29-EFT.WP.TBN.Measurement v1.0 | Chapter 4 | Phase & Frequency-Offset Estimation (PLL / IQ / FFT / Hilbert)

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Chapter 4 | Phase & Frequency-Offset Estimation (PLL / IQ / FFT / Hilbert)

• One-line objective: Normalize the bias/variance, measures, and units of three major estimator families—PLL tracking, IQ/Hilbert instantaneous phase, and FFT-based frequency estimation—in the TBN context; provide computable minimal equations and workflows that interoperate with S_phi/S_y/sigma_y and the dual-form T_arr, so that phase/frequency-offset estimates are auditable, replayable, and persistable.
  (All path integrals are taken along gamma(ell) with measure d ell.)
• I. Scope & Objects
• Inputs
• Acquisition & timebase: raw={samples, meta} (see Ch. 3), nominal/reference f_ref, analysis window W=[ts-Δt, ts] with window w(t), RBW/ENBW.
• Method selection: method ∈ {PLL, IQ, FFT, Hilbert} with configuration cfg (loop bandwidth/order/window type, etc.).
• Environment & path: RefCond (source/hash/validity), and the timebase coherence required by the two-form arrival-time integration (Chs. 2/3).
• Outputs
• Estimator sequences: phi(t), f_inst(t), y(t)=(f(t)-f_ref)/f_ref, frequency offset Δf̂, and uncertainties u(·).
• Spectral/statistical interface: directly convertible to S_phi/S_y/sigma_y (Ch. 2).
• Manifest: manifest.tbn.estimator.* (method/params/window/bandwidth/units/evidence URIs).
• Boundary
• Default assumptions: AWGN, narrowband, LTI; departures (non-Gaussian/nonstationary) are flagged via extended fields.
• II. Terms & Variables
• Signal model: x(t)=A(t) cos(2π f_c t + φ(t)) + n(t) or complex baseband z(t)=A(t) e^{jφ(t)} + n(t); f_c≈f_ref.
• PLL: loop-noise bandwidth B_L (unit="[1/T]"), phase-detector constant K_d, NCO constant K_0, closed-loop transfer H(s).
• FFT/CFO: observation length T_obs=Δt, resolution Δf≈1/T_obs, window sidelobe level L_sidelobe.
• Hilbert: analytic signal z(t)=x(t)+jH{ x(t) }, instantaneous phase φ(t)=arg z(t), instantaneous frequency f_inst=(1/2π)dφ/dt.
• Dimensions: unit(φ)="rad", unit(f_inst)="[1/T]", unit(y)=1, unit(B_L)="[1/T]".
• III. Postulates P504-*
• P504-1 (Explicit measures): Every estimate declares ( ∫_{t∈W} · dt ) and ( ∫_{f∈B} · df ); RBW/ENBW and the window are persisted.
• P504-2 (Unit consistency): All outputs must pass check_dim( y - f(x) ); dB↔linear and rad↔cycle conversions are written to scale.note.
• P504-3 (CRLB guardrails): Estimators must respect analytic lower bounds (CRLB/variance bounds). If not, downgrade/reject and trigger a strategy card.
• P504-4 (Dual-form compatibility): Estimator outputs must be window-aligned with T_arr^{form1/form2} (Ch. 2), and any method-dependent contribution to delta_form must be recorded (if applicable).
• P504-5 (Method annotation): method, window/order/bandwidth, de-leakage/de-bias steps are mandatory in the manifest; missing annotations → reject.
• IV. Minimal Equations S504-*
• Shared time-domain relations & statistics
• S504-1: f_inst(t) = (1/2π) dφ(t)/dt, y(t) = ( f_inst(t) - f_ref ) / f_ref.
• S504-2: S_y(f) = ( f / 2π f_ref )^2 S_phi(f); sigma_y^2(τ) = 2 ∫_0^∞ S_y(f) |H_τ(f)|^2 df (see Ch. 2).
• PLL estimation (phase-domain tracking)
• S504-3 (phase output, engineering approximation):
  S_phi,PLL(f) ≈ |H(j2π f)|^2 S_phi,in(f) + S_add(f), where S_add aggregates detector/quantization/jitter terms.
• S504-4 (frequency variance under AWGN):
  var( f̂ ) ≳ (N_L / 8π^2) B_L (equivalent-bandwidth approximation; record model and constants in the manifest).
• S504-5 (2nd-order loop): H(s)=ω_n^2/(s^2+2ζω_n s+ω_n^2), B_L≈ω_n/2π·g(ζ); publish ω_n, ζ, and B_L.
• IQ/Hilbert (phase-domain instantaneous)
• S504-6 (analytic signal): z(t)=x(t)+jH{ x(t) }, φ(t)=unwrap(arg z(t)), f_inst=(1/2π)dφ/dt.
• S504-7 (bias remedies):
• amplitude/DC removal: x'(t)=x(t)-mean(x(t));
• phase unwrapping: unwrap;
• frequency smoothing: f_inst = LPF( (1/2π)diff(φ) , B_smooth).
• S504-8 (variance bound; AWGN analytic approximation):
  var(f̂_inst) ≥ 6 / ( (2π)^2 · SNR · T_obs^3 ) (consistent with the CRLB; record window/bandwidth).
• FFT/CFO (spectral, batch)
• S504-9 (periodogram peak):
  f̂ = argmax_f | ∫_W x(t) e^{-j2π f t} dt |^2; Δf≈1/T_obs.
