08-EFT.WP.Core.Sea v1.0 | Chapter 4 — Signal Conditioning and Anti-Aliasing

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Chapter 4 — Signal Conditioning and Anti-Aliasing

I. Objectives and Scope

• Establish a conditioning-chain model H(f) from sensor output to ADC input, with clear definitions for f_c / BW / DR / ENOB and an explicit anti-aliasing budget.
• Provide postulates P84-, minimal equations S84-, and the design workflow Mx-4 for conditioning and anti-aliasing; align with Chapter 2 (Sampling & Quantization), Chapter 3 (Synchronization & Time Bases), and Chapter 8 (T_arr) for cross-chapter consistency.

II. Core Entities and Notation

• Conditioning chain and filters
  H_aa(f): analog anti-alias filter response; H_d(f): digital shaping/compensation response; composite H(f) = H_aa(f) * H_d(f).
  f_c: low-pass cutoff; BW: passband bandwidth; Delta_f_tr: transition bandwidth; A_s: stopband attenuation (linear amplitude or dB).
  Group delay tau_g(f) = - d arg H(f) / d ( 2 * pi * f ).
• Aliasing and folding
  Sampling rate fs, Nyquist f_N = fs / 2.
  Folding map f_alias = | f - k * fs |, with k ∈ Z such that f_alias ∈ [0, f_N].
• Dynamic range and gain staging
  FS: ADC full-scale peak; x_rms: input RMS; CF: crest factor; DR: dynamic range; ENOB: effective number of bits.
  Gain stages: A_gain (programmable gain), B_bias (bias), C_offset (zero shift).

III. Postulates P84- (Conditioning and Anti-Aliasing)*

• P84-1 Sampling-theorem postulate
  With target band upper limit f_max, enforce fs >= 2 * f_max. If fs < 2 * f_max, you must specify A_s/Delta_f_tr for H_aa(f) and provide a residual aliasing energy estimate.
• P84-2 In-band fidelity postulate
  Passband ripple delta_p and distortion from phase nonlinearity must be upper-bounded within BW. For arrival-time accuracy, prefer linear-phase FIR.
• P84-3 Budget closure postulate
  Aliasing, quantization, and jitter noise energies add in power and must satisfy
  SNR_total^{-1} ≈ SNR_alias^{-1} + SNR_q^{-1} + SNR_jitter^{-1} (sum in linear-power domain).
• P84-4 Gain and headroom postulate
  Any gain configuration must declare peak headroom M_headroom(dB), ensuring x_pk <= FS / 10^( M_headroom / 20 ). Default M_headroom >= 6 dB.
• P84-5 Group-delay uniformity postulate
  For time-sensitive measurements, require max_{f ∈ BW} | tau_g(f) - tau_g(f_c) | <= epsilon_tau, and record epsilon_tau in the manifest.

IV. Minimal Equations S84- (Estimates and Bounds)*

• S84-1 Residual aliasing energy bound
  With input PSD S_xx(f) and analog anti-alias filter H_aa(f), the residual aliasing power is approximated by
  E_alias ≈ ∑_{k≠0} ∫_{B_k} | H_aa(f) |^2 * S_xx(f) d f, where B_k = [ k*fs - f_N , k*fs + f_N ] \cap R^+.
  If the stopband PSD is bounded by S_sb_max and the minimum stopband attenuation satisfies |H_aa(f)| <= a_s, then
  E_alias <= a_s^2 * S_sb_max * ∑_{k≠0} |B_k|.
• S84-2 Folded in-band amplitude leakage
  For any f > f_N, leakage folded to f_alias is approximated by A_alias(f) ≈ | H_aa(f) |.
  Ensure A_alias(f) versus in-band noise floor meets the target aliasing SNR:
  SNR_alias_dB ≈ 10 * log10( P_signal / E_alias ) >= SNR_target_alias_dB.
• S84-3 Quantization and ENOB
  Quantization SNR: SNR_q_dB ≈ 6.02 * ENOB + 1.76; with ideal ADC_bits, ENOB <= ADC_bits.
  Full-scale to RMS relation:
  x_rms = FS / ( sqrt(2) * 10^( M_headroom / 20 ) * CF ) (for a sinusoid, CF ≈ sqrt(2)).
• S84-4 Jitter constraint (consistent with Chapter 3)
  SNR_jitter_dB ≈ -20 * log10( 2 * pi * f_in * J ); choose J such that SNR_jitter_dB >= SNR_target_jitter_dB.
• S84-5 Group delay and TOA deviation
  Delta_T_TOA <= max_{f ∈ BW} | tau_g(f) - tau_g(f_c) |; with linear-phase FIR, tau_g(f) ≈ const, so Delta_T_TOA is dominated by sampling-interpolation error.
• S84-6 Kaiser order estimate (FIR prototype)
  With normalized radian transition width Delta_omega ∈ (0, pi), FIR order
  N ≈ ( A_s_dB - 8 ) / ( 2.285 * Delta_omega ), with Delta_omega = 2 * pi * Delta_f_tr / fs.

