22-EFT.WP.Metrology.Instrument v1.0 | Chapter 6 — Noise and Sensitivity (SNR / Noise Figure / NEP)

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Chapter 6 — Noise and Sensitivity (SNR / Noise Figure / NEP)

One-Sentence Objective
Unify noise conventions for electrical and photonic instruments; provide measurable definitions, conversions, and uncertainty publication for SNR, Noise Figure, and NEP; and bind results to an auditable manifest.

I. Scope and Objects

1. Scope
   • Applies to analog front-ends (AFE), RF/IF chains, ADC/DAC evaluation, and sensitivity assessment of photonic detectors (photodiode/bolometer).
   • Covers baseband and RF, DC and modulated measurements; dynamic noise (phase/time noise) is incorporated under the input-referred convention—see cross-references.
2. Objects
   Default input-referred reporting; provide output-referred values when necessary, explicitly indicating the crossed gain G and transfer function H(f).
3. Outputs
   SNR_lin, SNR_dB, F (noise figure, linear), NF_dB, NEP, B_enbw, e_n, i_n, T_e, report.u(*), and manifest.instrument.noise.*.

II. Terms and Variables

• Power & bandwidth: P_sig, P_n, B, B_enbw; filter transfer H(f).
• Noise spectral densities: voltage S_e(f) [V^2/Hz], current S_i(f) [A^2/Hz]; input noises e_n = sqrt(S_e), i_n = sqrt(S_i).
• Quantization & jitter: sigma_q^2 = ( LSB^2 ) / 12, sigma_t (sampling jitter).
• Noise temperature & constants: T_0 = 290 K, equivalent noise temperature T_e, Boltzmann constant k_B.
• Photonic response: R (responsivity, A/W or V/W), P_min (minimum detectable power).
• Timebase & units: tau_mono, ts, offset/skew/J; unit(x), dim(x), check_dim(expr).

III. Postulates P706-*

• P706-1 (Input-referred first): publish all noise metrics input-referred by default; any cross-referencing must declare G, R_s/R_L, and H(f).
• P706-2 (Equivalent noise bandwidth): always use B_enbw; nominal bandwidths are not acceptable substitutes.
• P706-3 (Window correction): FFT-based spectra must publish the window w[n] and the corresponding B_enbw.
• P706-4 (Dimensional integrity): enforce unit/dimension correctness for power, spectral densities, and NEP with check_dim(all) before publication.
• P706-5 (Integrated timebase): when jitter and phase noise are involved, compute timing statistics on tau_mono; publish on ts with offset/skew/J.
• P706-6 (Uncertainty publication): attach combined standard uncertainty u(x) and expanded uncertainty U = k * u_c to all core metrics.

IV. Minimal Equations S706-*

• S706-1 (SNR definition)
  SNR_lin = P_sig / P_n, SNR_dB = 10 * log10( SNR_lin ).
  P_n = ( ∫ S_out(f) df ) / G^2 (converted to input-referred), or P_n = k_B * T_0 * B (thermal-noise approximation under matching).
• S706-2 (Equivalent noise bandwidth)
  Continuous: B_enbw = ( ∫ |H(f)|^2 df ) / |H(f_ref)|^2 (with f_ref at unit gain in-band).
  Discrete window: B_enbw = ( Fs * ∑ w[n]^2 ) / ( ( ∑ w[n] )^2 ).
• S706-3 (Noise Figure and noise temperature)
  F = SNR_in / SNR_out, NF_dB = 10 * log10(F).
  F = 1 + ( T_e / T_0 ), equivalently T_e = (F - 1) * T_0.
  Cascades (Friis): F_total = F1 + (F2 - 1)/G1 + (F3 - 1)/(G1*G2) + ....
• S706-4 (Input-referred noise)
  e_n_rms^2 = ( ∫ S_e(f) * |H_in(f)|^2 df ), i_n_rms^2 = ( ∫ S_i(f) * |H_in(f)|^2 df ).
  With source impedance R_s, P_n = e_n_rms^2 / (4 * R_s) (matching).
• S706-5 (ADC quantization and jitter bounds)
  Quantization: SNR_q_dB ≈ 6.02 * N + 1.76 (full-scale sine), ENOB = ( SNR_dB - 1.76 ) / 6.02.
  Jitter upper bound: SNR_jitter_dB = -20 * log10( 2 * pi * f_in * sigma_t ) (see timebase cross-reference).
• S706-6 (NEP and minimum detectable power)
  NEP = i_n / R (current mode) or NEP = v_n / R (voltage mode, units W/sqrt(Hz)).
  P_min(SNR=1) = NEP * sqrt( B_enbw ).
  More generally SNR = ( P_sig / NEP ) * ( 1 / sqrt( B_enbw ) ).
• S706-7 (Input↔output reference transform)
  S_out(f) = |H(f)|^2 * S_in(f) + S_add(f); back-referenced input: S_in_ref(f) = ( S_out(f) - S_add(f) ) / |H(f)|^2.
• S706-8 (Uncertainty propagation — linearized)
  u^2( SNR_lin ) ≈ ( ∂SNR/∂P_sig )^2 * u^2(P_sig) + ( ∂SNR/∂P_n )^2 * u^2(P_n);
  for NEP: u^2( NEP ) ≈ ( ∂NEP/∂i_n )^2 * u^2(i_n) + ( ∂NEP/∂R )^2 * u^2(R ).

