11-EFT.WP.Core.DrawingKinetics v1.0 | Chapter 7 Spectral Energy and Jitter Modeling

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Chapter 7 Spectral Energy and Jitter Modeling

I. Scope and Objectives

• Establish unified gauges for the power spectral density of key observables during drawing (v(ts), T_fil(ts), s(ts), etc.), namely S_xx(f), window energy U_w, and equivalent noise bandwidth ENBW.
• Define the equivalence between time-domain jitter and frequency-domain integration, specify the metric family and publication fields, and support stability diagnostics (Chapter 6), control strategy design, and regression thresholds.

II. Terminology and Symbols

• Time and sampling: ts, Delta_ts, f_s = ( 1 / Delta_ts ), N, f_k = ( k * f_s / N ), f_N = ( f_s / 2 ).
• Windows and energy: w[n], U_w = ( 1 / N ) * sum_{n=0}^{N-1} ( w[n]^2 ), ENBW = f_s * ( sum w[n]^2 ) / ( sum w[n] )^2.
• Spectrum: X_w[k] = sum_{n=0}^{N-1} ( w[n] * x[n] ) * exp( - i * 2 * pi * k * n / N ), S_xx(f_k) = ( 1 / ( U_w * f_s ) ) * | X_w[k] |^2.
• Cross-spectrum and coherence: S_xy(f), H_{y|x}(f) = S_{yx}(f) / S_{xx}(f), C^2_{xy}(f) = | S_{xy}(f) |^2 / ( S_{xx}(f) * S_{yy}(f) ).
• Jitter and variance: band-limited mean square sigma_x^2( B ) = ( ∫_{B} S_xx(f) d f ), x ∈ { v, T_fil, s }.

III. Postulates and Minimal Equations

1. P11-6 (weak stationarity within a time-invariant window)
   Within the analysis window T_obs = N * Delta_ts, the second-order statistics of x(ts) are approximately time-invariant, and S_xx(f) characterizes the energy distribution.
2. P11-7 (time-base consistency)
   All spectral computations operate on the aligned ts satisfying ts = alpha + beta * tau_mono (see Chapter 8 time-base alignment), with beta > 0.
3. Frequency–time equivalence (S12-10, Parseval/Plancherel gauge)
   • sigma_x^2 = ( 1 / T_obs ) * ( ∫_{0}^{f_N} 2 * S_xx(f) d f ) (one-sided real-signal spectrum).
   • If x = s = ( d/dt ) ( ln( lambda ) ), then sigma_{ln lambda}^2( B ) = ( ∫_{B} ( S_{ss}(f) / ( 2 * pi * f )^2 ) d f ).
4. Tension–velocity cross-verification (S12-11)
   • Under linear small perturbations, S_{TT}(f) = | H_{T|v}(f) |^2 * S_{vv}(f), where H_{T|v}(f) is determined by constitutive laws and boundaries (Chapters 4 and 5).
   • Coherence constraint: when C^2_{Tv}(f) -> 1, tension jitter is predominantly driven by velocity perturbations.

IV. Data Gauges and Manifest

1. Input traces
   x(ts), x ∈ { v, T_fil, s }; sampling metadata f_s, N, Delta_ts; window type and parameters win.kind, win.param; detrending method detrend.
2. Publication fields (per window or segment)
   • spec.S_xx(f_k) (units unit(x)^2 / Hz), spec.f_k, spec.U_w, spec.ENBW.
   • Band-limited jitter: spec.rms.[band] = sqrt( ∫_{band} S_xx(f) d f ).
   • Peaks and dispersion: spec.peak.f, spec.peak.S, spec.tonal_ratio = spec.peak.S / mean_band( S_xx ).
   • Cross-spectra and coherence (optional): spec.H_{T|v}(f), spec.C2_Tv(f).
3. Dimensional checks
   • check_dim( S_xx ) == unit(x)^2 / Hz; check_dim( U_w ) == 1; check_dim( ENBW ) == Hz.
   • Tolerances: inherit conservation gates eps_norm and eps_mass from Chapter 3. The spectrum–time variance discrepancy must satisfy | sigma_time^2 - sigma_spec^2 | / sigma_time^2 <= eps_spec.

