Angle modulation (FM/PM)

Clear through a thunderstorm

• AM crackles in a thunderstorm; FM stays clear. FM trades extra bandwidth for less noise.
• This lesson prices both sides of that trade: Carson's rule for the bandwidth, and the gain in signal-to-noise ratio.

The message moves the frequency

• Frequency modulation 频率调制 puts the message in the instantaneous frequency 瞬时频率: $f_{i} = f_{c} + k_{f}x(t)$.
• The largest swing is the frequency deviation 频偏 $\Delta f$. Against the message frequency it gives the index $\beta = \Delta f / f_{m}$.

Carson's rule

• Carson's rule 卡森公式 estimates the bandwidth: $B \approx 2(\Delta f + f_{m})$.
• It holds about 98% of the power. Narrowband FM ($\beta \ll 1$) shrinks to about $2f_{m}$, like AM.

Where the lines come from

• The phase is $2\pi f_{c}t + \beta\sin(\omega_{m}t)$. Expanding it gives lines at $f_{c} \pm kf_{m}$ with Bessel-function weights $J_{k}(\beta)$.
• The significant lines stop near $k \approx \beta + 1$, which is exactly Carson's $2(\Delta f + f_{m})$.

Worked example: FM broadcast

Worked example: the price of quiet

Three common slips

Recap