08-EFT.WP.Core.Sea v1.0 | Appendix C — Filter and Window Function Library

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Appendix C — Filter and Window Function Library

I. Purpose and Scope

• Purpose
  Assemble the window functions and digital filter prototypes used throughout this volume, with unified symbols, scaling, and acceptance criteria. The entries connect directly to I80-3 (conditioning & filtering) and I80-4 (spectra & features).
• Where it applies
  Sampling & anti-aliasing (see Chapter 4), PSD and feature extraction (see Chapter 5), and group-delay alignment for arrival-time estimation (see Chapter 8).
• Unified abbreviations
  fs (sample rate), N (FFT length or FIR tap count), w[n] (window), H(f) (transfer function), tau_g(H) (group delay), A_s_db (stopband attenuation in dB), A_p_db (passband ripple in dB).

II. Window Common Metrics and Scaling

• Coherent gain and amplitude correction
  CG = ( ∑_{n=0}^{N-1} w[n] ) / N
  C_amp = 1 / CG (amplitude correction factor)
• Equivalent noise bandwidth (in FFT-bin and Hz units)
  ENBW_bins = N * ( ∑ w[n]^2 ) / ( ∑ w[n] )^2
  ENBW_Hz = ( fs / N ) * ENBW_bins
• Main-lobe width (zero-to-zero approximation)
  Delta_f_main approx K_main * ( fs / N ), with K_main window-dependent.
• Ripple and scalloping loss
  First sidelobe suppression A_s_db; scalloping loss SL_db (max amplitude loss when a tone lies mid-bin).
• PSD calibration (one-sided)
  S_xx(f) = ( 2 / ( fs * U_w ) ) * |X_w(f)|^2, where U_w = (1/N) * ∑ w[n]^2 and X_w is the DFT of windowed data.

III. Window Catalog (Property Library v1.0)

• Rectangular
  K_main = 2.00; A_s_db approx 13; ENBW_bins = 1.00; SL_db approx 3.92.
  Use for: line amplitude fidelity when leakage is tolerable.
• Hann (raised cosine)
  K_main = 4.00; A_s_db approx 31.5; ENBW_bins approx 1.50; SL_db approx 1.42.
  Use for: general spectra with lower leakage.
• Hamming
  K_main = 4.00; A_s_db approx 41; ENBW_bins approx 1.36; SL_db approx 1.78.
  Use for: tighter sidelobe needs; leakage/resolution trade-off.
• Blackman (3-term)
  K_main = 6.00; A_s_db approx 58; ENBW_bins approx 1.73; SL_db approx 1.1.
  Use for: high sidelobe suppression at moderate resolution.
• Blackman–Harris (4-term)
  K_main = 8.00; A_s_db approx 92; ENBW_bins approx 2.00; SL_db approx 0.9.
  Use for: tone detection near the noise floor.
• Nuttall (4-term)
  K_main = 8.00; A_s_db approx 93; ENBW_bins approx 2.02; SL_db approx 0.9.
  Use for: BH-like very low sidelobes.
• Flat-top (5-term)
  K_main approx 10.0; A_s_db approx 93; ENBW_bins approx 3.77; SL_db approx 0.01.
  Use for: amplitude-accurate PSD when frequency resolution is secondary.
• Kaiser (parameter beta)
  Relation between A_s_db and beta (design convention):
  • beta = 0 → Rectangular
  • beta approx 0.1102 * ( A_s_db - 8.7 ) (for A_s_db > 50)
  • beta approx 0.5842 * ( A_s_db - 21 )^{0.4} + 0.07886 * ( A_s_db - 21 ) (for 21 <= A_s_db <= 50)
  K_main and ENBW_bins increase with beta.
  Use for: FIR design meeting specified A_s_db and transition bandwidth.
• Dolph–Chebyshev (parameter A_s_db)
  Equal-ripple sidelobes with minimum main-lobe width for a given A_s_db.
  Use for: extreme sidelobe control with tight main-lobe.

