22-EFT.WP.Metrology.Instrument v1.0 | Chapter 7 — Bandwidth and Response (Bode / Step / PSF / ImpResp)

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Chapter 7 — Bandwidth and Response (Bode / Step / PSF / ImpResp)

One-Sentence Objective
Under a unified LTI convention, characterize and measure electrical/photonic instruments’ frequency response H(f), step response s(t), impulse response h(t), and spatial PSF(x,y); report bandwidth, group delay, and MTF-class metrics; and publish an auditable manifest.

I. Scope and Objects

1. Scope
   • Amplitude–phase and time-domain responses of analog/RF links, sampled measurement paths, and photonic imaging chains within the linear time-invariant operating neighborhood.
   • Modalities: Bode frequency response, step (Step), impulse (ImpResp), and spatial point-spread PSF with its frequency-domain OTF/MTF.
2. Objects
   Low-pass and band-pass systems; single- and multi-channel configurations (matrix H(f) ∈ ℂ^{m×n}) with sub-band metrics.
3. Outputs
   H(f), Mag_dB, phi(f), tau_g(f), f_3dB, B_enbw, t_r, t_s, OS%, h(t), PSF, MTF, MTF50, and manifest.instrument.response.*.

II. Terms and Variables

• Frequency response & phase: H(f), Mag = |H(f)|, Mag_dB = 20 * log10( |H(f)| ) (voltage ratio), phi(f) = arg H(f).
• Group delay: tau_g(f) = - d phi / d ( 2 * pi * f ).
• Impulse/step: h(t), s(t) = ( h * u )(t); for causal systems ds/dt = h(t).
• Bandwidth: f_3dB (3 dB down point of Mag_dB), B_enbw (equivalent noise bandwidth).
• Spatial response: PSF(x,y), OTF(ξ,η) = ( ∫∫ PSF(x,y) e^{-j 2 pi ( ξ x + η y )} dx dy ), MTF = |OTF|, MTF50 is the spatial frequency where MTF = 0.5.
• Units & dimensions: unit(H) = 1 or V/V (must be stated), unit(PSF)=1/area, unit(MTF)=1; check_dim(all).

III. Postulates P707-*

• P707-1 (Valid LTI region): measurements and publications are limited to the approximated LTI operating point and RefCond; otherwise co-publish evidence of mismatch and nonlinearity.
• P707-2 (Input-referred first): publish H(f) and derived metrics input-referred by default; cross-stage conversions must state G and the H(f) chain.
• P707-3 (Phase & unwrapping): publish continuously unwrapped phase and the accompanying tau_g(f).
• P707-4 (Bandwidth convention): publish both f_3dB and B_enbw; the analysis window and B_enbw must match.
• P707-5 (Integrated timebase): acquire step/impulse data on tau_mono; publish on ts and record offset/skew/J.
• P707-6 (Spatial–frequency cross-proof): imaging chains publish both PSF and MTF, mutually verified by discrete Fourier transforms within an error tolerance.
• P707-7 (Uncertainty): attach u(x) and U = k * u_c to core metrics; for spectra, provide variances via segment averaging or bootstrap.

IV. Minimal Equations S707-*

• S707-1 (Convolution & frequency response)
  y(t) = ( h * x )(t); H(f) = ( ∫ h(t) e^{-j 2 pi f t} dt ); Mag_dB = 20 * log10( |H(f)| ) (voltage ratio).
  For power ratios, use 10 * log10( • ); the manifest must state the ratio convention.
• S707-2 (Group delay)
  tau_g(f) = - d phi / d ( 2 * pi * f ); discrete estimate tau_g[k] ≈ - ( phi[k+1] - phi[k-1] ) / ( 4 * pi * Delta_f ).
• S707-3 (Bandwidth definitions)
  f_3dB = { f : Mag_dB(f) = Mag_dB(0) - 3 } (for low-pass) or the passband reference;
  B_enbw = ( ∫ |H(f)|^2 df ) / |H(f_ref)|^2; for a discrete window B_enbw = ( Fs * ∑ w[n]^2 ) / ( ( ∑ w[n] )^2 ).
• S707-4 (Step–impulse duality)
  s(t) = ( h * u )(t); for a causal, stable system h(t) = ds/dt.
  First-order low-pass heuristic: t_r(10–90%) * f_3dB ≈ 0.35.
• S707-5 (PSF/OTF/MTF)
  OTF(ξ,η) = F2{ PSF(x,y) }, MTF(ξ,η) = | OTF(ξ,η) |.
  If PSF(x,y) is near-Gaussian, so is MTF; FWHM_psf * f_MTF50 ≈ 0.44 (approximate, must be labeled).
• S707-6 (Deconvolution FRF estimate)
  H_est(f) = Y(f) / X(f); regularized form H_reg(f) = ( X*(f) Y(f) ) / ( |X(f)|^2 + lambda ).
• S707-7 (Uncertainty propagation — linearized)
  u^2( f_3dB ) ≈ ( ∂f/∂Mag )^2 * u^2(Mag ) + ( ∂f/∂phi )^2 * u^2(phi );
  u^2( MTF50 ) is synthesized from spectral interpolation and window-leakage estimation errors.

