Module 11 · Lesson 11.2

The Fourier Transform

Any wiggly signal, no matter how jagged, can be built by adding together simple circles spinning at different speeds.

Intuition

A single point spinning at a constant speed traces a plain sine wave when you watch just its height over time. That's boring on its own — but attach a second, smaller spinning point to the tip of the first one, let it spin faster, and their combined height traces something far more interesting.

Keep attaching more spinning circles at different speeds and sizes, and you can build up almost any shape of wiggle. The Fourier transform is the reverse trick: given a complicated signal, it tells you exactly which spinning circles — which frequencies and amplitudes — you'd need to reconstruct it.

Core concept

An audio equalizer's bass, mid, and treble sliders are literally turning up or down the amplitude of different spinning circles inside the sound. JPEG compression works the same way for images: keep the slow, important circles and discard the fast, barely-noticeable ones.

f(θ) = sin(θ) + 0.5·sin(3θ)θ = 30°
sin(30°)0.5
0.5 · sin(3 × 30°) = 0.5 · sin(90°)0.5
Answer: f(30°) = 0.5 + 0.51.0

Interactive Lab

Adjust the second circle's speed (harmonic) and size (amplitude), then drag the angle slider to sweep through time. Watch the green dot trace the combined wave on the right.

f(θ) = 1.00

Key Idea

$$ f(\theta) = \sum_{k=1}^{N} a_k \sin(k\theta) $$

Every term in the sum is one spinning circle: $k$ sets how fast it spins relative to the others, and $a_k$ sets how big it is. The Fourier transform works backward from a signal to find these $a_k$ values — which circles, and how much of each, were used to build it.

$f(\theta)$The combined signal at angle (or time) $\theta$
$k$The harmonic number — how many times faster this circle spins
$a_k$The amplitude — how large the $k$-th spinning circle is
$N$The number of spinning circles being summed together

Think Further

  1. If you set the second circle's amplitude to 0, what does the combined wave look like, and why?
  2. Why does increasing the harmonic number make the wave look more jagged instead of just faster?
  3. An equalizer boosts "treble" — which typically means high harmonics. What would that do to the amplitudes $a_k$ for large $k$?
Show suggested answers
  1. It becomes a plain sine wave, since the second term drops out entirely and only the first spinning circle contributes to the height.
  2. A higher harmonic completes more full spins within the same stretch of θ, so it adds and subtracts from the base wave more often, creating extra wiggles rather than just speeding up the overall shape.
  3. Boosting treble increases the $a_k$ values for large $k$, giving the fast-spinning, high-frequency circles more influence on the final combined sound.