Fourier Series
Any reasonable periodic function is a sum of sines and cosines. The coefficients are recovered by integrating against each one, which is what lets a system driven by a complicated periodic input be solved one frequency at a time.
What a learner can do afterwards
- Compute the first Fourier coefficients of a square or sawtooth wave
- Explain how orthogonality of the sines and cosines isolates each coefficient
- Say what the partial sums do near a jump discontinuity
1 · Read
Any reasonable periodic function is a sum of sines and cosines. You stack them up: a0 over 2 plus a1 cosine x plus b1 sine x plus a2 cosine 2 x plus b2 sine 2 x, on and on. The constant term a0 over 2 is the average of the function over one full period, so a0 is twice that average. To get each an, multiply by cosine n x and integrate over a period, then double and divide by the period length. Do the same trick with sine n x for each bn.
Take the square wave that is -1 on (-pi, 0) and +1 on (0, pi), repeating every 2 pi. It is odd, so every an is 0 and only sine terms survive. For b1, multiply by sine x and integrate: b1 = (1 over pi) times the integral from -pi to pi of f(x) sin x dx, which is (2 over pi) times the integral from 0 to pi of sin x dx = (2 over pi) times 2 = 4 over pi. The same steps give bn = 4 over (n pi) for odd n and 0 for even n. A few terms already look like the wave, except near the jumps.
This works because sines and cosines of different frequencies are orthogonal: multiply two different ones and integrate over a full period, and you get zero. That is what isolates one coefficient at a time instead of solving for all of them together. Symmetry reads off whole families for free: odd functions keep sines only, and even functions keep cosines only.
Expect trouble at jump discontinuities. Near a jump the partial sums overshoot and undershoot by about 9 percent of the jump size, however many terms you keep; more terms only squeeze the wiggle closer to the jump. This is the Gibbs phenomenon. At the jump itself the series converges to the midpoint of the two sides: a wave that jumps from 1 to 3 gives 2 there. The split is still useful to engineers, because a system driven by a messy periodic input can be solved one frequency at a time and the answers added.
Multiply by each tone, integrate over a period, and stack the surviving sines and cosines.
Readings 'A recipe for Fourier coefficients' and 'Solving one frequency at a time' are original to this dataset. All 9 extra questions are original to this dataset.
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