28-EFT.WP.Propagation.PathRedshift v1.0 | Chapter 9 — Signal Chain & Observation Conventions (PLL / CFO / Spectral-Line Fitting)

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Chapter 9 — Signal Chain & Observation Conventions (PLL / CFO / Spectral-Line Fitting)

One-sentence goal: Convert carrier/line observations into an engineering-ready z_meas(t) with quantified uncertainty; unify the three observation conventions—PLL tracking, CFO estimation, and spectral-line fitting; and pair the published results with T_arr^{form1/form2} and z_path, with explicit cross-checks and manifests.

I. Scope & Objects

1. Inputs
   • Observation stream: complex baseband y(t)=A(t)e^{jφ(t)}+n(t) (coherent I/Q), or amplitude spectrum S_y(f) (non-coherent / line spectra); sampling rate Fs, observation window W=[t_0,t_1].
   • Nominal frequency & reference: f_nom or f_emit, reference oscillator offset / skew / J and calibration coefficients (recorded in RefCond).
   • Link priors: z_pred(t) or components z_kin / z_grav / z_med / z_cos (from Chapters 3–6), plus path and two-form arrival-time information (Chs. 2/7/8).
2. Outputs
   • Observation convention: a z_meas(t) series (from one of PLL / CFO / line-fit, or their fusion), windowed statistics, and uncertainty u/U;
   • Companion mapping: consistent phase↔group mapping z_φ ↔ z_g (Ch. 7), and the paired T_arr^{form1/form2};
   • Manifest: manifest.redshift.obs.* with contract outcomes.
3. Boundary
   Default AWGN and narrowband engineering assumptions; when strong nonlinearity, strong dispersion, or significant multipath are present, use the extensions and strategy cards in Chapters 6 / 12.

II. Terms & Variables

• Phase & instantaneous frequency: φ(t), f_inst(t) = (1/2π) dφ/dt; nominal / emitted frequencies f_nom, f_emit.
• Frequency offset & redshift: Δf(t)=f_inst(t)-f_emit (or relative to f_nom), 1+z = f_emit/f_obs, z_meas ≈ -Δf/f_emit (small-signal).
• CFO / PLL parameters: loop bandwidth B_L, PLL equivalent noise N_L, CFO observation window T_obs.
• Line fitting: peak estimate f̂_line and model (Lorentz / Gauss / Voigt), resolution Δf≈1/T_obs.
• Dimensions: unit(f)=1/[T], unit(z)=1, unit(B_L)=1/[T].

III. Postulates P65-9x

• P65-901 (Two-convention pairing): any published z_meas must be labeled as phase or group convention and accompanied by the consistent mapping (Ch. 7) and ΔT_map; also publish T_arr^{form1/form2} with delta_form.
• P65-902 (Analytic vs observation in parallel): each publication window must provide z_pred and z_meas, recording the difference resid_z = z_meas - z_pred and delta_form_z = |resid_z|.
• P65-903 (Explicit measures): every estimator declares window and measure ( ∫_{t∈W} • dt ) or frequency domain ( ∫_{f∈B} • df ), together with windowing/leakage corrections.
• P65-904 (Dimensions & references): pass check_dim( y - f(x) ); persist f_emit/f_nom and reference oscillator offset/skew/J in RefCond with hashes.
• P65-905 (SNR / resolution guardrails): SNR and T_obs must satisfy estimator bounds (CRLB / analytic limits); if not, degrade / reject and trigger strategy cards.

IV. Minimal Equations S65-9x

1. PLL observation (phase convention)

• S65-901 (Instantaneous frequency):
  f_inst(t) = (1/2π) dφ/dt, with discrete implementations via differencing / phase unwrap;
  redshift: z_φ(t) ≈ - ( f_inst(t) - f_emit ) / f_emit.
• S65-902 (PLL noise & variance) (AWGN, small phase noise):
  var(f̂) ≳ (N_L / 8π^2) B_L (engineering approximation, or use literature CRLB).
• S65-903 (CRLB reference) (pure sinusoid, amplitude A, AWGN PSD N0/2, observation T_obs):
  var(f̂) ≥ 6 / ( (2π)^2 • SNR • T_obs^3 ), with SNR = A^2/(N0 Fs) (state convention and constants).

2. CFO / period estimation (phase convention)

• S65-904 (Cycle counting) (zero-crossing / integer cycles):
  Δf̂ = k / T_obs, z_φ ≈ -Δf̂/f_emit; quantization error σ_f,quant ≈ 1/T_obs.
• S65-905 (Periodogram / spectral centroid) :
  f̂ = argmax_f | ∫_W y(t) e^{-j2π f t} dt |^2, with window-leakage corrections persisted.

3. Spectral-line fitting (phase or group, per model)

• S65-906 (Peak estimate): fit S_y(f) with Lorentz/Gauss/Voigt to obtain f̂_line;
  redshift: z_line ≈ ( f_emit / f̂_line ) - 1; resolution Δf ≈ 1/T_obs; fitting uncertainty via Fisher/covariance.

