27-EFT.WP.Packets.Light v1.0 | Chapter 6 — Propagation Impairments & Compensation (CD / PMD / Nonlinearity)

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Chapter 6 — Propagation Impairments & Compensation (CD / PMD / Nonlinearity)

One-sentence goal: Provide a unified modeling and compensation framework for chromatic dispersion (CD), polarization-mode dispersion (PMD), and nonlinearities (SPM/XPM/FWM) in optical links, with equivalent frequency-domain / time-domain realizations, residual metrics, and a publishable, auditable manifest of compensation and uncertainty.

I. Scope & Objects

1. Inputs
   • Path & media: segmented path gamma(ell) (fiber / fso / device); medium/device parameters beta(omega), beta2, D, alpha(f), nonlinearity gamma_nl, effective area A_eff, PMD statistics D_pmd or DGD samples.
   • Signals & metrology: TX/RX waveforms x(t) / y(t), spectra X(f) / Y(f), reference clocks tau_mono / ts, and RefCond.
   • Runtime policy: compensation mode and cascade order (pre-comp / in-line / post-comp), thresholds and tolerances.
2. Outputs
   • Compensation kernels & results: frequency-domain equivalent H_comp(f), time-domain equivalent h_eq(t), compensated waveform y_c(t) and residual metrics { CD_res, PMD_res, NL_res };
   • Two-form gap: frequency-domain (filtering/deconvolution) vs time-domain (equalization / DBP / Volterra) discrepancy delta_form_comp;
   • Uncertainty & manifest: u / U, manifest.packet.comp.*.
3. Boundary
   First-order engineering emphasis: weak nonlinearity and weak coupling (multi-channel XPM/FWM enter as equivalent noise/phase terms); FSO handled via equivalent group-delay / attenuation corrections.

II. Terms & Variables

• Dispersion & group delay: beta(omega), beta2, D = - ( 2*pi*c_ref / lambda^2 ) * beta2, group delay T_g = ( ∫ ( n_g / c_ref ) d ell ).
• PMD: DGD tau_DGD, statistics D_pmd (ps/√km), principal-state rotation R(ell).
• Nonlinearity: gamma_nl (1/W/km), A_eff (μm^2), SPM phase shift phi_SPM = gamma_nl * P * L_eff.
• Frequency-domain kernels & time-domain equivalents:
  H_CD(f) = exp( -j*pi*D*L * (f-f0)^2 ), H_PMDeq(f) (first-order PSP), H_loss(f) = exp( - ( ∫ alpha d ell ) ), total link H_link = H_CD * H_PMDeq * H_loss * …; time-domain equalizer h_eq, DBP operator B_DBP.
• Dimensional examples: unit(D) = "[s/m^2]" (state conversions from ps/(nm•km)), unit(alpha) = "[1/m]", unit(tau_DGD) = "[T]".

III. Postulates P606-*

• P606-1 (Two forms in parallel): Publish both frequency-domain (filtering/deconvolution) and time-domain (equalization/DBP/Volterra) compensation results, and record delta_form_comp = || y_f - y_t ||_2 / ||y_f||_2.
• P606-2 (Explicit measures): Every convolution/deconvolution/integral declares its domain and measure: ( ∫_{f∈B} • df ), ( ∫_{t∈W} • dt ), ( ∫_{gamma(ell)} • d ell ).
• P606-3 (Dimensional compliance): All physical quantities and kernel parameters entering equations must pass check_dim; any log↔linear conversions are disclosed in the manifest.
• P606-4 (Segment traceability): CD/PMD/loss parameters and compensation kernels are persisted per segment and aligned to path/device hashes.
• P606-5 (Weak-nonlinearity first with guardrails): Use linear-equivalent compensation while phi_SPM ≤ phi_max; beyond that, switch to DBP or reduce power, and annotate via contracts/panels.

IV. Minimal Equations S606-*

• CD frequency-domain compensation
  Link transfer (2nd order):
  H_CD(f) = exp( - j π D L * ( f - f0 )^2 )
  Inverse-phase compensation:
  H_comp,CD(f) = H_CD(f)^{-1} = exp( + j π D L * ( f - f0 )^2 )
  Time-domain FIR (approx.):
  h_eq,CD = IFFT( W(f) H_comp,CD(f) ), with windowing/band-limiting W(f).
• First-order PMD compensation
  Model:
  E_out(f) = R(ell) diag( e^{-j 2π f τ_1}, e^{-j 2π f τ_2} ) R(ell)^T E_in(f), with τ_2 − τ_1 = tau_DGD
  Frequency-domain compensation: estimate R̂, τ̂, then
  H_PMDeq(f) = R̂ diag( e^{+j 2π f τ̂_1}, e^{+j 2π f τ̂_2} ) R̂^T
  Time-domain MIMO equalization: y = W_eq * x, W_eq (2×2 LMS/RLS/CMA) minimizes ISI/polarization crosstalk.
• Nonlinearity (SPM/XPM/FWM)
  NLSE (weak-nonlinearity approximation):
  ∂A/∂z + (alpha/2)A + j (beta2/2) ∂^2 A/∂t^2 = j gamma_nl |A|^2 A
  Numerical compensation (single-channel DBP):
  Â(0,t) = B_DBP( A(L,t) ) ≈ ∏_{m=1}^M e^{-j gamma_nl |A|^2 Δz } e^{ -j (beta2/2) ∂^2/∂t^2 Δz } (reverse SSFM)
  Volterra equivalent (lower complexity):
  y(t) ≈ h1 * x + h3 * ( |x|^2 x ); identify h1, h3 for pre/post compensation.
• Loss & tilt
  Power: P_out = P_in * exp( - ∫ alpha(f) d ell ) * G(f)
  Tilt-equalization target: |H_link(f) * H_comp(f)| ≈ const in-band; error ε_tilt ≤ tilt_max.
• Residuals & penalties
  CD residual (baseband slope method): CD_res = | D_est - D_target | or pulse RMS broadening σ_out^2 - σ_in^2
  PMD residual: PMD_res = τ̂_DGD or MIMO cross-leakage XPol_leak
  Nonlinearity residual: NL_res = EVM_after - EVM_linear (or pre-/post-DBP delta).
• Two-form gap
  delta_form_comp = || y_f - y_t ||_2 / || y_f ||_2 (go-live per C60-6x).

