Fourier Series: Building Any Wave From Sines

Any periodic wave is a sum of sines

Joseph Fourier's 1807 claim — outrageous at the time — is that any reasonably well-behaved periodic function can be written as a sum of sines and cosines at integer multiples of its fundamental frequency. For a function f(x) with period 2π, that sum is:

f(x) = a₀/2 + Σ [aₙ·cos(nx) + bₙ·sin(nx)],  n = 1, 2, 3, ...

aₙ = (1/π) ∫ f(x)·cos(nx) dx   over one period
bₙ = (1/π) ∫ f(x)·sin(nx) dx   over one period

Each term is a harmonic: n = 1 is the fundamental, n = 2 is the first overtone at twice the frequency, and so on. The coefficients aₙ and bₙ are found by projecting f onto each sine and cosine using the integral above — exactly the same operation as decomposing a vector into orthogonal components, because sines and cosines of different integer frequencies are orthogonal functions over one period.

Epicycles: the geometric picture

Every sine term aₙcos(nx) + bₙsin(nx) can be redrawn as a rotating arm of fixed length spinning at n times the base frequency. Chain enough of these arms tip-to-tail, each spinning at its own harmonic rate, and the tip of the last arm traces out the target waveform. This is the epicycle construction — the same geometric idea, incidentally, that Ptolemy used to model planetary motion, and the reason a Fourier series animation of a square or sawtooth wave looks like a cascade of spinning wheels drawing the curve in real time.

Square, sawtooth, triangle: three canonical series

Different target shapes need different harmonic content. A square wave of amplitude 1 uses only odd harmonics, decaying as 1/n:

square(x) = (4/π) · [ sin(x) + sin(3x)/3 + sin(5x)/5 + sin(7x)/7 + ... ]

A sawtooth uses every harmonic, odd and even, decaying at the same 1/n rate, while a triangle wave uses only odd harmonics but decays much faster, as 1/n². That faster decay is not a coincidence: a triangle wave is continuous with a well-defined slope almost everywhere, while a square wave jumps discontinuously, and the general rule is that the smoother the function, the faster its Fourier coefficients shrink — smoothness in x-space buys you sparsity in frequency space.

The Gibbs phenomenon

Truncate the square-wave series at any finite number of terms and something odd happens right at the jump: the partial sum overshoots the true value by about 9% of the jump height, no matter how many terms you add. Adding more harmonics narrows the region where the overshoot occurs but never shrinks its height — this is the Gibbs phenomenon, a direct consequence of trying to approximate a discontinuity with smooth, bounded oscillations. It shows up anywhere a sharp edge meets a truncated frequency representation, from audio clipping to JPEG ringing artefacts around hard edges.

Parseval's theorem: energy adds up in frequency space

Parseval's theorem says the total energy of a periodic signal, averaged over one period, equals the sum of the energy in each harmonic:

(1/2π) ∫ f(x)² dx = (a₀/2)² + (1/2) Σ (aₙ² + bₙ²)

This is the reason a frequency spectrum plot is meaningful at all: each bar's squared height is literally that harmonic's share of the signal's total power, and the bars sum exactly to the time-domain energy. It is also the theoretical basis for lossy audio and image compression — throw away the harmonics with the least energy and you lose the least perceptible information per bit saved.

Frequently asked questions

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