Fourier Series

Think of a musical chord: it sounds like one complicated sound, but it's actually several pure notes played at once, each at its own frequency and volume. A Fourier series applies the same idea to any repeating (periodic) signal — daily seasonality in trading volume, an intraday volatility pattern, or a noisy oscillating sensor reading — by expressing it as a sum of pure sine and cosine waves of different frequencies, each scaled by a coefficient that says how much of that frequency is present.

Formally, a periodic function $f(t)$ with period $T$ can be written as

$$f(t) = a_0 + \sum_{k=1}^{\infty} \left[ a_k \cos\!\left(\frac{2\pi k t}{T}\right) + b_k \sin\!\left(\frac{2\pi k t}{T}\right) \right],$$

where $a_0$ is the average level of the signal, and each $a_k, b_k$ measures how strongly a wave that repeats $k$ times per period contributes to the shape of $f$. In plain terms: any repeating pattern is a mixture of a constant plus waves that wiggle once per cycle, twice per cycle, three times per cycle, and so on, each weighted by how prominent that particular wiggle is in the data.

A trading desk analyzing intraday volume might find that the $k=1$ term (one full cycle per day) captures the broad U-shape of higher volume at the open and close, while smaller higher-$k$ terms capture finer bumps around scheduled data releases — decomposing one messy curve into a short list of clean, interpretable frequency components.