• S504-10 (high-resolution refinements; Kay/Quinn–Macleod/Jacobsen):
  with peak-bin index k̂ and neighbor bins X[k̂±1],
  δ ≈ (|X[k̂+1]| - |X[k̂-1]|) / (2|X[k̂]| - |X[k̂+1]| - |X[k̂-1]|);
  f̂ ≈ (k̂ + δ) Fs/N_fft (window-dependent correction constants are persisted).
• S504-11 (window/leakage correction): Ŝ(f) = P(f)/ENBW; sidelobe leakage L_sidelobe contributes to u(f̂).
• Bias & drift (de-bias / de-trend)
• S504-12: de-bias φ'(t)=φ(t)-⟨φ⟩_W; de-trend φ''(t)=φ'(t)-polyfit_1(t); linearization errors are mapped into y(t).
• Estimator → statistics/spectra interface
• S504-13: pipeline from f_inst(t)/y(t) to S_phi/S_y/sigma_y:
  S_y(f) ← PSD{y(t), RBW, ENBW, w(t)}; S_phi(f) ← (2π f_ref/f)^2 S_y(f); sigma_y(τ) ← Allan{y(t)}.
• V. Metrology Flow M50-4 (Ready → Estimate → Check → Persist)
• Ready: bind RefCond; choose method and parameters {B_L | window_fn, N_fft | B_smooth}; set W, RBW/ENBW.
• Estimate:
• PLL: lock and produce φ_PLL(t), f̂_PLL(t).
• IQ/Hilbert: φ_H(t), f̂_H(t).
• FFT/CFO: batch Δf̂_FFT or sliding-window offsets.
• Normalize to y(t) and derive S_phi/S_y/sigma_y.
• Check:
• check_dim(*); compare empirical variances to CRLB; enforce consistency of window/leakage/bandwidth.
• Window-align with T_arr^{form1/2} (Ch. 2) and record any method-coupled error component affecting delta_form.
• Persist:
• manifest.tbn.estimator = {method, cfg, window:{Δt,w,RBW,ENBW}, outputs:{phi,f_inst,y}, spectra:{S_phi,S_y}, stats:{sigma_y}, crlb:{var_min}, residuals:{var_emp}, twoform_ref:{ts_sync, link}, RefCond, contracts.*, signature}.
• VI. Contracts & Assertions C50-4x (Suggested thresholds)
• C50-401 (CRLB/SNR/window length): var_emp(f̂) ≤ α · CRLB(f̂) (e.g., α≤3); SNR ≥ SNR_min, T_obs ≥ T_min.
• C50-402 (Window/bandwidth compliance): persisted RBW/ENBW matches w(t); L_sidelobe within threshold.
• C50-403 (Units/dimensions): check_dim(φ)="rad", check_dim(f_inst)="[1/T]", check_dim(y)=1.
• C50-404 (Dual-form compatibility): estimator sequences aligned with the two-form T_arr and method-related error components recorded.
• C50-405 (Method annotation): method/cfg/window/leakage correction/high-res interpolation fields are mandatory.
• VII. Implementation Bindings I50-4* (Interface prototypes, I/O, invariants)
• I50-41 pll_estimate(iq, f_ref, cfg:{ω_n,ζ,B_L}) -> {phi, f_inst, y, crlb, meta}
• I50-42 hilbert_instantaneous(x, w, B_smooth) -> {phi, f_inst, y, crlb, meta}
• I50-43 fft_cfo_estimate(x, window_fn, N_fft) -> {Δf̂, var, meta} (optional Kay/Quinn–Macleod/Jacobsen post-processing)
• I50-44 spectra_from_y(y, w, RBW, ENBW) -> {S_phi, S_y, sigma_y, meta}
• I50-45 assert_estimator_contracts(outputs, rules) -> {report, pass}
• I50-46 emit_estimator_manifest(results, policy) -> {uri, status}
• Invariants: two_forms_present=true; check_dim(*) passes; window/bandwidth/params and reference f_ref, RefCond are traceable.
• VIII. Cross-References
• Mathematical baseline & dual forms: Ch. 2.
• Acquisition & timebase: Ch. 3.
• Link & environmental corrections: Chs. 6–7.
• Instrument/loopback: Ch. 8.
• Analytics vs. observation vs. anchors: Ch. 10.
• Uncertainty & release gates: Chs. 11–12.
• IX. Quality & Risk Control
• SLIs/SLOs: var_emp/CRLB ratio, S_phi/S_y threshold compliance, sigma_y(τ) quantiles, ΔT_obs_p95 window-alignment rate, panel_freshness.
• Fallback strategies: low SNR or high leakage → extend window/change window/increase averaging; unstable PLL → narrow loop bandwidth or switch to Hilbert; unstable FFT coarse estimate → enable high-res interpolation or switch to PLL; poor dual-form alignment → unify on form2 and widen guardband.
• Audit: method/params/window/bandwidth, CRLB vs empirical variance, leakage and corrections, manifest signature chain and replay scripts.
• Summary
• This chapter unifies PLL/IQ/FFT/Hilbert under a single executable framework of measures–units–bounds–persistence, and codifies the mapping from phase/frequency-offset estimation to S_phi/S_y/sigma_y with dual-form T_arr compatibility.
• With M50-4 / C50-4x / I50-4* and manifest.tbn.estimator as anchors, the estimation stage becomes measurable, auditable, and rollback-ready—providing stable inputs for noise statistics (Ch. 5) and link metrology (Chs. 6–7).