V. Metrology Procedure M70-7 (Sweep → FRF → Time → Spatial → Cross-Proof → Persistence)

• Readiness & calibration
  Fix RefCond (temperature, load, source/measurement impedance, illumination); calibrate a reference channel and transfer standard; lock tau_mono and offset/skew/J.
• Frequency response (Bode)
  Choose chirp/MLS/VNA sweep and window w[n]; acquire x(t), y(t); estimate H_est(f) with phase unwrapping; compute f_3dB, B_enbw, tau_g(f).
• Time-domain (Step/ImpResp)
  Inject a step or impulse; measure s(t) and h(t)=ds/dt; compute t_r, t_s, OS%; cross-validate with the inverse transform of H_est(f).
• Spatial response (PSF/MTF)
  Capture point/knife-edge/line targets to obtain PSF; denoise/register as needed; compute OTF/MTF and MTF50, anisotropy; compare with pixel-domain equivalent bandwidth from the electrical H(f).
• Cross-proof & uncertainty
  Verify FFT(h) ≈ H_est and d/dt s ≈ h; estimate u( f_3dB ), u( MTF50 ); record methods and coverage factor k.
• Compliance & rollback
  Enforce C70-7*; on failure, revert to the prior calibration or de-rate the passband; record disposition and impact.
• Persistence & signing
  Emit manifest.instrument.response.*: digest of H(f), f_3dB, B_enbw, tau_g, t_r/t_s/OS%, PSF/MTF/MTF50, RefCond, u/U, signature, and version.

VI. Contracts & Assertions C70-7*

• C70-71 f3dB.min: f_3dB ≥ f_min (or band-pass edges within tolerance).
• C70-72 passband.ripple: passband ripple ≤ ripple_max_dB.
• C70-73 stopband.attn: stopband attenuation ≥ attn_min_dB.
• C70-74 groupdelay.ripple: max(tau_g) - min(tau_g) ≤ tau_ripple_max over the passband.
• C70-75 rise.settle: t_r ≤ t_r_max, t_s ≤ t_s_max, OS% ≤ os_max.
• C70-76 mtf.floor: MTF50 ≥ mtf50_min, MTF@Nyquist ≥ mtf_nyq_min (imaging).
• C70-77 dim.check: check_dim(all)=true; window–bandwidth consistency |B_enbw - B_target|/B_target ≤ tol_enbw.
• C70-78 crossproof: || FFT(h) - H_est ||_2 / || H_est ||_2 ≤ tol_xproof.

VII. Implementation Bindings I70-7* (Interface Prototypes)

• measure_bode(x, y, method, window) -> {H(f), Mag_dB, phi, meta}
• unwrap_phase(phi) -> phi_cont
• estimate_group_delay(phi_cont, f) -> tau_g
• measure_step(trace) -> {s(t), t_r, t_s, overshoot}
• deconvolve_impulse(x, y, lambda) -> h(t)
• estimate_enbw(H, f, window) -> B_enbw
• extract_f3dB(Mag_dB, f, ref) -> f_3dB
• compute_psf(image, method) -> PSF, meta
• psf_to_mtf(PSF) -> {OTF, MTF, MTF50}
• crossproof_time_freq(h, H) -> tol_metric
• emit_response_manifest(results, policy) -> manifest.instrument.response
  Invariants: window matches B_enbw; RefCond explicit; unit/dim checks pass; tau_mono/ts fields complete.

VIII. Cross-References

• Noise and B_enbw: see Chapter 6 of this volume.
• Linearity impacts and compensation in frequency-response tests: see Chapter 5.
• Timebase and jitter errors in Bode/Step: see EFT.WP.Metrology.TimeBase v1.0, Chapters 6–8.
• Imaging PSF/MTF linkage to task-level metrics: see EFT.WP.Methods.Imaging v1.0, Chapters 5 and 14.
• Cleaning & release freeze; manifests & signatures: see EFT.WP.Methods.Cleaning v1.0, Chapter 10.

IX. Quality & Risk Control

• SLI/SLO
  f_3dB_p95, ripple_p95_dB, tau_g_ripple_p95, t_r_p95, MTF50_p95, xproof_err_p95.
• Drift & audit
  Track Δf_3dB/Δt, ΔMTF50/Δt with RefCond; threshold excursions trigger re-tests and calibration rollback.
• Rollback strategy
  De-rate passband, update H(f) calibration, switch to a redundant channel, or restore a prior response file; publish impact assessment and mitigation window.

Summary
Through P707-* / S707-* / M70-7 / C70-7* / I70-7*, this chapter unifies measurement and publication of Bode/Step/ImpResp/PSF/MTF, ensuring bandwidth and response for electrical and photonic chains are comparable, controllable, and traceable.