Signals, spectra and bandwidth
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| bandwidth/ˈbændwɪdθ/ | băng thông |
| sideband | Băng bên |
The tail you keep
- A square wave is odd harmonics at $4/\pi \cdot 1/k$; bandwidth is the design choice of how much tail you keep. The fundamental alone carries about 81% of the power; through a 5 kHz low-pass the 1st and 3rd harmonics survive and the waveform rounds over.
- For modulated signals the sideband rule: mixing shifts the spectrum, never reshapes it — baseband width $W$ on a carrier occupies $2W$.
{exercise:uni-ee1111-tail}
A 1 kHz square wave passes a 5 kHz ideal low-pass. Which harmonics survive?
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The filter passes everything below 5 kHz: the 1 kHz fundamental and the 3 kHz third harmonic survive; the 5 kHz component sits at the edge and the 7th is gone.
A 4 kHz message on a 1 MHz carrier occupies —
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Mixing shifts and mirrors: the carrier plus two sidebands span 996 to 1004 kHz — occupancy 2W is geometry, not electronics.
Parseval as the meter
- Power fractions come from amplitude squares: keeping harmonics 1 and 3 keeps about 90% of a square wave's power — the ear hears the rounding, the meter barely moves. Every bandwidth claim is a power claim in disguise.
{exercise:uni-ee1112-parseval}
Keeping only the fundamental and third harmonic keeps what fraction of a square wave power?
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Powers are amplitude squares: 1.273 squared plus 0.424 squared against the full tail gives about 90% — the meter barely moves while the ear hears rounding.