20-EFT.WP.Metrology.TimeBase v1.0 | Chapter 8 — Buffering, Resampling, and Time-Base Conversion

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Chapter 8 — Buffering, Resampling, and Time-Base Conversion

One-line objective: Using tau_mono as the kernel, construct an auditable buffering and resampling chain, perform time-base conversion for ts publication, and—while preserving causality and dimensional consistency—map the servo estimates of offset/skew/J into a reproducible time warping warp(t) with controllable distortion bounds.

I. Scope & Targets

1. Covered
   • Buffering & replay for data/event streams: buffer occupancy O(t), watermarks, overflow/underflow control.
   • Resampling & time-base conversion: fs_in → fs_out, including fixed ratios and variable-rate sampling (VRS).
   • Time warping warp(t): a monotone mapping driven by estimated offset/skew/J for tau_mono → ts.
   • Event timestamps & arrival compatibility: preserve the two T_arr formulations and delta_form.
2. Inputs
   Raw sequence x_in[k] @ fs_in or events (v_i, t_i); synchronization traces hat{offset}(t), hat{skew}(t), hat{J}(t); target publication ts and fs_out.
3. Outputs
   x_out[n] @ fs_out or transformed events (v'_j, t'_j); warp(t) and filter specification; manifest.time.conv.* with quality gauges.

II. Terms & Variables

• Rates & deviations
  fs_in, fs_out, r = fs_out / fs_in; relative frequency offset skew = ( fs_in - fs_ref ) / fs_ref.
• Buffering
  O(t): buffer occupancy (samples or seconds); H_lo/H_hi: low/high watermarks; arrival/consumption rates lambda_in/lambda_out.
• Time warping
  Linear approximation t_out = a + b * t_in with b = 1 + s_rel; general case t_out = warp(t_in), d warp / d t ≥ 0.
• Resampling kernels
  Band-limited kernel h(•); fractional delay d[n] = frac( n / r ); interpolation operators ZOH/FOH/Lagrange/sinc-windowed.
• Arrival & publication
  Dual-form T_arr, delta_form, mapping tau_mono → ts, and offset/skew/J.

III. Axioms P508- **

• P508-1 (Monotonicity & causality): warp(t) must be non-decreasing with d warp / d t ∈ [b_min, b_max]; no backward time or look-ahead sampling.
• P508-2 (Anti-alias & energy conservation): upsampling requires low-pass prefiltering; downsampling requires prior band-limiting; check_dim(expr)=pass.
• P508-3 (Buffer stability): steady-state E[ lambda_in ] ≈ E[ lambda_out ]; publish JB.overflow/underflow.
• P508-4 (Dual-form preservation): no time-base conversion may alter the two T_arr definitions; mirror them back and verify delta_form.
• P508-5 (Traceability): warp(t) must be built from the published hat{offset/skew/J}, include parameters and version, and be reconstructible via TraceID.
• P508-6 (Slip & insert/delete protection): detect sample slip; if correction is required, use time-scale modification (TSM), never raw sample drop/insert.
• P508-7 (Window consistency): compute statistics on tau_mono, publish on ts, and attach offset/skew/J.

IV. Minimal Equations S508- **

• S508-1 (Linear time-base mapping)
  t_out = a + b * t_in，where b = 1 + s_rel；s_rel = skew_src - skew_tgt。
• S508-2 (General time warping)
  t_out = warp(t_in) = a + ∫_{0}^{t_in} ( 1 + s_rel( u ) ) d u，with d warp / d t ≥ 0。
• S508-3 (Buffer occupancy dynamics)
  Discrete: O_{k+1} = O_k + A_k - C_k；continuous: d O / d t = lambda_in(t) - lambda_out(t)；steady-state E[A]=E[C]。
• S508-4 (Band-limited resampling)
  x_out[n] = ( ∑_{k} x_in[k] * h( n / r - k ) )，with low-pass h (e.g., windowed sinc), r = fs_out / fs_in。
• S508-5 (ZOH/FOH/Lagrange)
  ZOH: x_out[n] = x_in[ floor( n / r ) ]；
  FOH: x_out[n] = (1 - d[n]) * x_in[k] + d[n] * x_in[k+1]，k = floor( n / r )；
  Lagrange(p): coefficients are closed forms in d[n] for order p。
• S508-6 (Variable-rate resampling, VRS)
  x_out[n] = ( ∑_{k} x_in[k] * h( tau_out[n] - k ) )，tau_out[n] = warp( n / fs_out ) * fs_in。
• S508-7 (Anti-alias constraints)
  With signal bandwidth B_sig, require B_sig ≤ min( 0.5 * fs_in, 0.5 * fs_out ); before decimation, use h_lp(f_c) with f_c ≤ 0.5 * fs_out / r_margin。
• S508-8 (Jitter–buffer coupling)
  For input jitter J_rms, occupancy variance approximates Var[O] ≈ ( J_rms^2 ) * ( d lambda_in / d t )^2 (small-perturbation linear regime)。
• S508-9 (Timestamp conversion)
  Events t'_i = warp( t_i ); error propagation u( t'_i ) ≈ | d warp / d t | * u( t_i ) ⊕ u_warp。

