ISEGORIA / MATH ENCYCLOPEDIA
Fourier analysis: functions made of waves
Harmonics, convolution, and the limits of sampling.
Before you begin: Trigonometry and integration
Predict, manipulate, then check your reasoning against the example and question. Graphs illustrate the mathematics; they do not replace a proof.
1. Reconstruct a square wave
Add odd harmonics to a square wave. Smooth regions converge, but the overshoot next to each jump settles near 8.95% of the jump while becoming narrower, as the zoomed panel shows. At the jump itself the series converges to the midpoint 0, and the coefficient panel shows the 1/k decay behind all of this.
Worked example. The first approximation is 4 sin(x)/π. Its amplitude exceeds one.
Watch out. More terms do not remove the limiting Gibbs overshoot relative to the jump size.
Does convergence have to be uniform?
No. Continuous partial sums cannot converge uniformly to a discontinuous function.
2. Convolution as overlap
Slide a reflected copy of g across the unit pulse f by dragging either handle. The product f(τ)g(t − τ) is shaded, and its area is the convolution at t, traced below. Two unit pulses give a triangle; the ramp makes the reflection visible, and the one-sided decay shows convolution as smoothing with memory.
Worked example. At displacement t=0.25 the overlap is 0.75.
Watch out. Convolution includes a reversal before shifting. The reversal is invisible for these symmetric pulses.
Why is the output zero at displacement 1.2?
The two supports no longer overlap.
3. Sampling and aliasing
Compare a continuous cosine with equally spaced samples. When f exceeds half the sampling rate, the samples also fit the dashed cosine at the folded frequency. The frequency axis below shows why: sampling copies the spectrum to every k·fₛ ± f, and only the copy inside [0, fₛ/2] is seen.
Worked example. A 7 Hz cosine sampled at 10 Hz shares its samples with a 3 Hz cosine.
Watch out. The sampling theorem assumes a band-limited signal and ideal reconstruction. Finite samples do not uniquely identify arbitrary signals.
What sampling rate avoids aliasing for a signal band-limited to 7 Hz?
Choose a rate strictly greater than 14 Hz.