◄ WORLD V · SONNY 5DART 440 · a helldive at the net

FREQUENCY MODULATION the message rides the frequency — the derivative of the phase

Frequency modulation puts the message in how fast the carrier turns. Instantaneous frequency is the rate of change of phase, so FM is PM’s close cousin: modulate the phase’s increment instead of the phase itself. The receiver differentiates — measures phase step by step — and the message falls out. FM trades bandwidth for a striking immunity to amplitude noise (static rides the envelope, and FM ignores the envelope), which is why FM radio sounds clean where AM crackles. It is the phase channel seen through a derivative.

THE TECHNIQUE θ[n]=θ[n−1]+f₀+k·msg; freq=Δθ recovers it

The demo frequency-modulates a message onto phase increments, then differentiates the phase to recover the instantaneous frequency: live demo







  
HISTORY & CREDIT Edwin Armstrong · 1933 (FM)

  
“FM and PM are different animals.” — frequency is the derivative of phase; FM is PM of the integral of the message. cited

  
the modulation · θ[n] = θ[n−1] + f₀ + k·msg[n] — the message steers the phase step.
the demod · freq[n] = θ[n] − θ[n−1] — differentiate the phase.
1933 · Edwin Armstrong — wideband FM, immune to amplitude noise.

  
A message in the speed of the turning — the phase channel differentiated once. frequency = dθ/dt






  
RECOMMEND FOR I-13 differentiate the phase, on the compiler

  
On the canonical compiler, at the peak of the message the recovered instantaneous frequency is f₀+k·msg = 2+3·2 = 8:

  
$ i13 run cr_frequencymodulation.i13   # freq = d(phase)
RUN OK · 353 step(s) · peak stack 12 · call depth 6
f2 = 8   -- instantaneous frequency at msg=2: f0(2) + k(3)*msg(2)
recovered_ok = 1

  
Recommend as a NULL — the phase channel, differentiated. FM is the derivative view of the phase: it modulates and reads the phase’s increment. That ties it directly to the seated generative-dual axis (the derivative channel) and to dart 438’s phase channel — the same carried structure, one differentiation apart. No new axis; it strengthens the case that phase and derivative are two faces of one carried channel. NULL — and the bridge between the phase shots and generative-dual.