Fourier Series
A Fourier series represents a periodic function as an infinite sum of sines and cosines (or complex exponentials) at integer multiples of a fundamental frequency.
Definition
For a function $f(x)$ with period $T$, the Fourier series is:
$$ f(x) = \sum_{n=-\infty}^{\infty} c_n \, e^{j 2\pi n x / T} $$where the Fourier coefficients are:
$$ c_n = \frac{1}{T} \int_{-T/2}^{T/2} f(x) \, e^{-j 2\pi n x / T} \, dx $$Real Form
Using Euler’s formula, the series can be written with sines and cosines:
$$ f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[ a_n \cos\left(\frac{2\pi n x}{T}\right) + b_n \sin\left(\frac{2\pi n x}{T}\right) \right] $$where:
$$ a_n = \frac{2}{T} \int_{-T/2}^{T/2} f(x) \cos\left(\frac{2\pi n x}{T}\right) dx $$$$ b_n = \frac{2}{T} \int_{-T/2}^{T/2} f(x) \sin\left(\frac{2\pi n x}{T}\right) dx $$Relationship to Fourier Transform
| Fourier Series | Fourier Transform |
|---|---|
| Periodic functions | Aperiodic functions |
| Discrete spectrum (harmonics) | Continuous spectrum |
| Coefficients $c_n$ | Transform $F(f)$ |
| Sum over integers | Integral over all frequencies |
The Fourier transform of a periodic function yields Dirac delta functions at the harmonic frequencies, with weights equal to the Fourier coefficients:
$$ \mathcal{F}\{f(x)\} = \sum_{n=-\infty}^{\infty} c_n \, \delta\left(f - \frac{n}{T}\right) $$Convergence
The Fourier series converges to $f(x)$ under various conditions:
| Condition | Type of Convergence |
|---|---|
| $f$ is piecewise smooth | Pointwise (to midpoint at jumps) |
| $f$ is continuous | Uniform |
| $f \in L^2$ | Mean-square (Parseval’s theorem) |
At points of discontinuity, the series converges to the average of left and right limits.
Parseval’s Theorem
The total “energy” in a period equals the sum of squared coefficients:
$$ \frac{1}{T} \int_{-T/2}^{T/2} |f(x)|^2 \, dx = \sum_{n=-\infty}^{\infty} |c_n|^2 $$Examples
Square Wave
$$ f(x) = \text{sgn}(\sin(2\pi x/T)) $$$$ f(x) = \frac{4}{\pi} \sum_{n=1,3,5,...}^{\infty} \frac{1}{n} \sin\left(\frac{2\pi n x}{T}\right) $$Only odd harmonics are present (due to symmetry).
Triangle Wave
$$ f(x) = \frac{8}{\pi^2} \sum_{n=1,3,5,...}^{\infty} \frac{(-1)^{(n-1)/2}}{n^2} \sin\left(\frac{2\pi n x}{T}\right) $$Coefficients decay as $1/n^2$ (faster than square wave) because the function is smoother.
Applications in Optics
- Diffraction gratings: Periodic structures analyzed via Fourier series
- Periodic objects: Produce discrete diffraction orders at harmonic frequencies
- Holography: Periodic fringe patterns encode Fourier coefficients
2D Generalization
For a function periodic in both $x$ and $y$ with periods $X$ and $Y$:
$$ f(x, y) = \sum_{n=-\infty}^{\infty} \sum_{m=-\infty}^{\infty} c_{nm} \, e^{j 2\pi (nx/X + my/Y)} $$