◄ WORLD V · SONNY 5DART 444 · a helldive at the net

THE HETERODYNE multiply two waves, shift a frequency — the trick under every radio

Heterodyning is how a radio tunes: multiply the incoming signal by a local oscillator and the product contains the sum and difference frequencies. Pick off the difference and a station at 100 MHz slides down to a fixed, easy-to-filter intermediate frequency — the superheterodyne receiver, unchanged in principle since 1918. In phasor terms it is nothing but multiplication: e^(iω₁t)·e^(iω₂t) = e^(i(ω₁+ω₂)t) — frequencies add (and, with real signals, also subtract). Mixing is the phasor law put to work.

THE TECHNIQUE multiply phasors ⇒ frequencies add (real mix: sum & difference)

The demo multiplies two unit phasors and shows the product is a unit phasor whose angle is the sum — the mixing that shifts frequency: live demo







  
HISTORY & CREDIT Fessenden 1901 · superheterodyne 1918

  
“To change a signal’s frequency you must regenerate it.” — just multiply by another wave; the product carries the sum and difference. cited

  
the mix · cos(ω₁t)cos(ω₂t) = ½[cos(ω₁−ω₂)t + cos(ω₁+ω₂)t] — sum & difference.
the phasor view · e^(iω₁t)e^(iω₂t)=e^(i(ω₁+ω₂)t) — frequencies add.
1901 / 1918 · Reginald Fessenden invented and named the heterodyne principle (1901); Edwin Armstrong and Lucien Lévy the superheterodyne (1918; Lévy’s French patent has priority).

  
Two waves multiplied into their sum and difference — the frequency shift that lets a radio tune. frequencies add






  
RECOMMEND FOR I-13 phasor mixing, on the compiler

  
On the canonical compiler, multiplying (0.6,0.8) by (0.8,0.6) gives (0,1) — a unit phasor at 90°, the summed angle:

  
$ i13 run cr_heterodyne.i13   # phasor multiply = frequency add
RUN OK · 58 step(s) · peak stack 5 · call depth 1
px = 0   py = 1        -- product phasor at 90 degrees (angles summed)
mag2 = 1   unit = 1    -- still on the unit circle
# real mixing yields both sum and difference frequencies

  
Recommend as a NULL — the phasor law applied. Heterodyning is multiplication of phasors, so “frequencies add” is the same forall-pinned identity as dart 441 (B39/confluence-shaped). It is a beautiful use of the phase channel, not a new invariant. NULL — and the working half of the phasor: mixing, the move that tunes every receiver.