V. Metrology Pipeline M60-6 (Ready → Estimate → Compensate → Verify → Persist)

• Ready: freeze path segments & device inventories (hash / id), RefCond; unify units (Hz, s, m, dB) and frequency grid.
• Parameter estimation:
  CD: sweep/phase fitting for D, beta2;
  PMD: frequency-domain Jones/PSP estimates R̂, τ̂_DGD;
  Nonlinearity: from P_tx, A_eff, gamma_nl plus EVM/phase rotation—or learn h3 from training.
• Compensation:
  Frequency-domain: generate H_comp(f) and filter;
  Time-domain: design h_eq or run DBP / Volterra / MIMO equalizers;
  record computational complexity and latency.
• Verification & uncertainty:
  Compute { CD_res, PMD_res, NL_res } and delta_form_comp;
  evaluate OSNR / Q / EVM improvement and tilt error ε_tilt;
  propagate uncertainty u / U (parameters / measurements / algorithmic approximations) and persist provenance.
• Persistence:
  manifest.packet.comp.* = { segments.hash, RefCond, H_comp.hash | h_eq.hash, method:{freq|time}, DBP/Volterra/MIMO.cfg, CD_res, PMD_res, NL_res, delta_form_comp, u/U, contracts.*, signature }.

VI. Contracts & Assertions C60-6x (Suggested Thresholds)

• C60-601 (Two-form consistency): delta_form_comp_p95 ≤ tol_comp (recommend tol_comp = 0.02 or modulation/EVM-target–based).
• C60-602 (CD residual): CD_res ≤ D_max_res or σ_out − σ_in ≤ σ_max_res.
• C60-603 (PMD residual): PMD_res ≤ PMD_max_res or XPol_leak ≤ XPol_max.
• C60-604 (Nonlinearity residual): NL_res ≤ NL_max_res or EVM_after ≤ EVM_target.
• C60-605 (Tilt & in-band flatness): ε_tilt ≤ tilt_max; power-equalization plan persisted.
• C60-606 (Dimensions & freshness): all compensation parameters and residuals pass check_dim; parameter/manifest freshness within configured windows.

VII. Implementation Bindings I60-6* (interfaces, I/O, invariants)

• I60-61 estimate_cd(y, X, grid) -> { D, beta2, fit_err }
• I60-62 compensate_cd_freq(X, D, L, window) -> Xc, H_comp.hash
• I60-63 equalize_cd_time(y, taps, method) -> yc, h_eq.hash
• I60-64 estimate_pmd(y, method) -> { R̂, τ̂_DGD, qual }
• I60-65 equalize_pmd_mimo(y, cfg) -> yc, report
• I60-66 dbp_ssfm(y, link, steps) -> yc, complexity
• I60-67 volterra_eq(y, order3_cfg) -> yc, h1h3.hash
• I60-68 compute_residuals(x, y, yc) -> { CD_res, PMD_res, NL_res }
• I60-69 compare_dual_comp(y_f, y_t) -> delta_form_comp
• I60-6A emit_comp_manifest(results, policy) -> manifest.packet.comp
  Invariants: two_forms_present = true; check_dim(*) passes; parameters path/device hashes are traceable; methods and RefCond persisted.

VIII. Cross-References

• Physical baselines & parameter definitions: Chapter 2.
• Framing & guard margins (CD/PMD contributions to T_guard): Chapter 3.
• Modulation/detection & EVM/BER metrics: Chapter 4.
• Arrival-time harmonization & path correction: Chapter 8.
• Metrology panels & OSNR / Q / EVM: Chapter 11.
• WDM/SDM labeling & power equalization: Chapter 5.

IX. Quality & Risk Control

• SLI / SLO: delta_form_comp_p95, CD_res, PMD_res, NL_res, ε_tilt, EVM_after, complexity / latency.
• Fallback strategies: high CD/PMD residuals → raise filter order / change estimators; strong nonlinearity → switch to DBP or reduce power; large tilt → recompute equalization; large two-form gap → unify to either time- or frequency-domain path and back-write thresholds.
• Audit: persist H_comp / h_eq versions, parameters & evidence, manifest signature chain, residual/performance curves, and replay consistency.

Summary

• This chapter unifies CD / PMD / nonlinearity modeling and compensation into a dual-form executable framework—frequency-domain filtering and time-domain equalization—together with residual and uncertainty publication.
• Via P606 / S606 / M60-6 / C60-6x / I60-6*, compensation outcomes are comparable, traceable, and rollback-ready, and are delivered in manifest.packet.comp.*.