IV. FIR Filter Library and Design Recipes

1. Unified spec & naming
   FIR.lp.<window|remez>.<N>.<f_p>.<f_s>; likewise FIR.hp.*, FIR.bp.*, FIR.bs.*. Frequencies recorded as f_hat = f / fs.
2. Kaiser window method (low-pass example)
   • Specs: f_p, f_s, A_s_db, A_p_db. Let Delta_ω = 2 * pi * ( f_s - f_p ).
   • Order estimate: N_min approx ceil( ( A_s_db - 8 ) / ( 2.285 * Delta_ω ) ) + 1.
   • Coefficients: h[n] = sinc( 2 * f_c * ( n - (N-1)/2 ) ) * kaiser(n, N, beta), with f_c = ( f_p + f_s ) / 2.
   • Linear-phase group delay: tau_g(H) = ( N - 1 ) / ( 2 * fs ).
3. Remez (Parks–McClellan) equiripple FIR
   Pass/stop weights W_p, W_s → achieves equal ripple A_p_db, A_s_db.
   Use for: compact order when the transition band is most constrained.
4. Multi-band bandpass and notch
   Construct via spectral algebra: FIR.bp = LP(f_hi) - LP(f_lo); notch FIR.bs = 1 - FIR.bp, then normalize and level-match.
5. Acceptance
   max_pass_ripple <= A_p_db; min_stop_atten >= A_s_db; alias bound E_alias per Chapter 4 Mx-C2.

V. IIR Prototypes and Biquad Realization

• Naming & implementation
  IIR.bw.lp.<n>.<f_c> (Butterworth), IIR.ch1.lp.<n>.<f_p,A_p_db> (Chebyshev I), IIR.ch2.lp.<n>.<f_s,A_s_db> (Chebyshev II), IIR.ell.lp.<n>.<f_p,f_s,A_p_db,A_s_db> (Elliptic).
  Digitize via bilinear transform; pre-warp Omega = tan( pi * f_hat ).
• Order estimate (example: Butterworth LP)
  n_min = ceil( log10( (10^{0.1*A_s_db} - 1) / (10^{0.1*A_p_db} - 1) ) / ( 2 * log10( Omega_s / Omega_p ) ) ).
• Biquad (RBJ form; LP)
  omega0 = 2 * pi * f_c / fs, alpha = sin(omega0) / ( 2 * Q ).
  b0 = (1 - cos(omega0)) / 2, b1 = 1 - cos(omega0), b2 = (1 - cos(omega0)) / 2;
  a0 = 1 + alpha, a1 = -2 * cos(omega0), a2 = 1 - alpha;
  normalize: b* /= a0, a1 /= a0, a2 /= a0.
• Second-order notch
  omega0 = 2 * pi * f0 / fs, r = exp( - pi * BW / fs ).
  H(z) = ( 1 - 2 * cos(omega0) * z^{-1} + z^{-2} ) / ( 1 - 2 * r * cos(omega0) * z^{-1} + r^2 * z^{-2} ).
• Group delay
  IIR tau_g(H) is frequency-dependent; arrival-time estimation must reference tau_mono and log compensation in trace (see Chapter 8).

VI. Multirate Filtering and CIC Pipelines

• CIC N-stage, R-rate decimator
  H_cic(z) = ( ( 1 - z^{-R} ) / ( 1 - z^{-1} ) )^N.
  Passband droop: |H_cic(f)| approx | sin(pi * f * R / fs) / ( R * sin(pi * f / fs) ) |^N.
  Typical chain: [CIC] -> [FIR.compensation] -> [IIR.smoothing].
• Guidance
  Prefer CIC for large decimation; add a short FIR to flatten the passband and achieve A_s_db.

VII. Group Delay, Phase, and Impact on Arrival Time

• FIR linear-phase compensation
  tau_align = ( N - 1 ) / ( 2 * fs ); subtract from ts or shift the waveform by an integer number of samples.
• IIR phase nonlinearity
  Measure tau_g(H) offline; apply band-limited average compensation to the time base used by estimate_toa; preserve evidence via trace.

