22-EFT.WP.Metrology.Instrument v1.0 | Chapter 3 — Instrument Classes & Signal Chains (AFE/ADC/DAC/RF/Photonic)

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Chapter 3 — Instrument Classes & Signal Chains (AFE/ADC/DAC/RF/Photonic)

One-Sentence Objective
Establish unified signal-chain conventions, noise and linearity budgets, and coherent arrival-time/timebase handling across AFE/ADC/DAC/RF/photonic instruments, providing a computable baseline for subsequent calibration and compliant publication.

I. Scope and Objects

1. Scope
   • End-to-end chain modeling and metrology from source signal through front-end AFE, RF frequency conversion, ADC/DAC sampling/conversion, photonic transduction and readout (PD/TIA).
   • Harmonized expressions for band-pass/low-pass chains, super-heterodyne and zero-IF architectures, and time-/frequency-domain instruments.
2. Objects
   Signal-chain transfer function H_chain(f) = H_src(f) * H_afe(f) * H_rf(f) * H_bb(f) with equivalent noise from quantization/jitter, and the two arrival-time forms T_arr.
3. Outputs
   Total gain G_tot, noise figure NF_tot, dynamic range DR_dB, ENOB, SFDR, linearity (IIP3/OIP3), and publication-grade uncertainty U = k * u_c.

II. Terms and Variables

• Front-end & conversion: AFE, ADC, DAC, PD (photodiode), TIA, Mixer, LO, IF, IQ.
• Frequencies & bandwidths: f_s, B_sig, RBW, VBW, ENBW.
• Gain & noise: G_k (linear gain), NF_k, S_n(f), Vn_rms, i_shot, i_thermal.
• Linearity & dynamic range: IIP3, OIP3, P1dB, SFDR, DR_dB, CrestFactor.
• Quantization & jitter: N (bits), LSB, ENOB, sigma_t (jitter), SNR_j.
• Photonics & path: P_opt, R (responsivity), I_sig = R * P_opt, G_TIA; path gamma(ell) and n_eff.
• Dimensions & units: unit(x), dim(x), check_dim(expr); arrival-form difference delta_form.

III. Postulates P703-*

• P703-1 (Chain decomposability): within the linear operating region, the chain is approximated as cascaded linear blocks plus additive noise, H_chain(f) = ∏ H_k(f).
• P703-2 (Friis convention): compute noise figure by Friis’ formula with linear gains; NF_tot is dominated by the first stage.
• P703-3 (Anti-alias first): the sampling chain must include H_aa(f) such that B_sig ≤ 0.45 * f_s, with image and leakage kept within thresholds.
• P703-4 (Integrated timebase): windowing and spectral estimation on tau_mono; publication on ts with offset/skew/J recorded.
• P703-5 (Parallel arrival-time forms): when a transmission medium is involved, compute both T_arr forms in parallel and persist delta_form.
• P703-6 (Dimensional conservation): check_dim( y - f(x) ) = 0 must pass; unit(H_k) = 1.
• P703-7 (Declared linear region): the publication scope is valid only within P1dB and thermally stable RefCond; beyond that, provide derating or nonlinear correction.

IV. Minimal Equations S703-*

• S703-1 (Chain gain):
  G_tot = ∏_k G_k, Y_rms = G_tot * X_rms (linear region).
• S703-2 (Friis noise figure):
  NF_tot = NF_1 + ( NF_2 - 1 ) / G_1 + ( NF_3 - 1 ) / ( G_1 * G_2 ) + ... (linear quantities, not dB).
• S703-3 (Quantization & ENOB):
  LSB = FS / 2^N, sigma_q^2 = ( LSB^2 ) / 12, ENOB = ( SNR_total_dB - 1.76 ) / 6.02.
• S703-4 (Jitter-limited SNR bound):
  SNR_j = - 20 * log10( 2 * pi * f_in * sigma_t ).
• S703-5 (SFDR approximation):
  SFDR ≈ min( SNR_total_dB, 2/3 * ( IIP3_dBm - P_in_dBm ) ) (baseband-equivalent).
• S703-6 (Mixing & image frequency):
  f_if = | f_rf - f_lo |, image at f_im = f_lo ± f_if, image rejection IRR = |H_rf(f_im)| / |H_rf(f_rf)|.
• S703-7 (Photonic readout):
  I_sig = R * P_opt, V_out = G_TIA * I_sig; noise i_shot^2 = 2 * q * I_dc * B, i_thermal^2 = 4 * k_B * T / R_f * B, Vn_rms^2 = ( G_TIA^2 ) * ( i_shot^2 + i_thermal^2 ).
• S703-8 (Anti-alias margin):
  margin_aa = f_s/2 - B_sig, require margin_aa ≥ margin_min.
• S703-9 (Difference between two arrival-time forms):
  delta_form = | ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell ) - ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell ) |.
• S703-10 (Dimensional checks):
  unit(G_k)=1, unit(NF_k)=1, unit(Vn_rms) = unit(Y_rms), check_dim(all)=true.

