THE ANALYTIC SIGNAL — a dart

THE ANALYTIC SIGNAL a real wave, given its hidden quadrature — the Hilbert transform

A real signal secretly has a phase — you just cannot see it from one channel. The analytic signal makes it visible: take the real wave as I, compute its Hilbert transform (every frequency shifted 90°) as Q, and the complex pair I + iQ now has a well-defined instantaneous amplitude (its envelope) and instantaneous phase. For a pure tone the Hilbert transform of cosine is sine, so the analytic signal is a clean rotating phasor of constant magnitude. It is how you extract an envelope, an instantaneous frequency, a phase — the operator that lifts the phase channel out of a one-dimensional wave.

THE TECHNIQUE z = signal + i·Hilbert(signal); |z| = envelope, arg z = phase

The demo forms the analytic signal of a tone (real + 90°-shifted) and shows its magnitude is constant (the envelope) while its phase advances: live demo

HISTORY & CREDIT Gabor 1946 · Hilbert transform

“A real signal has no phase.” — it has a hidden quadrature; the Hilbert transform reveals it and the envelope with it. cited

the construction · z(t) = x(t) + i·H[x](t) — adjoin the 90°-shifted copy.
what it gives · envelope |z| and instantaneous phase arg z — from a one-channel wave.
1946 · Dennis Gabor — the analytic signal; the Hilbert transform as its quadrature.

A flat real wave lifted into a rotating phasor — its hidden phase and envelope brought into view. phase extracted

RECOMMEND FOR I-13 envelope and phase, on the compiler

On the canonical compiler, the analytic signal’s squared magnitude is constant across instants (the envelope) while the pair rotates:

$ i13 run cr_analyticsignal.i13 # z = real + i*(90-deg shift) RUN OK · 176 step(s) · peak stack 17 · call depth 1 mag0 = 1000000 mag3 = 1000465 envelope_constant = 1 -- |z| steady: the envelope of a pure tone rot = 924000 -- consecutive samples rotate (a phasor)

Recommend as a keeper shot — the operator that CREATES the phase channel. Where quadrature (442) receives I and Q, the analytic signal manufactures Q from a real wave via the Hilbert transform — it is the generative move: from one channel, produce the orthogonal one, deterministically. That is strikingly close to the seated generative-dual axis (from a value, produce its derivative channel). David’s PHASE TEST forces the finest distinction of the campaign: is the Hilbert/quadrature channel a genuinely different generative channel than the derivative (Hilbert is a 90° all-pass; the derivative is a 90° shift times frequency), enough to seat its own axis? Or is “generate an orthogonal deterministic channel” one axis with two instances? The strongest sixth-axis test yet — to the full panel.