Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis can be used to analyze the frequency characteristics of various filters. In this section, we explain how Fourier analysis lets us determine these characteristics.
A Fourier series is a way of representing a periodic function as a (possibly infinite) sum of sine and cosine functions. It is analogous to a Taylor series, which represents functions as possibly infinite sums of monomial terms.
<aside> 📖 Suppose we have a function $f: [-L, L] \rightarrow \mathbb R$. The Fourier series for $f$ relative to the interval $[-L, L]$ is:
$$ f(x) \approx \frac{a_0}{2} + \sum_{n = 1}^\infty a_n \cos\left(\frac{n \pi}{L}x\right) + b_n\sin\left(\frac{n \pi}{L}x\right) $$
Here, $a_n$ and $b_n$ are known as the Fourier coefficients and are defined as:
$$ a_n = \frac{1}{L}\int_{-L}^Lf(x)\cos\left(\frac{n\pi}{L}x\right)dx $$
$$ b_n = \frac{1}{L}\int_{-L}^Lf(x)\sin\left(\frac{n\pi}{L}x\right)dx $$
</aside>
Signal Processing: A Mathematical Approach, Second Edition (oapen.org)
https://eee.guc.edu.eg/Courses/Communications/COMM401 Signal & System Theory/Alan V. Oppenheim, Alan S. Willsky, with S. Hamid-Signals and Systems-Prentice Hall (1996).pdf