Fourier Analysis and the Wave Packet
Any repeating shape is a sum of sines, and a pulse is a sum over a continuous range of frequencies. The shorter the pulse, the wider the range of frequencies it needs.
What a learner can do afterwards
- Writes a square wave as a sum of harmonics and describes what adding more terms does
- Relates the width of a pulse in time to the spread of frequencies it contains
- Applies the idea to bandwidth in a communication channel
1 · Read
Any repeating shape is a sum of sines at whole multiples of a base frequency. A square wave, for instance, is built from odd multiples stacked together. Superposition does the work: waves add point by point wherever they meet.
Each new harmonic sharpens the corners of the square wave sum. In the limit the sum reproduces the jumps of the square wave. More terms always mean a closer match.
Squeeze a pulse shorter in time and it demands a wider range of frequencies. A needle thin spike needs a broad band, while a long gentle swell needs only a narrow one. Short in time means wide in frequency.
A channel must carry every frequency its signal contains. A narrow bandwidth rounds off fast changes, so sharp pulses arrive smeared. Match the bandwidth to the fastest change you must preserve.
Sums of sines build any shape, and shorter pulses always cost a wider band.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.