37-EFT.WP.EDX.HighSpeed v1.0 | Chapter 7 — Coherence Window & In-Band KPIs (S20-HF-5)

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Chapter 7 — Coherence Window & In-Band KPIs (S20-HF-5)

I. Chapter Objectives & Structure

1. Objective: Define the coherence window and in-band KPIs for high-speed scenarios; give selection/evaluation procedures, record fields, and release gates so S20-HF / S30-HF / S50-HF / I30-HF can be measured and accepted under a unified dialect.
2. Structure: Window definition & selection → KPI definitions → Gate tiers → Sampling & estimation → Records & QA → Falsifiability → Compliance templates → Cross-chapter closure.
3. Shared time-of-arrival dialect (equivalent; explicit gamma(ell) & d ell; record delta_form):
   • Constant-factored: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
   • General: T_arr = ( ∫ ( n_eff / c_ref ) d ell )

II. Definition & Selection of the Coherence Window

1. Definition: A coherence window Ω = [ω₁, ω₂] is a frequency span for a given device/configuration where linear-phase and stable group-delay hold and coherent summation is admissible; outside the window use energy composition.
2. Selection criteria (all must hold):
   • Linear-phase acceptable: E_phase(Ω) ≤ E_phase_gate;
   • Group-delay ripple acceptable: GDR(Ω) ≤ GDR_gate;
   • Propagation consistency: max_Ω | T_group − ( L·dβ/dω ) | ≤ u_Tgroup (if β is available);
   • Hard QA: check_dim=pass, passivity (Re{Z_eft} ≥ 0), KK_consistency=pass.
3. Disjoint/multi-window: If a single window fails, select segmented windows Ω₁, Ω₂, …; record and release per window.

III. In-Band KPI Definitions

• Phase linearity error
• E_phase(Ω) = max_{ω∈Ω} | arg Z_eft(ω) − ( ω·T_arr + φ0_opt ) |

with φ0_opt the optimal constant in-window; apply sync correction first: arg Z_corr = arg Z_raw − ( ω·Δt_sync ).

• Group-delay ripple
• T_group(ω) = d/dω( arg Z_eft(ω) ), GDR(Ω) = max_{ω∈Ω} | T_group(ω) − median_{Ω}(T_group) |
• Path/mode weight drift
• ΔW(Ω) = Σ_{p,m} | w_{p,m}(ω₂) − w_{p,m}(ω₁) |, 0 ≤ ΔW ≤ 1
• Radiation positive-real gate (if enabled)
• Re{ΔZ_rad(ω)} ≥ 0 (∀ ω∈Ω)
• Characteristic-impedance deviation (optional)
• ΔZ_c(Ω) = max_{ω∈Ω} | Z_c(ω) − Z_{c,ref} |

IV. Engineering Gates & Tiers

• Engineering (validation): E_phase ≤ 0.08 rad, GDR ≤ 0.25 ns, ΔW ≤ 0.30; if radiation channel enabled, Re{ΔZ_rad} ≥ 0.
• Release (benchmark): E_phase ≤ 0.05 rad, GDR ≤ 0.20 ns, ΔW ≤ 0.20; all hard QA gates passed.
• Gates may be specialized per product line in dataset cards, but hard QA must not be relaxed.

V. Sampling & Estimation (execution dialect)

• Sampling grid: use uniform (linear/log) sampling in a candidate band, N_ω ≥ 200 (typ. 1–2k); unwrap arg Z if needed.
• Phase correction: apply timebase correction Δt_sync before linear fits and KPI computation.
• Derivatives & smoothing: compute T_group = d(arg Z)/dω via three-point or Savitzky–Golay derivatives; avoid over-smoothing that biases KPIs.
• Consistency: if β(ω) is available, verify T_group ≈ L·dβ/dω; otherwise use T_arr together with E_phase/GDR.
• Out-of-window handling: use energy composition outside the window:
• R_inc(ω) = ( Σ_{p,m} w_{p,m}(ω) · | r_{p,m}(ω) |^2 )^{1/2}

VI. Records & QA (aligned with dataset cards)

• Required fields: band_GHz, coherence_window{w1,w2}, arrival{form,gamma,measure,c_ref,Tarr,u_Tarr,delta_form}, E_phase, GDR, T_group_s[], (optional) beta_per_m[], Zc_ohm[], and (if enabled) ΔZ_rad(ω) with Re_Zrad_min, ΔW, qa_gates{check_dim, passivity, KK}.
• Hard release gates: two-dialect T_arr agreement; Re{Z_eft} ≥ 0; KK_consistency = pass; if radiation enabled then Re{ΔZ_rad} ≥ 0.

VII. Falsifiability (chapter-specific)

• J-HF-7-1 (linear phase): If E_phase exceeds gate or residuals show systematic second-order curvature in any candidate window, reject the window or the linear-phase approximation.
• J-HF-7-2 (group-delay consistency): If max | T_group − L·dβ/dω | > u_Tgroup, reject the β estimate or the window.
• J-HF-7-3 (weight drift): If design perturbations are minor but ΔW exceeds gate, reject the path/mode stability assumption or binding.
• J-HF-7-4 (radiation gate): If Re{ΔZ_rad} < 0 after enabling the radiation channel, or if sealing does not reduce Re{ΔZ_rad}, reject the mapping or geometry records.

VIII. Compliance Templates (copy-ready)

• Record template
• highspeed:
• band_GHz: [f_min, f_max]
• coherence_window: {w1: ω1, w2: ω2}
• arrival:
• form: "n_over_c" # or "one_over_c_times_n"
• gamma: "explicit"
• measure: "d_ell"
• c_ref: 299792458.0
• Tarr_s: 1.234e-09
• u_Tarr_s: 6.0e-12
• delta_form: "n_over_c"
• kpis:
• E_phase_rad: 0.043
• GDR_s: 1.8e-10
• ΔW: 0.17
• dispersion:
• T_group_s: [ ... ] # over ω-grid within Ω
• beta_per_m: [ ... ] # optional
• Zc_ohm: [ ... ]
• radiation_gate:
• enabled: true
• Re_Zrad_min: 0.0
• KK_consistency: "pass"
• qa_gates: {check_dim:"pass", passivity:"pass", KK:"pass"}
• Computation flow (pseudocode)
• # Phase (after sync correction) on Ω = [ω1, ω2]
• phi = argZ_raw - omega*Δt_sync
• phi = unwrap(phi)
• # KPIs
• T_group = grad(phi, omega)
• E_phase = max_abs(phi - (omega*Tarr + phi0_opt))
• GDR = max_abs(T_group - median(T_group))
• # Optional β-consistency
• if beta_available:
• assert max_abs(T_group - L*grad(beta, omega)) <= u_Tgroup
• # Gates
• assert E_phase <= E_phase_gate and GDR <= GDR_gate
• assert min(Re(Z_eft)) >= 0.0 and KK_consistency(Z_eft)
• if radiation_enabled:
• assert min(Re(ΔZ_rad)) >= 0.0

IX. Cross-Chapter Links & Closure

• Dependencies: Chapter 2 (Terms & Symbols), Chapter 4 (Minimal Equations & Dispersion Relations), Chapter 5 (Radiation & Leakage Correction), Chapter 6 (Modes & Interconnect Primitives).
• Downstream: Chapter 12 (Layout & Process Rules — use KPIs for acceptance); Chapter 14 (SimStack & cases — use KPIs for regression); Chapter 16 (Design Protocol & Checklist — include KPI gates in sign-off).