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Fourier Series · Sawtooth Wave Decomposition and Harmonic Superposition

A sawtooth wave can be decomposed into the superposition of all sine harmonics (both odd and even, with alternating signs): \(f(t) = \frac{2}{\pi}\left(\sin\omega t - \frac{1}{2}\sin 2\omega t + \frac{1}{3}\sin 3\omega t - \frac{1}{4}\sin 4\omega t + \cdots\right)\)
The charts below show the fundamental wave, and the process of progressively adding 2nd, 3rd, 5th, 10th, 20th, and 50th harmonics to approximate a sawtooth wave.

Fourier Series (Sawtooth Wave)

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

\[ f(t) = \frac{2}{\pi} \sum_{n=1}^{\infty} (-1)^{n+1} \frac{\sin(n\omega t)}{n} \]

• Fundamental component: \(\frac{2}{\pi}\sin(\omega t) \approx 0.637\sin(\omega t)\)
• 2nd harmonic: \(-\frac{2}{2\pi}\sin(2\omega t) \approx -0.318\sin(2\omega t)\)
• 3rd harmonic: \(\frac{2}{3\pi}\sin(3\omega t) \approx 0.212\sin(3\omega t)\)
• 4th harmonic: \(-\frac{2}{4\pi}\sin(4\omega t) \approx -0.159\sin(4\omega t)\)
• Coefficients decay as \(\frac{2}{n\pi}\), with alternating signs (+, -, +, -...).
Unlike the square wave, the sawtooth wave contains all harmonics (odd + even), with alternating coefficient signs. As harmonics increase, the synthesized waveform approaches the ideal sawtooth wave, but Gibbs phenomenon (~9% overshoot) appears at discontinuities.

📌 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 ~0.637, same frequency as the original sawtooth wave.
  • Adding the 2nd harmonic, the waveform begins to show asymmetric tilt, presenting a sawtooth outline.
  • Continuing to add 3rd, 4th, and 5th harmonics, the rising edge approaches a straight line, falling edge steepens.
  • Adding the 10th harmonic, the sawtooth shape is very clear with a steep falling edge.
  • Adding 20th and 50th harmonics, the waveform nearly perfectly approximates the ideal sawtooth wave, with only Gibbs overshoot remaining.
  • Higher harmonics yield more precise synthesis, but the overshoot cannot be fully eliminated (Gibbs phenomenon).
  • The sawtooth wave contains even harmonics, so its convergence differs from the square wave (odd harmonics only).

① Ideal Sawtooth Wave (Reference)

Amplitude ±1 V

Ideal sawtooth wave, linear rise, instantaneous jump back to -1

② Fundamental Component (1st Harmonic)

Amplitude 2/π ≈ 0.637

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

③ Fundamental + 2nd Harmonic

1st + 2nd

Waveform begins to tilt, showing asymmetric outline

④ Fundamental + 2nd + 3rd Harmonic

1+2+3rd

Sawtooth outline more pronounced, rising edge approaches straight line

⑤ Fundamental ~ 5th Harmonic

1+2+3+4+5th

Falling edge steepens, sawtooth shape gradually becomes clear

⑥ Fundamental ~ 10th Harmonic

1+2+3+...+10th

Rising edge nearly straight, falling edge steep, Gibbs overshoot appears

⑦ Fundamental ~ 20th Harmonic

1+2+3+...+20th

Waveform edges steeper, top ripple refined, approaching ideal sawtooth wave

⑧ Fundamental ~ 50th Harmonic

1+2+3+...+50th

Higher-order harmonics closely reproduce the waveform; only fine ripples and Gibbs overshoot remain

⚡ Gibbs Phenomenon: At discontinuities, a finite Fourier series exhibits ~9% overshoot that does not vanish with increasing terms. The sawtooth wave contains all harmonics (odd + even), with alternating coefficient signs.