29-EFT.WP.TBN.Measurement v1.0 | Chapter 5 | Phase Noise & the Allan Family

Home ／ Docs-Technical WhitePaper ／ 29-EFT.WP.TBN.Measurement v1.0

Chapter 5 | Phase Noise & the Allan Family

• One-line objective: Provide computable conventions, units, and measures for S_phi(f) ↔ S_y(f) ↔ sigma_y(τ) together with ADEV/MDEV/HDEV/TDEV; cover power-law noise identification, bias correction, and confidence intervals; and ensure consistency, auditability, and publishability with the dual-form T_arr and the manifest workflow.
• I. Scope & Objects
• Inputs
• Spectra & sequences: S_phi(f), S_y(f) (from Ch. 2); frequency-deviation sequence y(t_k) or time-error sequence x(t_k) (with x the phase-time).
• Windowing & sampling: W=[ts-Δt,ts], sampling interval τ_0, thresholds & bandwidth RBW/ENBW, window function w(t).
• Reference conditions: RefCond (f_ref/Fs/RBW/ENBW/window_fn, time tags tau_mono/ts, source hash/validity/coverage).
• Outputs
• Allan family: ADEV sigma_y(τ), Modified ADEV mod_sigma_y(τ), Hadamard HDEV(τ), Time Deviation TDEV(τ), and power-law coefficients h_α.
• Statistics & uncertainty: effective degrees of freedom ν_eff(τ), confidence U = k·u_c, bias/dead-time corrections.
• Manifest: manifest.tbn.allan.* (methods/measures/units/confidence & evidence URIs).
• Boundaries
• Default engineering regime: AWGN, stationary, overlapped (no dead time). Nonstationary or dead-time sampling must be explicitly flagged and corrected.
• II. Terms & Variables
• Frequency- & time-domain: S_phi(f) (unit="rad^2/Hz"), S_y(f) (unit="1/Hz"), y(t) = (f(t)-f_ref)/f_ref (dimensionless).
• Allan family: sigma_y(τ) (ADEV), mod_sigma_y(τ) (MDEV), HDEV(τ), TDEV(τ).
• Power-law noise: S_y(f) = ∑_{α=-2}^{+2} h_α f^{α} (white PM corresponds to S_φ ∝ f^0, but exponents differ in S_y; see below).
• Units: unit(sigma_y)=1, unit(TDEV) = "[T]"; unit(h_α) depends on α (recorded in the manifest).
• III. Postulates P505-*
• P505-1 (Explicit measures): Computations of ADEV/MDEV/HDEV/TDEV must declare the τ grid, overlap vs. non-overlap, dead-time vs. no-dead-time, weighting window, and the integration/summation convention ( ∫_{t∈W} · dt ) or the discrete equivalent.
• P505-2 (Unit consistency): Units/dimensions of S_phi/S_y/sigma_y/HDEV/TDEV must pass check_dim( y - f(x) ); log↔linear conversions and window/ENBW corrections are written to scale.note.
• P505-3 (Dual-form compatibility): When publishing Allan-family results, align to the same timebase/window W used for T_arr^{form1/form2}; if needed, record any method-dependent component that influences delta_form.
• P505-4 (Power-law recognition): The slope↔noise-type mapping inferred from sigma_y(τ) must be persisted (for gating & traceability), together with fitted h_α.
• P505-5 (Confidence & bias): ADEV/MDEV/HDEV/TDEV must include ν_eff(τ) and U=k·u_c; dead time or short-sequence biases must be corrected or clearly annotated.
• IV. Minimal Equations S505-*
• Core links among spectra, statistics, and time domain
• S505-1 (from Ch. 2): S_y(f) = ( f / 2π f_ref )^2 S_phi(f);
  ADEV in the frequency domain:
  sigma_y^2(τ) = 2 ∫_0^∞ S_y(f) |H_τ(f)|^2 df, with
  |H_τ(f)|^2 = sin^4(π f τ) / (π f τ)^2.
