Back to Home

Fourier Series · Triangle Wave Decomposition and Harmonic Superposition

A triangle wave can be decomposed into the superposition of odd sine harmonics (coefficients decay as \(n^2\)):
\(f(t) = \frac{8}{\pi^2}\left(\sin\omega t - \frac{1}{9}\sin 3\omega t + \frac{1}{25}\sin 5\omega t - \frac{1}{49}\sin 7\omega t + \cdots\right)\)
The charts below show the fundamental wave, and the process of progressively adding 3rd, 5th, 7th, 11th, 33rd, and 55th harmonics to approximate a triangle wave.

Fourier Series (Triangle Wave)

Fourier series of a periodic triangle wave (amplitude \(\pm1\), peak-to-peak 2, angular frequency \(\omega=2\pi f\)):

\[ f(t) = \frac{8}{\pi^2} \sum_{n=1,3,5,\ldots}^{\infty} \frac{(-1)^{\frac{n-1}{2}}}{n^2} \sin(n\omega t) \]

• Fundamental component: \(\frac{8}{\pi^2}\sin(\omega t)\) ≈ \(0.8106\sin(\omega t)\)
• 3rd harmonic: \(-\frac{8}{9\pi^2}\sin(3\omega t)\) ≈ \(-0.09007\sin(3\omega t)\)
• 5th harmonic: \(\frac{8}{25\pi^2}\sin(5\omega t)\) ≈ \(0.03242\sin(5\omega t)\)
• 7th harmonic: \(-\frac{8}{49\pi^2}\sin(7\omega t)\) ≈ \(-0.01654\sin(7\omega t)\)
As harmonics increase, the synthesized waveform approaches the ideal triangle wave. Since coefficients decay as \(1/n^2\), convergence is faster than for the square wave, and the Gibbs phenomenon is less pronounced.

📌 Horizontal axis is time \(t\) (seconds), fundamental frequency \(f = 1\,\text{Hz}\), showing two complete periods.

Key Observations

  • The fundamental is a sine wave with amplitude ~0.81, same frequency as the original triangle wave.
  • Adding the 3rd harmonic, the waveform begins to show a "triangle wave" outline with curved top.
  • Continuing to add 5th and 7th harmonics, the waveform approaches linear rising/falling segments, top becomes flatter.
  • Adding 11th, 33rd, and 55th harmonics, the waveform edges become steeper, top linearity improves, and Gibbs overshoot is extremely small.
  • Higher harmonics yield more precise synthesis, with convergence much faster than for the square wave.

① Ideal Triangle Wave (Reference)

Amplitude ±1 V

Ideal triangle wave, symmetric linear rise/fall, peak-to-peak 2V

② Fundamental Component (1st Harmonic)

Amplitude 8/π² ≈ 0.8106

\(f_1(t)=\frac{8}{\pi^2}\sin(\omega t)\), a single sine wave

③ Fundamental + 3rd Harmonic

1st + 3rd

Top begins to flatten, rise/fall segments curve

④ Fundamental + 3rd + 5th Harmonics

1+3+5th

Closer to triangle wave, straight segments approach linearity

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

1+3+5+7th

Further approximation, flat top, improved linearity of rising edge

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

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

Rising edge straighter, top ripple extremely fine

⑦ Fundamental ~ 33rd Harmonic (Odd)

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

The triangle 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 triangle wave

💡 The triangle wave Fourier series decays as \(1/n^2\), converging much faster than the square wave (\(1/n\)). Therefore, the finite-term synthesis waveform is much closer to the original waveform, and the Gibbs phenomenon is nearly invisible.