V. Metrology Procedure M70-3 (Signal-Chain Budget → Verification → Persistence)

• Readiness
  Bind operating mode and RefCond; complete time_align(tau_mono); finalize FS, f_s, B_sig, and the cascade topology.
• Chain budgeting
  Compute G_tot, NF_tot (S703-1/2); estimate Vn_rms, DR_dB, SFDR.
• Frequency planning
  Select f_lo and H_rf(f) to meet IRR and image/leakage suppression; evaluate margin_aa.
• Photonic & front-end
  From P_opt, R, G_TIA, compute V_out and noise terms (S703-7); check linear region and headroom to saturation.
• Sampling & timebase
  Evaluate sigma_t and SNR_j; synthesize SNR_total_dB and ENOB; record offset/skew/J.
• Arrival time
  For gamma(ell), compute both T_arr forms in parallel and verify delta_form.
• Verification & persistence
  Enforce contracts C70-3*; generate manifest.instrument.chain.* and signature.

VI. Contracts & Assertions C70-3*

• C70-31 gain.window: G_tot within [G_min, G_max].
• C70-32 nf.friis: NF_tot ≤ NF_spec_max (compare as linear quantity).
• C70-33 aa.margin: margin_aa ≥ margin_min, and |H_aa(f)| meets stop-band suppression A_stop ≥ A_min.
• C70-34 jitter.snr: SNR_j ≥ snr_min_jitter.
• C70-35 enob.floor: ENOB ≥ enob_min; DR_dB ≥ dr_min.
• C70-36 linearity.ip3: IIP3_dBm ≥ iip3_min and P_in_dBm ≤ P1dB_margin.
• C70-37 image.reject: IRR_dB ≥ irr_min.
• C70-38 photonic.noise: Vn_rms ≤ Vn_spec_max; I_sig below the TIA linear-range limit.
• C70-39 arrival.forms: delta_form ≤ tol_Tarr.
• C70-3A dim.check: check_dim(all)=true.

VII. Implementation Bindings I70-3* (Interface Prototypes)

• compute_chain_gain(blocks) -> G_tot, Y_rms
• friis_noise(blocks) -> NF_tot, Vn_rms
• cascade_ip3(blocks, Pin) -> IIP3_dBm, SFDR
• plan_lo_and_filters(spec) -> f_lo, IRR_dB, margin_aa
• photodiode_tia_model(P_opt, R, G_TIA, R_f, T, B) -> V_out, Vn_rms
• adc_budget(fs, FS, N, sigma_t, CrestFactor) -> ENOB, SNR_total_dB, DR_dB
• compute_arrival_two_forms(gamma, n_eff, c_ref) -> T_arr_1, T_arr_2, delta_form
• assert_chain_contracts(metrics, rules) -> report
  Invariants: check_dim(all)=true; delta_form ≤ tol_Tarr; idempotent interfaces; TraceID and RefCond span the full chain.

VIII. Cross-References

• Timebase & jitter estimation: see EFT.WP.Metrology.TimeBase v1.0, Chapters 6–7.
• Synchronization & delay asymmetry: see EFT.WP.Metrology.Sync v1.0, Chapters 7–8.
• Imaging photonic chain & OTF: see EFT.WP.Methods.Imaging v1.0, Chapters 5 and 8.
• Cleaning & release freeze: see EFT.WP.Methods.Cleaning v1.0, Chapter 10.

IX. Quality & Risk Control

• SLI/SLO recommendations
  contract_violation_rate, calibration_age_days, alias_event_rate, ENOB_p5, SFDR_p5, latency_ms_p99.
• Drift & rollback
  If NF_tot, ENOB, IRR_dB, or delta_form exceeds limits, trigger down-sampling/increased suppression/re-calibration or switch to redundant chains, preserving DR_dB and P1dB_margin.

Summary
This chapter presents unified signal-chain equations S703-*, postulates P703-*, workflow M70-3, contracts C70-3*, and interfaces I70-3* across AFE/ADC/DAC/RF/photonic systems. With consistent dimensions, timebase, and arrival-time handling, it furnishes an executable baseline for calibration, operation, and compliant publication.