VIII. Leakage, Aliasing, and Spectral Accuracy Notes

• Anti-aliasing (see Chapter 4)
  Prioritize analog H(f); digital filt.chain follows. Windows reduce leakage but cannot stop super-Nyquist aliases.
• Three-metric window pick for line spectra
  Choose by A_s_db, ENBW_bins, SL_db. Flat-top for amplitude metrology; Blackman–Harris/Nuttall for detection; Hann/Hamming for localization.
• Frequency and amplitude uncertainty
  Resolution Delta_f = fs / N; main-lobe bound Delta_f_main.
  Amplitude uncertainty U_amp approx f( SL_db, SNR, ENBW_bins ), reported as part of U = k * u_c.

IX. Selection Guidance (By Scenario)

• Laboratory amplitude metrology
  Window: Flat-top; Filter: FIR.lp denoise then peak-pick; report ENBW_bins and C_amp.
• Field weak-signal detection
  Window: Blackman–Harris or Nuttall; Filter: FIR.bp on target band; add IIR.notch for mains if needed.
• Streaming online monitoring
  Window: Hann; Filtering: CIC + FIR.comp during decimation; keep rho < 1 for real-time stability (see Core.Threads).

X. I80 Bindings and Usage Snippets

• Design & apply

fref = design_filter(kind="fir_lp", params={"f_p":0.18,"f_s":0.22,"A_s_db":80,"method":"kaiser"})

sig_f = filter_apply(sig=block, filt=fref)

S = psd(sig=sig_f, method="welch", seg=8, overlap=0.5, window="hann")

• Resampling & anti-aliasing

sig_ds = resample(sig=sig_f, fs_target=fs/4, mode="polyphase") # filt.chain includes anti-alias LP

XI. Acceptance and Recording Template

• Required fields
  filt.kind, filt.N, filt.params, filt.A_p_db, filt.A_s_db, filt.Delta_f, filt.tau_g, window.name, window.ENBW_bins, window.C_amp.
• Quality thresholds
  A_s_db >= A_s_min; A_p_db <= A_p_max; Delta_f_main <= Delta_f_budget; |tau_g - tau_align| <= eps_tau.

XII. Versioning and Naming

• Anchors
  filt.H_rev, spec.window_rev, chain.rev increment on parameter change.
• Freezing rule
  Any change of window, N, or A_s_db constitutes a new version; record in io.manifest_rev (see Chapter 7).

XIII. Quick Reference (Typical Window Numbers, agreed convention)

• Rectangular: K_main=2.00, A_s_db=13, ENBW_bins=1.00, SL_db=3.92.
• Hann: K_main=4.00, A_s_db=31.5, ENBW_bins=1.50, SL_db=1.42.
• Hamming: K_main=4.00, A_s_db=41, ENBW_bins=1.36, SL_db=1.78.
• Blackman: K_main=6.00, A_s_db=58, ENBW_bins=1.73, SL_db=1.1.
• Blackman–Harris(4): K_main=8.00, A_s_db=92, ENBW_bins=2.00, SL_db=0.9.
• Flat-top(5): K_main≈10.0, A_s_db=93, ENBW_bins=3.77, SL_db≈0.01.
• Kaiser: beta from A_s_db (above); K_main and ENBW_bins increase with beta.

XIV. Cross-Volume Anchors and Citations

• Anti-aliasing & path: H(f), f_c, BW, tau_g(H) (see Chapter 4).
• Arrival time: T_arr, gamma(ell), n_eff(x,t), c_ref (see Chapter 8 and the Energy Filaments companion).
• Concurrency & stability: chan, cap, bp, W_q, rho < 1 (see EFT.WP.Core.Threads v1.0).

XV. Closing Note

This library fixes the design, calibration, and recording conventions for windows and filters in a uniform symbol system and plain-text formulas, enabling comparability for S_xx(f), time consistency for T_arr, and reproducibility across scenarios.