• S505-2 (phase/time jitter): σ_φ^2 = ∫ S_φ(f) df, σ_t = σ_φ / (2π f_ref) (bandwidth and leakage corrections are recorded in the manifest).
• Allan-family discrete formulas (overlapped, no dead time)
• ADEV:
  sigma_y^2(τ) = (1/(2(M-2m))) ∑_{k=1}^{M-2m} ( ȳ_{k+2m} - 2ȳ_{k+m} + ȳ_k )^2,
  where ȳ_k is the average over m=τ/τ_0 samples and M is the total sample count.
• MDEV (Modified Allan):
  mod_sigma_y^2(τ) = (1/(2 m^2 (M-3m+1))) ∑_{k=1}^{M-3m+1} ( ∑_{i=0}^{m-1} (ȳ_{k+2m+i} - 2ȳ_{k+m+i} + ȳ_{k+i}) )^2.
• HDEV (Hadamard; suppresses frequency drift):
  HDEV^2(τ) = (1/(6(N-3))) ∑_{i=1}^{N-3} ( x_{i+3} - 3x_{i+2} + 3x_{i+1} - x_i )^2 / τ^2 (with x the time error).
• TDEV:
  TDEV(τ) = τ · sqrt( MVAR(τ) / 3 ), where MVAR = mod_sigma_y^2(τ).
• Power-law noise types & sigma_y(τ) scalings (engineering approximations)
• S505-3 (noise type ↔ slope on log10 sigma_y vs. log10 τ):
• white PM → sigma_y(τ) ∝ τ^{-1}
• flicker PM → ∝ τ^{-1} (different constant; MDEV helps distinguish)
• white FM → ∝ τ^{-1/2}
• flicker FM → ∝ τ^{0}
• random-walk FM → ∝ τ^{+1/2}
  Fit multi-segment slopes to identify noise types and estimate h_α.
• Effective degrees of freedom & confidence intervals
• S505-4 (ν_eff, engineering approximations):
• ADEV (overlapped, no dead time): ν_eff(τ) ≈ 0.5 · (M / m).
• MDEV/HDEV use tabulated or simulation-based approximations (record source).
  Confidence: U(τ) = k · sigma_y(τ) / √ν_eff(τ), with k≈2 for ~95%.
• Dead-time sampling & bias correction
• S505-5: If dead time t_dead exists, use non-overlapped formulas or dead-time-removal weighting; apply a bias factor b(τ,t_dead) to correct:
  sigma_y,unbias(τ) ≈ b(τ,t_dead) · sigma_y,meas(τ) (document the source and constants in the manifest).
• Synthesizing Allan stats directly from S_phi/S_y
• S505-6 (numerical integration):
  sigma_y^2(τ) = 2 ∫_{f_L}^{f_H} S_y(f) |H_τ(f)|^2 df,
  with f_L,f_H set by the measurement bandwidth and window; apply ENBW correction Ŝ_y(f) = P(f)/ENBW.
• V. Metrology Flow M50-5 (Ready → Transform/Estimate → Identify → Check → Persist)
• Ready: Freeze RefCond and RBW/ENBW/window_fn; define a (log-uniform) τ grid and window W; select overlapped/no-dead-time conventions and bias-correction strategy.
• Transform/Estimate:
• From y(t), compute ADEV/MDEV/HDEV/TDEV using the discrete formulas (S505-2).
• Or integrate from S_y(f) (S505-6) and cross-check the results.
• Identify/Fit:
• Fit slopes across multiple τ segments to identify noise types and estimate h_α; report noise shares.
• Check:
• check_dim(sigma_y)=1, check_dim(TDEV)='[T]'; record dead-time/overlap conventions and ν_eff.
• Ensure Allan ↔ spectral/CRLB consistency (Chs. 2 & 4).
