak matters. A hump in the spectrum at some \omega_0 announces a dominant period of 2\pi/\omega_0 time steps — the series "prefers" to oscillate at that rate. A peak at \omega_0 = \pi/6, say, means a favoured cycle of length 12; find such a peak in monthly data and you have found an annual season, purely from the shape of the spectrum.

A worked example: splitting the variance

Suppose x_t = \tfrac{1}{2}x_{t-1} + \varepsilon_t with \sigma^2 = 1. Its variance is \gamma(0) = \sigma^2/(1-\phi^2) = 1/(1-0.25) = 4/3 \approx 1.33. The spectral density is f(\omega) = \frac{1}{2\pi}\cdot\frac{1}{1.25 - \cos\omega}. Evaluate it at the ends:

• At \omega = 0: f(0) = \frac{1}{2\pi}\cdot\frac{1}{0.25} = \frac{2}{\pi} \approx 0.637 — the tall low-frequency shoulder.
• At \omega = \pi: f(\pi) = \frac{1}{2\pi}\cdot\frac{1}{2.25} \approx 0.0707 — the low high-frequency floor.

Low frequencies carry roughly nine times the power of high ones: this series is decisively "red". And if you integrated f across the whole band from -\pi to \pi, you would recover exactly 1.33 — the variance, whole again.