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Fourier Series · Trapezoidal Wave Decomposition and Harmonic Superposition

A trapezoidal wave can be decomposed into the superposition of odd sine harmonics, with spectral decay rate between that of the square wave and triangle wave.
Trapezoidal wave parameters: rising/falling edges each occupy 20% of the half-period (\(0.1\,\text{s}\)), flat top occupies 60% (\(0.3\,\text{s}\)), amplitude \(\pm1\).
The charts below show the fundamental wave, and the process of progressively adding 3rd, 5th, 7th, 11th, 33rd, and 55th harmonics to approximate a trapezoidal wave.

Fourier Series (Trapezoidal Wave)

Fourier series of a periodic trapezoidal wave (amplitude \(\pm1\), rise time \(\tau_r\), flat top time \(\tau_p\), fall time \(\tau_f\)) contains only odd sine terms:

\[ f(t) = \sum_{n=1,3,5,\ldots}^{\infty} b_n \sin(n\omega t) \]

Coefficients \(b_n\) decay as \(n^2\) (when rise/fall edges are non-zero), converging faster than the square wave. This demo uses numerical integration to precisely compute each harmonic coefficient.
Parameters: period \(T=1\,\text{s}\), rising edge length \(0.1\,\text{s}\), flat top length \(0.3\,\text{s}\), falling edge length \(0.1\,\text{s}\).

📌 Horizontal axis is time \(t\) (seconds), fundamental frequency \(f = 1\,\text{Hz}\), showing two complete periods. Trapezoidal wave edges are smooth, harmonic decay is fast, and finite-term synthesis already closely approximates the waveform.

Key Observations

  • The fundamental is a sine wave with amplitude ~0.93 (slightly larger than the triangle wave fundamental).
  • Adding the 3rd harmonic, the waveform begins to show a "trapezoidal" outline with small ripples in the flat top region.
  • Adding 5th and 7th harmonics reduces flat top ripples, and rising/falling edges steepen.
  • Adding 11th, 33rd, and 55th harmonics makes the waveform nearly coincide with the ideal trapezoidal wave, with negligible Gibbs overshoot.
  • Due to the presence of rising/falling edges, the trapezoidal wave's high-frequency components decay rapidly, and synthesis convergence is superior to the square wave.

① Ideal Trapezoidal Wave (Reference)

Amplitude ±1 V

Rise/fall edges 0.1s each, flat top 0.3s, symmetric trapezoid

② Fundamental Component (1st Harmonic)

Amplitude ≈ 0.905

\(f_1(t)=b_1\sin(\omega t)\), dominant frequency component

③ Fundamental + 3rd Harmonic

1st + 3rd

Flat top outline begins to appear, rise/fall segments curve

④ Fundamental + 3rd + 5th Harmonics

1+3+5th

Flat top ripples decrease, rise/fall edges steepen

⑤ Fundamental + 3rd + 5th + 7th Harmonics

1+3+5+7th

Further approximation, flatter flat top region

⑥ Fundamental + 3+5+7+9+11th Harmonics

1+3+5+7+9+11th

Flat top ripples essentially disappear, rising edge straighter

⑦ Fundamental ~ 33rd Harmonic (Odd)

1+3+5+...+33rd

The trapezoidal wave is nearly reproduced; higher-order terms contribute very little

⑧ Fundamental ~ 55th Harmonic (Odd)

1+3+5+...+55th

With many higher-order terms, the waveform nearly coincides with the ideal trapezoidal wave

💡 The trapezoidal wave Fourier series decays as \(1/n^2\) (when rise/fall edges are non-zero), converging as fast as the triangle wave and much faster than the square wave. Therefore, the finite-term synthesis waveform already closely approximates the original waveform.