S Channels¶

Three extraction channels convert domain signals into phase states.

P -- Physical Channel¶

Input: Continuous waveform (voltage, pressure, displacement).

Method: Hilbert transform. The analytic signal a(t) = s(t) + i*H[s(t)] yields instantaneous phase theta = arg(a) and amplitude |a|. Instantaneous frequency from the phase gradient.

Quality: SNR estimate: sig_power / (sig_power + noise_power), saturating at 1.

When to use: The signal is a real-valued time series with a dominant oscillatory component. EEG, vibration, AC current, heartbeat waveform.

Extractor: PhysicalExtractor in oscillators.physical.

I -- Informational Channel¶

Input: Event timestamps (sorted ascending). Spike trains, request arrivals, heartbeat R-peaks.

Method: Inter-event intervals yield instantaneous frequency f = 1/IEI. Phase via cumulative integral theta = cumsum(2*pi*f*dt), taken modulo 2*pi.

Quality: Inverse coefficient of variation of intervals: 1 / (1 + CV). Regular events score high; bursty or irregular events score low.

When to use: The observable is a point process, not a continuous signal. Packet arrivals, neuron spikes, heartbeat R-peaks, job completions.

Extractor: InformationalExtractor in oscillators.informational.

S -- Symbolic Channel¶

Input: Sequence of discrete state indices s in {0, 1, ..., N-1}.

Method: - Ring mode: theta = 2*pi*s/N. Maps discrete state to equally spaced phases on the unit circle. - Graph mode: normalise sequential position to [0, 2*pi). Frequency from phase differences between successive states.

Quality: Transition regularity. Single-step transitions score 1.0. Stalls (repeated state) score 0.2. Large jumps penalised proportionally to step/N.

When to use: The observable is a finite state machine, protocol state, or categorical variable that cycles. TCP states, workflow stages, Markov chain position, graph random walk.

Extractor: SymbolicExtractor in oscillators.symbolic.

Mixed-Channel Systems¶

A single domain can use all three channels. The oscillator_families dict in the binding spec maps family names to channels:

oscillator_families:
  heart_wave:
    channel: P
    extractor_type: physical
  spike_train:
    channel: I
    extractor_type: informational
  protocol_state:
    channel: S
    extractor_type: symbolic
    config:
      n_states: 8

All channels output PhaseState with the same fields. The UPDE engine does not distinguish channels -- it operates on the unified phase vector.

Quality Gating¶

Extracted phases with quality < min_quality (default 0.3) are down-weighted by PhaseQualityScorer.downweight_mask(). Collapsed oscillators (quality < 0.1 for majority of states) trigger detect_collapse().