Mathematics.

Harmonic Analysis

Fourier Series

Real Analysis55 minDifficulty8 out of 10

You should know: taylor series

Overview

A Fourier series represents a periodic function as an infinite sum of sines and cosines. Where a Taylor series approximates a function using powers of x near a single point, a Fourier series approximates using waves of different frequencies across the whole period — making it the natural tool whenever a signal repeats.

Intuition

Any repeating sound — a musical note, a heartbeat, an AC current — can be built by adding together pure tones (sine waves) of different frequencies and loudness. A Fourier series is the recipe: it tells you exactly how much of each frequency to mix in to reconstruct the original repeating signal. A square wave sounds harsh because it needs infinitely many high-frequency sine waves layered together to approximate its sharp corners.

Interactive Graph

Watch a square wave build up from sine harmonics

Loading visualization…

Formal Definition

Definition

For a function f with period 2π, the Fourier series is:

f(x)=a02+∑n=1∞[ancos⁡(nx)+bnsin⁡(nx)]f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left[a_n \cos(nx) + b_n \sin(nx)\right]
Fourier series
an=1π∫−ππf(x)cos⁡(nx) dx,bn=1π∫−ππf(x)sin⁡(nx) dxa_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\cos(nx)\,dx, \quad b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\sin(nx)\,dx

The Fourier coefficients, computed by projecting f onto each sine/cosine

Notation

NotationMeaning
an,bna_n, b_nFourier coefficients for the cosine and sine terms
ω=2π/T\omega = 2\pi/TAngular frequency for a function of period T

Properties

Square wave series

square(x)=4π∑k=0∞sin⁡((2k+1)x)2k+1\text{square}(x) = \frac{4}{\pi}\sum_{k=0}^{\infty} \frac{\sin((2k+1)x)}{2k+1}

Example: Uses only odd harmonics

Gibbs phenomenon

Near a jump discontinuity, partial sums overshoot by about 9%, no matter how many terms are used.\text{Near a jump discontinuity, partial sums overshoot by about 9\%, no matter how many terms are used.}

Parseval's theorem

12π∫−ππ∣f(x)∣2 dx=a024+12∑n=1∞(an2+bn2)\frac{1}{2\pi}\int_{-\pi}^{\pi} |f(x)|^2\,dx = \frac{a_0^2}{4} + \frac{1}{2}\sum_{n=1}^{\infty}(a_n^2+b_n^2)

Example: Energy in the signal equals energy in its frequency components

Applications

Audio compression (MP3), image compression (JPEG's DCT is a close relative), and every equalizer or spectrum analyzer are direct applications.

Worked Examples

  1. The square wave series only has odd-numbered sine harmonics, with amplitude inversely proportional to n.

    4π(sin⁡x+sin⁡3x3+sin⁡5x5+⋯ )\frac{4}{\pi}\left(\sin x + \frac{\sin 3x}{3} + \frac{\sin 5x}{5} + \cdots\right)

Answer: Only odd harmonics (1st, 3rd, 5th, ...), decaying as 1/n.

Practice Problems

Difficulty 7/10

Why does the Gibbs phenomenon never disappear, no matter how many terms are added?

Difficulty 6/10

An audio equalizer boosts or cuts specific frequencies. How does the Fourier series/transform make this possible?

Difficulty 6/10

In structural/mechanical engineering, why is decomposing a complex periodic load into a Fourier series useful for vibration analysis?

Common Mistakes

Common Mistake

Assuming a Fourier series always converges to f(x) exactly, everywhere.

At a jump discontinuity, the series converges to the AVERAGE of the left and right limits, not to either one-sided value -- and the Gibbs phenomenon means nearby points overshoot.

Quiz

A Fourier series decomposes a periodic function into a sum of:

Flashcards

1 / 2

Historical Background

Joseph Fourier introduced the idea in 1807 while studying heat conduction, claiming any periodic function could be represented as a sum of sines and cosines — a claim so controversial at the time (Lagrange objected strongly) that his paper was initially rejected by the French Academy of Sciences. Fourier's work was eventually published in 1822 as Théorie analytique de la chaleur, and the rigorous conditions for convergence were worked out over the following century by Dirichlet, Riemann, and others.

  1. 1807

    Fourier submits his heat conduction paper with the series idea, initially rejected

    Joseph Fourier

  2. 1822

    Théorie analytique de la chaleur is published

    Joseph Fourier

  3. 1829

    Dirichlet proves rigorous convergence conditions

    Peter Gustav Lejeune Dirichlet

Summary

  • A Fourier series represents a periodic function as a sum of sines and cosines of different frequencies.
  • Coefficients aₙ, bₙ are computed by integrating f against cos(nx)/sin(nx).
  • Gibbs phenomenon: ~9% overshoot near jump discontinuities that persists no matter how many terms are used.
  • Foundation of audio/image compression, spectral analysis, and signal processing broadly.

References

  1. BookFourier, J. (1822). Théorie analytique de la chaleur.
  2. WebsiteWikipedia — Fourier series