V. Design Strategies and Selections

1. Analog first, digital compensation second
   Use H_aa(f) to achieve primary anti-alias roll-off to target A_s; use H_d(f) to flatten passband ripple and equalize sensor response.
   Ensure H(f) = H_aa(f) * H_d(f) meets delta_p and epsilon_tau within BW, and A_s in the stopband.
2. Filter families (guidance)
   • Low-order passive RC: low cost, non-linear phase; suitable when fs is generously oversampled.
   • Active MFB / Sallen–Key: steeper roll-off; watch amplifier noise and headroom.
   • Linear-phase FIR: digital compensation/shaping; Kaiser or equiripple are practical.
   • Low-latency IIR: Butterworth (flat magnitude) or Chebyshev (tighter transition at ripple cost) for real-time constraints.
3. Multistage decimation (polyphase)
   Sample at a high fs_in with initial H_aa suppression; follow with multistage H_d decimation:
   • Stage j has decimation M_j and transition Delta_f_tr_j; total M = ∏ M_j.
   • Allocate A_s_j such that ∑ E_alias_j <= E_alias_budget.
4. Passband equalization
   For known sensor/acoustic ripples G_sens(f), set H_eq(f) ≈ 1 / G_sens(f) constrained within BW and regularized to avoid noise amplification.

VI. Gain Staging and Dynamic-Range Management

• Headroom and noise posture
  With preamp noise density e_n and sensor noise S_sens, total input noise S_tot ≈ S_sens + e_n^2.
  Choose A_gain so x_rms approaches FS / 10^( M_headroom / 20 ) without driving saturation to meet SNR_total.
• Peak protection and AGC
  Ensure the AGC’s effective transfer is ~constant across BW, or deconvolve with H_d offline.Peak detect window Delta_t_pk and release time T_rel; constrain AGC’s frequency-domain footprint:
• Bias and zero shift
  Use C_offset for single-supply headroom; remove DC in the digital domain with high-pass or differencing, taking care not to erode near-DC content.

VII. Anti-Aliasing Budget and Verification

1. Budget decomposition
   With target SNR_target_total_dB, map to linear power and apportion across alias/q/jitter:
   SNR_target_total^{-1} = w_alias / SNR_alias + w_q / SNR_q + w_j / SNR_jitter, with weights w_* summing to 1.
2. Verification methods
   • Sweep injection: for f ∈ [f_N, K*fs+f_N], measure folded amplitude at f_alias to validate A_s and E_alias.
   • Noise floor contrast: inject out-of-band white noise; compare baselines with/without H_aa.
   • Group delay: chirp or multitone to validate tau_g(f) flatness and compute epsilon_tau.

VIII. Arrival Time and Group Delay (Cross-chapter Consistency)

• When T_arr depends on in-band phase, require near-linear phase in BW:
  If tau_g(f) ≈ const = tau_g0, then Delta_T_TOA ≈ | tau_g0 - tau_ref | + Delta_interp.
• If H(f) is non-linear phase, apply minimum- or linear-phase compensation via H_d before the Chapter 8 path integral, and record epsilon_tau and compensation version in the manifest.

IX. Workflow Mx-4 (Conditioning and Anti-Aliasing)

• Set targets: BW / f_max / SNR_target_total_dB / M_headroom / epsilon_tau; bring ENOB from Chapter 2 and J from Chapter 3.
• Coarse H_aa(f) design: choose topology and f_c / Delta_f_tr / A_s; estimate FIR order with S84-6 for digital compensation.
• Budgeting: decompose SNR_target_total into alias/q/jitter and derive E_alias_budget.
• Multistage decimation planning: choose number of stages and {M_j}; set A_s_j and Delta_f_tr_j per stage.
• Gain staging: compute A_gain and M_headroom to satisfy S84-3 and P84-4; set B_bias / C_offset.
• Simulate & sweep: estimate E_alias, SNR_total, and tau_g(f); iterate steps 2–5 if unmet.
• Bench verify: sweep/white-noise floor tests and group-delay measurements; freeze H_aa / H_d parameters.
• Manifest & release: record {H_aa_kind, f_c, Delta_f_tr, A_s, FIR_N, delta_p, epsilon_tau, A_gain, M_headroom}.

X. Implementation Bindings and Interface Hints (I80-3, I80-4)

• design_filter(kind:str, params:dict) -> FiltRef
  kind ∈ {"aa_rc","aa_mfb","fir_equiripple","fir_kaiser","iir_butter","iir_cheby1","iir_cheby2"};
  params must include at least {fs, f_c, Delta_f_tr, A_s_dB, delta_p_dB}.
• filter_apply(sig:any, filt:FiltRef) -> any
  Ensure delay compensation: linear-phase FIRs allow whole-sample delay N/2 correction; for IIRs, record phase to compensate later in Chapter 8.
• resample(sig:any, fs_target:float, mode:str="polyphase") -> any
  In multistage decimation/interpolation, ensure each stage’s A_s_j and transition width meet Step 4 budgets.

XI. Manifest — Minimal Fields

• {H_aa_kind, H_d_kind, f_c, BW, Delta_f_tr, A_s_dB, delta_p_dB, FIR_N, tau_g_flatness:epsilon_tau, fs, M_decim, A_gain, M_headroom_dB, ENOB, J, SNR_target_total_dB, E_alias_budget}.
• Any experiment involving T_arr must cite this manifest version and annotate the tau_g compensation status.