V. Algorithms and Implementation Bindings

1. I10-5 emit_metrics_drawing(state) -> dict (spectral and jitter metrics)
   • Time-base and preprocessing: map tau_mono to ts, apply detrend (mean or linear), choose window w[n].
   • Window energy and bandwidth: compute U_w = ( 1 / N ) * sum w[n]^2, ENBW = f_s * ( sum w[n]^2 ) / ( sum w[n] )^2.
   • FFT and calibration: X_w[k] = sum ( w[n] * x[n] * exp( - i * 2 * pi * k * n / N ) ); S_xx(f_k) = ( 1 / ( U_w * f_s ) ) * | X_w[k] |^2 (double the one-sided spectrum except for k=0,N/2).
   • Band-limited integration: for specified bands = { [0,f_LF], [f1,f2], [f_HF,f_N] } compute RMS values.
   • Optional cross-spectra: for x=v and y=T_fil, estimate S_{vv}, S_{TT}, S_{Tv}, output H_{T|v}, C^2_{Tv}.
   • Conservation and consistency checks: reconcile sigma_spec^2 with time-domain variance; for x=s, verify sigma_{ln lambda}^2 via frequency-domain integration matches the time-domain integral.
   • Return: spectral arrays, band-limited RMS, peak information, U_w, ENBW, optional coherence and transfer functions, and TS.spec.* metrics.
2. Recommended windows
   Low leakage: Hann; high resolution: Blackman-Harris; burst detection: Tukey(alpha). Always report the ENBW to account for effective bandwidth penalties.

VI. Metrology Workflow and Run Graph

• Synchronize: align ts and perform missing-data interpolation (Chapter 8).
• Preprocess: detrend and, where needed, de-dimension (normalize x to a reference scale).
• Choose window and overlap: win.kind, overlap; set N_seg and T_obs.
• Estimate spectra: use Welch or multi-segment averaging (variance reduction), retain U_w, ENBW.
• Cross-verification: using Chapter 6’s k_peak mapped frequency band, check whether spec.peak.f is consistent.
• Publish: write the manifest and figure artifacts; enforce gate.spec.* and the conservation gates.

VII. Verification and Test Matrix

1. Minimum required
   • White-noise input: verify flat S_xx(f) and correct amplitude scaling with ENBW.
   • Single-tone signal x = A * sin( 2 * pi * f0 * ts ): confirm the peak at f0, high tonal_ratio, and integrated variance recovers A^2 / 2.
   • Step–ramp mixture: low-frequency band-limited RMS dominates; spec.peak.f shifts lower.
   • Maxwell viscoelastic synthesis: compare S_{TT}(f) with | H_{T|v}(f) |^2 * S_{vv}(f) residuals.
   • Multi-segment averaging: as segment count increases, spectral variance converges as 1 / N_seg.
2. Edge and extreme cases
   Near-Nyquist narrowband noise, high-leakage short windows, resampling errors, saturation and quantization noise; for each, state expected bias direction and fallback (expand window, raise sampling rate, add pre-filtering).

VIII. Cross-References and Dependencies

• Chapter 2: s = ( d/dt ) ( ln( lambda ) ) and mapping along gamma(ell) define spectral targets.
• Chapter 4: source of constitutive T_fil and the transfer function H_{T|v}(f).
• Chapter 5: modulation of spectral peaks by boundary impedance and winding frequencies.
• Chapter 6: k_peak ↔ spec.peak.f cross-validation and localization of unstable bands.
• Core.Sea: time base; Core.Metrology: windows, bandwidth, and uncertainty; Core.Errors: costs of spectral gate false alarms/misses; Core.DataSpec: ingestion fields.

IX. Risks, Limits, and Open Questions

• Strong non-stationary bursts violate the weak-stationarity assumption; extend to time–frequency or wavelet energy densities when needed.
• Window leakage and edge effects can underestimate narrowband peaks; ENBW and frequency interpolation mitigate but impose an upper-bias bound.
• Constitutive nonlinearity makes H_{T|v}(f) amplitude-dependent; consider piecewise-linear or adaptive identification.
• Sampling jitter and time-base uncertainty introduce high-frequency spurs; calibration must be unified in Chapter 8.

X. Deliverables and Version Management

1. Artifacts
   • psd_{x}.npz (containing f_k, S_xx, U_w, ENBW), cpsd_{vT}.npz, spec_report.pdf (plots, band-limited RMS, peaks, and coherence).
   • Metric and gate files: TS.spec.*, gate.spec.rms_max, gate.spec.tonal_max, gate.spec.coherence_min.
2. Version strategy
   Changes to window families or calibration gauges are marked MOD; addition of cross-spectra/coherence outputs is marked ADD; maintain compat.spec.v1 to preserve historical comparability.