4. Observation → group-convention mapping (weak dispersion, narrowband)

• S65-907: if z_meas is phase while publication is group, then
  z_g ≈ z_φ - (1/n_g) ( d n_g / d ln f ) • z_φ (see Ch. 7, S65-703), and publish u(ΔT_map).

5. Arrival-time pairing

• S65-908: within the window, group arrival time t̂_cont is obtained via correlation/centroid; co-publish T_arr^{form1/form2} and delta_form, and report
  ΔT_obs = |T_arr* - t̂_cont| (Ch. 8) with its uncertainty.

6. Fusion (optional)

• S65-909: for multiple observation conventions, form a weighted fusion
  z_meas = (∑ w_i z_i)/(∑ w_i), with w_i ∝ 1/u_i^2; persist u(z_meas) and per-convention weights.

V. Metrology Pipeline M65-9 (Ready → Estimate → Verify → Persist)

1. Ready: lock f_emit/f_nom and RefCond (oscillator/calibration, SNR gates, window function and T_obs); load z_pred and T_arr^{form1/form2}.
2. Estimate:
   • Choose a convention (PLL / CFO / LINE) to obtain z_i(t) and u_i(t);
   • Map phase→group if needed to get z_meas and u(z_meas);
   • Compute t̂_cont and ΔT_obs.
3. Verify:
   • check_dim(z) = 1; delta_form ≤ tol_Tarr;
   • Analyze resid_z = z_meas - z_pred with windowed quantiles;
   • Check SNR/CRLB sufficiency; persist leakage/window/offset residuals.
4. Persist:

manifest.redshift.obs = { method∈{ PLL, CFO, LINE, FUSED },

window:{ T_obs, win, window_fn },

refs:{ f_emit, f_nom, osc.hash },

z_meas, u/U, z_pred, resid_z,

mapping:{ φ↔g, ΔT_map, u(ΔT_map) },

Tarr:{ form1, form2, delta_form, ΔT_obs },

RefCond, contracts.*, signature }

VI. Contracts & Assertions C65-9x (suggested thresholds)

• C65-901 (Two-form gap): delta_form_p95 ≤ tol_Tarr; and ΔT_obs_p95 ≤ tol_align (per Ch. 8).
• C65-902 (Analytic vs observation): |resid_z|_p95 ≤ tol_z; violations trigger fallback (shorter windows / change convention / raise SNR).
• C65-903 (SNR / resolution): SNR ≥ SNR_min and T_obs ≥ T_min meeting CRLB; otherwise reject or degrade.
• C65-904 (Convention labeling & mapping): method, phase/group label, and ΔT_map / u(ΔT_map) must be persisted; unlabeled → reject.
• C65-905 (Dimensions & freshness): check_dim(z) = 1; panel/manifest update intervals ≤ Δt_panel_max.

VII. Implementation Bindings I65-9* (interfaces, I/O, invariants)

• I65-91 pll_track(iq, f_nom, cfg) -> { f_inst(t), z_phi(t), u, meta }
• I65-92 cfo_estimate(iq, T_obs, method) -> { Δf̂, z_phi, u, meta }
• I65-93 fit_spectral_line(S_y, model, init) -> { f̂_line, z_line, cov, meta }
• I65-94 build_z_series(z_list, weights) -> { z_meas, u, meta }
• I65-95 map_phase_to_group(z_phi, n_phi, n_g, band) -> { z_g, ΔT_map, u(ΔT_map) }
• I65-96 compare_pred_meas(z_pred, z_meas) -> { resid_z, pass }
• I65-97 assert_obs_contracts(ds, rules) -> { report, pass }
• I65-98 emit_obs_manifest(results, policy) -> manifest.redshift.obs
  Invariants: two_forms_present = true; check_dim(*) passes; reference frequency / oscillator / windowing / mapping method and applicable band are traceable.

VIII. Cross-References

• Redshift baseline & two forms: Chapter 2; kinematics / gravity / media: Chs. 3–5 (source of z_pred); phase↔group mapping: Chapter 7;
• Worldlines / path integrals & arrival time: Chapter 8; fusion & calibration: Chs. 10 / 11; uncertainty & publication: Chapter 13 and Appendices C / E.

IX. Quality & Risk Control

• SLI / SLO: resid_z_p95, delta_form_p95, ΔT_obs_p95, SNR_p95, panel_freshness.
• Fallback strategies: low SNR → extend T_obs / denoise; high leakage → change window / suppress sidelobes; convention inconsistency → dual-convention publication; persistent resid_z bias → back-write z_pred model or re-calibrate oscillator.
• Audit: PLL parameters and lock state, CFO/line-fit residuals, CRLB vs observed variance, and the manifest signature chain with replay scripts.

Summary

• This chapter unifies PLL / CFO / spectral-line fitting into a publishable z_meas, paired with phase/group and two-form arrival-time conventions.
• With M65-9 / C65-9x / I65-9* and manifest.redshift.obs, observations become measurable, auditable, and rollback-ready, supporting the fusion & calibration chapters (10–11) and the end-to-end composition of z_path.