V. Processing Flow M50-8 (Ready → Design → Implement → Verify → Persist)

• Readiness & checks
  Normalize fields & units; import hat{offset/skew/J}; ensure both T_arr forms and delta_form are available.
• Warp construction
  Build warp(t) from servo traces: linear or spline segments; constrain b ∈ [b_min, b_max]; publish version and parameters.
• Buffering & watermarks
  Set H_lo/H_hi, initial O(0); choose overflow/underflow policies (TSM speed-up/slow-down, backpressure, or ZOH degradation).
• Filters & kernels
  Design h (window, length, A_stop, group delay) to match r and B_sig; use polyphase or Farrow structures for VRS.
• Real-time execution
  Pull input fragments via tau_out[n] = warp( n / fs_out ) * fs_in; at boundaries use mirroring or masks m ∈ {0,1}.
• Integrity & quality
  Compute SNR_resamp, A_stop, group_delay_var, buffer overflow/underflow rates; bridge tdev/J_rms to resampling distortion.
• Persist & publish
  Write manifest.time.conv.*: warp, fs_in/out, kernel, O.*, quality metrics, and signature.

VI. Contracts & Assertions

• C50-81 (Monotone mapping): warp(t1) ≤ warp(t2) for any t1 ≤ t2; d warp / d t ∈ [b_min, b_max].
• C50-82 (Anti-alias suppression): A_stop ≥ A_stop_min and B_sig & sampling rates satisfy S508-7.
• C50-83 (Buffer stability): P( O(t) > H_hi ) ≤ p_overflow_max; P( O(t) < H_lo ) ≤ p_underflow_max.
• C50-84 (Distortion bounds): SNR_resamp ≥ SNR_min, group_delay_var ≤ gd_var_max.
• C50-85 (Dual-form consistency): delta_form_after = delta_form_before within tolerance tol_Tarr.
• C50-86 (Slip protection): Upon sample slip detection, only TSM-based correction is allowed; raw insert/delete violates the contract.
• C50-87 (Dimensional integrity): Units/dimensions are unchanged after resampling and conversion; check_dim(expr)=pass.

VII. Implementation Bindings I50-8*

• build_warp(sync_trace, policy) -> warp
• design_resampler(r | warp, B_sig, A_stop, L) -> kernel
• run_vrs(x_in, fs_in, fs_out, warp, kernel, masks) -> x_out
• convert_timestamps(ts_in, warp) -> ts_out
• jitter_buffer(H_lo, H_hi, policy) -> JB (supports push(chunk), pull(n), and reports O(t) with overflow/underflow events)
• detect_and_tsm_slip(stream, warp, budget) -> stream'
• audit_timebase_conv(x_in, x_out, metrics) -> report
• emit_conv_manifest(meta, warp, kernel, metrics) -> manifest.time.conv
• Invariants: unique(TraceID); warp is invertible & monotone; kernel matches fs_in/out; JB is stable; delta_form preserved.

VIII. Cross-References

• Synchronization estimates & servos: Chapters 5–6 (sources & updates for offset/skew/J).
• Phase/time noise & gauges: Chapter 7 (bridging tdev/J_rms to resampling distortion).
• Timestamp chains & latency budget: Chapter 4 (topology/link constraints on buffering policy).
• Cleaning & publication contracts: Methods.Cleaning v1.0 Chapters 5 & 10 (ts publication and manifest).
• Streaming & backpressure: Methods.Cleaning v1.0 Chapter 11 (channel stability and idempotent retries).

IX. Quality Gauges & Risk Control

1. Metrics
   lat_e2e_p99, JB.overflow/underflow_rate, SNR_resamp, A_stop, group_delay_mean/var, slip_count, psi(intervals).
2. Risk actions
   • Soft-degrade ladder: sinc → Lagrange → FOH → ZOH; if necessary, band-limit then decimate.
   • Fall back to linear warp and freeze parameters; enable backpressure or reduce fs_out; publish audit with TraceID.

Summary