Fourier series

# tag · @ author · ! scientist

A Fourier series is a representation of a periodic function f(x) with period T as a sum of trigonometric functions: f(x) = a0/2 + Σ[a_n cos(2π n x/T) + b_n sin(2π n x/T)]. The coefficients a_n and b_n are calculated as projections of the function onto the corresponding cosine and sine waves and show how strongly each harmonic is expressed in the original signal. The expansion is possible due to the orthogonality of sines and cosines. Fourier series are a powerful tool for solving differential equations (e.g., the heat equation) and underpin spectral analysis.

History

French mathematician Jean-Baptiste Joseph Fourier developed this method in the early 19th century while studying how heat propagates in solid bodies.

How it works

Imagine swinging on a swing: our motion repeats (is periodic). The Fourier series says: this motion can be assembled from several pendulums swinging at different speeds — one slow, another twice as fast, a third three times as fast, and so on. If we correctly adjust the amplitudes (weights) of these pendulums, their sum will give our trajectory.

💡 When expanding a rectangular signal into a Fourier series, the sum of all odd harmonics yields a staircase that approaches a rectangle as the number of terms increases, but near the discontinuity, 'bumps' of about 9% of the signal amplitude remain — this is the Gibbs phenomenon, occurring in any series with discontinuities.

The classical Fourier series is the expansion of a function f(x), integrable on the interval [-π, π], into a complete orthogonal system of functions {1, cos(nx), sin(nx)}n∈N, possessing the property of best approximation in the sense of mean square deviation. The coefficients are calculated by the formulas a_n = (1/π)∫ f(x) cos(nx) dx, b_n = (1/π)∫ f(x) sin(nx) dx. In complex form, this is equivalent to expansion in exponentials e^{inx}, linking the Fourier series to the Fourier transform. For the series to converge to f(x) at point x, Dirichlet conditions must be satisfied (piecewise monotonicity and a finite number of discontinuities). In practice, the Fast Fourier Transform (FFT) is widely used — an algorithm that computes coefficients in O(N log N) operations. Fourier series are generalized to arbitrary orthonormal systems in Hilbert spaces, applied in quantum mechanics (expansion of the wave function over eigenstates) and signal processing.

History

The origins of the idea go back to Daniel Bernoulli (1753), who represented string motion as a sum of harmonics. Jean-Baptiste Joseph Fourier in 1807, in his memoir on heat conduction, systematically used trigonometric series and proposed a method for computing coefficients. His work 'The Analytical Theory of Heat' (1822) established this apparatus, despite criticism from contemporaries for lack of rigor. A rigorous convergence theory was built by Peter Gustav Lejeune Dirichlet in 1829, and later developed by Riemann, Lebesgue, and others.

How it works

When expanding f(x), harmonics with numbers n form a spectrum. Low n correspond to slow changes, high n to fast details. The integrals for coefficients a_n and b_n are the scalar product of the function and the cosine/sine; thanks to the orthogonality of the basis ∫ cos(nx) cos(mx) = 0 for n≠m, each harmonic 'picks out' only its part of the signal. The sum of all harmonics gives the original function almost everywhere, and when truncated to a finite number, ensures minimum mean square error. The Fast Fourier Transform exploits the symmetry of complex roots of unity, recursively reducing the computation of an N-point DFT to two N/2-point ones, dramatically accelerating processing. In physics, harmonic decomposition is used for time series analysis (astronomy), mode composition of laser beams, plasma diagnostics, and solving Schrödinger equations.

f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos\left(\frac{2\pi n x}{T}\right) + b_n \sin\left(\frac{2\pi n x}{T}\right) \right)

f(x) — periodic function with period T, a_0/2 — average value of the function over the period, a_n and b_n — Fourier coefficients, x — argument (e.g., time or coordinate), n — harmonic number (integer).

a_n = \frac{2}{T} \int_0^T f(x) \cos\left(\frac{2\pi n x}{T}\right) dx, \quad b_n = \frac{2}{T} \int_0^T f(x) \sin\left(\frac{2\pi n x}{T}\right) dx

a_n and b_n — amplitudes of the nth cosine and sine components respectively, T — period of the function, f(x) — original function, n — integer (harmonic number). The integral is taken over one full period.

Open problems & fun facts

Gibbs phenomenon: with a finite number of terms, overshoots remain at the discontinuity and do not vanish with increasing N; Expansion of functions not satisfying Dirichlet conditions may diverge at individual points; Need to adapt to non-periodic signals via windowed Fourier transform or wavelets

💡 The fast Cooley–Tukey algorithm (1965) revolutionized digital processing; according to some estimates, FFT is among the top five most used algorithms in the world — every time you listen to music or watch a video, FFT is at work.

No articles yet

Fourier series

History

How it works

History

How it works

History

How it works

Open problems & fun facts

Related articles