Fourier series
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| harmonic/hɑːˈmɒnɪk/ | harmonic |
Fourier: every periodic signal is a sum of sinusoids
- Coefficients are averages against each sinusoid; symmetry deletes half of them before integrating (odd → no cosines).
- The square wave's $4/π(1 - 1/3 + …) = 1$ at the peak is the check that convinces; harmonic amplitudes fall as
1/n— sharp edges need many harmonics.
{exercise:uni-ee2x-fourier}
The square wave's harmonic amplitudes follow 1/n. A channel passing only n ≤ 3 outputs —
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Sharp corners need the cancellation of MANY harmonics; truncating at n = 3 leaves ~15% undershoot at the peak and overshoot near edges — 4/π(1 − 1/3) = 0.849, not 1.
At t = T/4 the square-wave series 4/pi (1 - 1/3 + 1/5 - ...) evaluates to —
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The alternating reciprocal-odd series sums to pi/4; scaled by 4/pi it gives exactly 1 - the value the square wave demands at its peak. The Gibbs overshoot lives at the jump, not at the peak.
Of a +-1 V square wave power delivered to 1 ohm, the fundamental alone carries about —
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The fundamental amplitude is 4/pi; its mean-square share is (4/pi)^2/2 = 8/pi^2 = 0.811. The third harmonic adds just 9% - a low-pass filter keeping only the fundamental still delivers most of the power.