Fourier Series Coefficients Calculator

A Fourier series rebuilds a periodic signal as a sum of sines and cosines at multiples of the base frequency 1/T. Each coefficient is an inner product: multiply the signal by the harmonic, average over one period. For the odd square wave of amplitude ±1 and period 2π, the cosine coefficients vanish (aₙ = 0) and integration by parts yields bₙ = 4/(πn) for odd n only: b₁ = 4/π ≈ 1.273, b₃ = 4/(3π) ≈ 0.424, b₅ ≈ 0.255. Partial sums converge with the characteristic overshoot near discontinuities — the Gibbs phenomenon, about 9% of the jump regardless of term count. The calculator evaluates the integrals for standard waveforms (square, sawtooth, triangle, pulse), lists coefficients and amplitudes √(aₙ²+bₙ²), and reports the harmonic-by-harmonic RMS reconstruction error.

1. 1

   Choose or define the waveform

   Standard shapes or piecewise definition over one period

2. 2

   Integrate for each coefficient

   a₀, aₙ, bₙ formulas are evaluated with steps

3. 3

   Assemble the series

   f(t) ≈ a₀/2 + Σ aₙcos + bₙsin

4. 4

   Measure reconstruction

   Amplitude spectrum and Gibbs overshoot are reported

• 1

  Odd square wave: only odd sine terms, bₙ = 4/(πn)

• 2

  Symmetries kill whole coefficient families (even → no bₙ)

• 3

  Gibbs overshoot stays near 9% of the jump no matter the terms

• Forgetting the 2/T normalization on the integrals

• Including even harmonics for a half-wave symmetric square wave

• Reporting a₀ without halving in the series convention

• Expecting the overshoot to vanish by adding terms — it narrows but persists