• Publish co-windowed with T_arr^{form1/form2}; if Allan processing contributes to delta_form, record its method-dependent component.
• Persist:
• manifest.tbn.allan = {method:{ADEV|MDEV|HDEV|TDEV}, window:{W,τ_grid,deadtime,overlap}, spectra:{RBW,ENBW,window_fn}, results:{sigma_y,mod_sigma_y,HDEV,TDEV}, ν_eff,k,U, noise_fit:{h_α,slopes}, refs:{S_y|y(t)}, RefCond, contracts.*, signature}.
• VI. Contracts & Assertions C50-5x (Suggested thresholds)
• C50-501 (Units/dimensions): check_dim(S_phi)="rad^2/Hz", check_dim(S_y)="1/Hz", check_dim(sigma_y)="1", check_dim(TDEV)='[T]'.
• C50-502 (Measure conventions): Overlap/no-dead-time, window/bandwidth, τ grid, and ENBW corrections must be recorded; missing annotations → reject.
• C50-503 (ν_eff & confidence): Provide ν_eff(τ) and U(τ) with coverage ≥ 95%; if ν_eff is too low (short sequences), downgrade or extend acquisition.
• C50-504 (Slope & noise type): Fitted slopes and noise-type classification must be physically plausible; if not, flag anomalies and trigger a strategy card.
• C50-505 (Dual-form alignment): Align with T_arr; if Allan statistics show method-dependent impact on delta_form, reflect it in the guardband.
• VII. Implementation Bindings I50-5* (Interface prototypes, I/O, invariants)
• I50-51 allan_dev(y_series, τ_grid, overlap, deadtime) -> {sigma_y, ν_eff, U, meta}
• I50-52 mod_allan_dev(y_series, τ_grid, overlap) -> {mod_sigma_y, ν_eff, U, meta}
• I50-53 hadamard_dev(x_series, τ_grid) -> {HDEV, ν_eff, U, meta}
• I50-54 tdev_from_mdev(mod_sigma_y, τ_grid) -> {TDEV, meta}
• I50-55 allan_from_Sy(S_y, τ_grid, RBW, ENBW, window_fn) -> {sigma_y, meta}
• I50-56 fit_powerlaw_noise(sigma_y, τ_grid) -> {h_α, slopes, model_fit}
• I50-57 assert_allan_contracts(results, rules) -> {report, pass}
• I50-58 emit_allan_manifest(results, policy) -> {uri, status}
• Invariants: two_forms_present=true; check_dim(*) passes; RefCond/hash and RBW/ENBW/window_fn are traceable; dead-time/overlap/ν_eff/confidence fields are mandatory.
• VIII. Cross-References
• Mathematical baseline & dual forms: Ch. 2.
• Acquisition & timebase: Ch. 3.
• Estimators: Ch. 4.
• Link & environmental corrections: Chs. 6–7.
• Analytics vs. anchors: Ch. 10.
• Uncertainty & release gates: Chs. 11–12.
• IX. Quality & Risk Control
• SLIs/SLOs: sigma_y(τ) quantiles & slope consistency, TDEV_p95, ν_eff coverage, RBW/ENBW/window compliance, panel_freshness.
• Fallback strategies: short sequences / large dead time → merge windows or extend acquisition; window/ENBW mismatch → recompute; anomalous noise-type fit → inspect front-end/reference and path corrections; dual-form coupling anomaly → unify on form2 and widen guardband.
• Audit: methods & parameters, τ grid and overlap/dead-time settings, ν_eff/U curves, noise-type fit evidence, manifest signature chain & replay scripts.
• Summary
• This chapter unifies phase-noise and Allan-family analysis into an executable convention—measures/units/formulas/confidence/bias and noise-type recognition—compatible with dual-form T_arr publication.
• Anchored by M50-5 / C50-5x / I50-5* and manifest.tbn.allan, it enables measurable, auditable, and rollback-ready metrology for Time-Base Backbone Networks.