Fourier Analysis and the Wave Packet

For a piecewise smooth periodic signal, a Fourier series adds a mean to sine and cosine terms at whole multiples of the base frequency. Each sine-cosine pair sets harmonic size and phase.

Linear waves obey superposition: at each position and time, add the disturbances, including their signs. Opposite signs may partly or fully cancel.

For the odd square wave that is -1 on (-pi, 0) and +1 on (0, pi), repeating every 2 pi, the series is (4/pi)[sin x + sin(3x)/3 + ...]. Here x is phase in radians. Finite sums are smooth. Away from jumps they converge to the wave. At jumps they tend to the midpoint, 0; nearby Gibbs overshoot approaches 9% of the jump size in a narrowing region.

An isolated pulse has a continuous Fourier spectrum. For a fixed shape, dividing duration by a positive factor a multiplies frequency spread by a, using consistent width measures. A shorter pulse of a different shape need not have a wider spread.

An ideal low-pass channel removes frequencies above its cutoff. Removing components needed for sharp edges slows their transitions. Filtering a jump this way also produces Gibbs overshoot and ringing near it.

Practical bandwidth retains detail to an agreed accuracy; rectangular pulses have unlimited spectral tails. For a fixed pulse shape and the same accuracy, dividing duration by a multiplies the required frequency band by a. Compare its new upper frequency with the low-pass channel cutoff: needed components above the cutoff are removed; those below pass.