Question 1

Why can any periodic function be represented as a sum of sines and cosines?

Accepted Answer

Sines and cosines of different frequencies form an orthogonal basis for the space of periodic functions—analogous to how x, y, z unit vectors form a basis for 3D space. Just as any 3D vector can be decomposed into x, y, z components, any periodic function can be decomposed into sine and cosine components at each frequency. The orthogonality property means each coefficient can be computed independently by multiplying by the corresponding sine or cosine and integrating over one period.

Question 2

What is the Gibbs phenomenon?

Accepted Answer

The Gibbs phenomenon is the ~9% overshoot that occurs near a jump discontinuity in a function when approximated by a finite Fourier series. As you add more terms, the overshoot gets sharper and narrower but never disappears—it converges to a spike of constant height (~9% of the jump) at the discontinuity. This is not an error in Fourier theory but a genuine mathematical property: pointwise convergence of the partial sums fails at discontinuities even as the mean-squared error goes to zero.

Question 3

How is Fourier series related to Fourier transform?

Accepted Answer

The Fourier series applies to periodic functions and decomposes them into discrete harmonics at frequencies nω₀. The Fourier transform generalizes this to non-periodic functions, replacing the discrete sum with a continuous integral over all frequencies: F(ω) = ∫f(t)e^(-iωt)dt. The Fourier series can be seen as the special case where the period T → ∞ and the discrete spectrum becomes continuous. Both share the same mathematical structure: representing a function in frequency space.

Question 4

What are even and odd functions and how do they simplify Fourier series?

Accepted Answer

An even function has only cosine terms in its series; an odd function has only sine terms. A square wave centered at the origin is odd (only odd sine harmonics), a triangular wave is also odd. Recognizing symmetry halves the computation needed.

Question 5

How is Fourier series used in music and audio?

Accepted Answer

Every instrument produces a characteristic harmonic spectrum defining its timbre. MP3 compression uses psychoacoustic models combined with Fourier analysis to discard inaudible frequency components, and equalizers adjust the amplitude of Fourier coefficients across frequency bands.

Question 6

What does the N harmonics slider do, and why does the waveform look better with more terms?

Accepted Answer

N harmonics (1-50) sets how many terms are summed. Max error shrinks and Energy captured approaches 100% as N grows, except right at a jump discontinuity, where Gibbs overshoot persists no matter how large N gets.

Question 7

Why do square and sawtooth waves need so many more harmonics than a triangle wave to look clean?

Accepted Answer

A square wave has a jump discontinuity, so its coefficients decay slowly as 1/n and it always shows Gibbs overshoot. A triangle wave has only a kink in slope, so coefficients decay faster as 1/n^2, needing far fewer harmonics to look smooth.

Question 8

What does the epicycle diagram on the left actually represent?

Accepted Answer

Each harmonic is drawn as a rotating arm whose length is its amplitude and rotation rate is its frequency multiple n. Arms chain tip-to-tail like Ptolemy's epicycles; the final arm's tip traces the same waveform shown on the right.

Question 9

What does the Speed slider change, and does it affect the accuracy of the approximation?

Accepted Answer

Speed (0.1x-3x) only scales how fast the epicycle arms rotate and the animation plays; it has no effect on the coefficients, Max error, or Energy captured, which are fixed once you choose a waveform and N.

Question 10

What is the Leading harmonic stat, and why does it stay at f1 for some waveforms even at high N?

Accepted Answer

Leading harmonic reports which term has the largest Fourier coefficient — usually the fundamental f1, since coefficients generally decrease with harmonic number for the presets offered. It stays fixed at f1 across most N because higher harmonics refine fine detail near discontinuities but rarely overtake the fundamental's dominant contribution.

🎵 Fourier Series Decomposition

What did you find?