V. Metrology Procedure M70-6 (Spectrum → ENBW → Input-Ref → SNR/NF/NEP → Uncertainty → Persistence)

1. Readiness
   • Fix RefCond (temperature, supply, impedances, optical background) and select the measurement chain (hardware front-end/probes/windows).
   • Lock triggering on tau_mono; record offset/skew/J.
2. Noise spectrum measurement
   Choose w[n] and Fs/N_fft; measure S_out(f); compute B_enbw (S706-2) and apply window/scaling corrections.
3. Input-reference conversion
   Using H(f) and G, obtain S_in(f) and e_n, i_n; verify with unit/dim checks.
4. Metric computation
   • P_n = ( ∫ S_in(f) df ); measure or inject P_sig to compute SNR.
   • For cascaded devices, evaluate F and T_e; if needed, treat ADC quantization as an additional “stage”.
   • In photonics, compute NEP and P_min (S706-6).
5. Uncertainty
   Use Welch segment averaging or bootstrap to estimate u( SNR_dB ), u( NF_dB ), u( NEP ); publish U = k * u_c.
6. Compliance & rollback
   Enforce contracts C70-6*; on failure, rollback to the prior calibration or narrow the bandwidth (reduce B_enbw) and record rationale.
7. Persistence & signing
   Write manifest.instrument.noise.* (including B_enbw, w[n], H(f) digest, G, RefCond, u/U, signature, and version).

VI. Contracts & Assertions C70-6*

• C70-61 snr.min: SNR_dB ≥ snr_min_dB at the specified B_enbw and RefCond.
• C70-62 nf.max: NF_dB ≤ nf_max_dB or T_e ≤ T_e_max.
• C70-63 nep.max: NEP ≤ nep_max, and P_min ≤ pmin_max.
• C70-64 enbw.spec: | B_enbw - B_target | / B_target ≤ tol_enbw.
• C70-65 jitter.bound: SNR_jitter_dB ≥ snr_jitter_floor_dB (where applicable).
• C70-66 dim.check: check_dim(all)=true.
• C70-67 stability: cross-RefCond drift ΔSNR_dB ≤ drift_snr_max, ΔNEP ≤ drift_nep_max.

VII. Implementation Bindings I70-6* (Interface Prototypes)

• measure_noise_spectrum(stream, window, nfft) -> S_out(f), meta
• estimate_enbw(window, Fs, nfft) -> B_enbw
• refer_to_input(S_out, H, G) -> S_in, e_n, i_n
• compute_snr(Psig, S_in, B_enbw) -> SNR_lin, SNR_dB
• compute_noise_figure(SNR_in, SNR_out) -> F, NF_dB
• friis_cascade(F_list, G_list) -> F_total, NF_total_dB
• compute_nep(e_or_i_n, R, B_enbw) -> NEP, P_min
• jitter_snr_bound(f_in, sigma_t) -> SNR_jitter_dB
• uncertainty_welch(Sxx, segs) -> u_metric
• emit_noise_manifest(results, policy) -> manifest.instrument.noise
  Invariants: unit/dimension checks pass; B_enbw matches the window; H(f) versions are traceable; RefCond and timebase fields are recorded.

VIII. Cross-References

• Range, dynamic range, and ENOB: see Chapter 4 of this volume.
• Interference of linearity/residuals on noise estimation: see Chapter 5.
• Jitter and phase/time noise: see EFT.WP.Metrology.TimeBase v1.0, Chapters 6–7.
• Cleaning & release freeze; manifests & signatures: see EFT.WP.Methods.Cleaning v1.0, Chapter 10.
• Task-level mapping from imaging NEP to SNR: see EFT.WP.Methods.Imaging v1.0, Chapter 14 and Appendix D.

IX. Quality & Risk Control

• SLI/SLO
  snr_dB_p95, nf_dB_p95, nep_p95, pmin_p95, B_enbw_dev, jitter_floor_margin_dB.
• Drift & audit
  Track long-term ΔSNR_dB/Δt, ΔNEP/Δt with environmental RefCond; threshold violations auto-trigger re-measurement and version rollback.
• Rollback policy
  Prefer narrowing B_enbw, lowering noisy gain stages, switching to lower-noise devices, or enabling the previous H(f) calibration.

Summary
Through P706-* / S706-* / M70-6 / C70-6* / I70-6*, this chapter standardizes the measurement, conversion, and uncertainty publication of SNR/NF/NEP, ensuring that noise and sensitivity metrics are comparable, controllable, and auditable across instruments and modalities.