Frequency
How many oscillations occur per unit time or distance.
Analysis & Signals · Accessible first encounter
Fourier Analysis:
Building Signals from Waves
Fourier analysis asks how functions and signals can be decomposed into sines, cosines, or complex exponentials. It links vibration, heat flow, sound, images, partial differential equations, and data processing.
01 · Opening mystery
How can a sharp corner be made from perfectly smooth waves?
Every sine wave is smooth. A square wave has sudden jumps. Yet adding carefully chosen sine waves produces an increasingly accurate square-wave approximation.
This surprising fact suggests that frequency is a coordinate system for functions: just as a vector has components along basis directions, a signal can have components along oscillating directions.
Before exploringWhen two tones are played together, can you recover the hidden frequencies from the combined graph?
Make a prediction. The laboratory is designed to challenge or refine it.
02 · Interactive laboratory
Mix tones or build a square wave from harmonics.
Choose a two-tone signal or a square-wave approximation. The upper plot shows the signal in time; the lower bars expose its frequency components.
03 · The big idea
Functions can have coordinates along waves.
Over a full period, sine and cosine waves of different integer frequencies are orthogonal: their products average to zero. This lets us isolate one frequency by multiplying the signal by that wave and integrating.
The resulting Fourier coefficients play the role of vector components. Large coefficients mark frequencies strongly present in the signal; small coefficients mark weak contributions.
Central definition
A Fourier series represents a periodic function as a sum of sine and cosine waves with integer-multiple frequencies.
f(x) ≈ a₀/2 + Σ[aₙ cos(nx) + bₙ sin(nx)]
Amplitude
The strength of a frequency component.
Orthogonality
A generalized perpendicularity that allows components to be separated by integration.
04 · A beautiful result
A square wave emerges from odd harmonics.
A symmetric square wave has only odd sine harmonics. The nth odd harmonic has amplitude proportional to 1/n, so higher frequencies contribute progressively finer corrections.
Near a jump, the partial sums overshoot. Adding more terms narrows the overshoot region but does not eliminate its peak immediately. This Gibbs phenomenon is a warning that convergence can behave differently near discontinuities.
- 1
Symmetry eliminates the cosine coefficients and all even sine coefficients.
- 2
Integrating the square wave against sin(nx) yields coefficients 4/(πn) for odd n.
- 3
Adding the first several odd harmonics reproduces the flat regions and rapid transitions increasingly well.
05 · Why this subject matters
Frequency turns differential equations into algebra.
Differentiation of a sine or complex exponential only multiplies it by a simple frequency factor. Therefore Fourier transformation can convert differential equations into algebraic equations frequency by frequency.
This is why Fourier methods appear in heat flow, acoustics, optics, image filtering, medical imaging, communications, and quantum mechanics.
Signal Processing
Filters noise and separates frequency bands.
Partial Differential Equations
Solves heat, wave, and diffusion models mode by mode.
Medical Imaging
Reconstructs spatial structure from measured frequency data.
06 · Friendly assessment
Check the central ideas without pressure.
The questions focus on the main insights, not obscure details. Each response receives an explanation immediately.
Where this idea leads
Continue through the mathematical atlas.
Connected field
Wavelet Analysis
Add location and scale to frequency-like decomposition.
Explore →Connected fieldPartial Differential Equations
Use modes to solve evolution equations.
Explore →Connected fieldSignal Processing
Design filters and analyze measured data.
Explore →Connected fieldHarmonic Analysis
Extend Fourier ideas to groups and general spaces.
Explore →You have now experienced
You have built a signal from smooth waves, read its spectrum, and seen how infinitely many harmonics can encode a jump.
This is an invitation to continue, not a compressed substitute for a full university course.