08-EFT.WP.Core.Sea v1.0 | Chapter 5 — Spectral Analysis and Features

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Chapter 5 — Spectral Analysis and Features

I. Objectives and Scope

• Provide a coherent family of estimators that take discrete-time x[n] to power spectral density S_xx(f) (periodogram / Welch / multitaper), with unified units and normalizations.
• Define a feature family—peaks, band energy, spectral centroid, spectral flatness, harmonic clusters, coherence and phase—together with robustness assessment, consistent with Chapter 2 (sampling & quantization), Chapter 4 (H(f) conditioning), and Chapter 3 (tau_mono/ts time bases).
• Deliver postulates P85-, minimal equations S85-, and the execution workflow Mx-5 for spectral analysis; align with interfaces I80-4 (fft/psd/feature_extract).

II. Core Objects and Unit Conventions

• DFT and frequency axis
  X[k] = ∑_{n=0}^{N-1} x[n] * w[n] * exp( -j * 2 * pi * k * n / N ).
  f[k] = k * fs / N, Delta_f = fs / N, with k ∈ {0,...,N-1}.
  Window power-normalization U_w = ( 1 / N ) * ∑_{n=0}^{N-1} w[n]^2; coherent gain G_c = ( 1 / N ) * ∑_{n=0}^{N-1} w[n].
• One- vs two-sided spectra and units
  One-sided PSD: S_xx^1(f) ≈ 2 * S_xx^2(f) (except f=0 and f=fs/2); unit remains unit(x)^2 / Hz.
  Amplitude spectral density ASD(f) = sqrt( S_xx(f) ), with unit unit(x) / sqrt(Hz).
  Parseval consistency: ∑_{n} |x[n]|^2 / N ≈ ∑_{k} S_xx(f[k]) * Delta_f under proper normalization.

III. Postulates P85- (Consistency of Spectral Estimation)*

• P85-1 Units and normalization
  Every PSD report must explicitly state {one_sided | two_sided}, U_w, and whether the factor of 2 was applied; feature computations must use the same convention.
• P85-2 Leakage and windowing
  When windowing, provide mainlobe width and sidelobe suppression A_sl_dB; for in-band peak measurement, ensure the peak falls within the mainlobe or apply interpolative correction.
• P85-3 Statistical independence
  With Welch or multitaper averaging, report effective independent averages K_eff (or equivalently nu_eff degrees of freedom); all variance/confidence bounds are based on this.
• P85-4 Time–frequency consistency
  For STFT, declare window length L and hop H, and record Delta_t ≈ L / fs and Delta_f ≈ fs / L; align time with tau_mono, publish with ts.
• P85-5 Arrival time and phase
  Any phase- or group-delay-based feature must compensate the Chapter 4 phase of H(f); otherwise label as uncompensated and quantify potential bias.

IV. Minimal Equations S85- (Estimators and Bounds)*

• S85-1 Periodogram normalization
  P_xx[k] = | X[k] |^2 / ( N * fs * U_w ) (two-sided).
  One-sided correction: P_xx^1[k] = 2 * P_xx[k], k ∈ (0, N/2).
• S85-2 Welch averaging and variance
  Split into K segments of length N_s with overlap o ∈ [0,1):
  S_hat_xx(f[k]) = ( 1 / K ) * ∑_{i=1}^{K} | X_i[k] |^2 / ( N_s * fs * U_w ).
  Independent DOF nu_ind ≈ 2 * K * ( ( ∑ w^2 )^2 / ∑ w^4 ); with overlap correlation nu_eff = nu_ind / c_ovlp, c_ovlp >= 1.
  Variance (approx.): Var[ S_hat_xx(f) ] ≈ 2 * S_xx(f)^2 / nu_eff.
• S85-3 Multitaper estimate
  With P DPSS tapers: S_hat_xx(f) = ( 1 / P ) * ∑_{p=1}^{P} | FFT( v_p * x ) |^2 / ( N * fs * U_{v_p} ), and nu_eff ≈ 2 * P.
• S85-4 Spectral-peak interpolation (parabolic)
  With powers P_- , P_0 , P_+ at bins k-1, k, k+1:
  delta = ( P_+ - P_- ) / ( 2 * ( 2 * P_0 - P_- - P_+ ) ),
  f_peak ≈ ( k + delta ) * fs / N; amplitude correction A_peak ≈ P_0 - 0.25 * ( P_- - P_+ ) * delta.
• S85-5 Band energy and bandwidth
  BandPower( f1,f2 ) = ∑_{k: f1<=f[k]<=f2} S_hat_xx(f[k]) * Delta_f.
  f_centroid = ( ∑ f[k] * S_hat_xx[k] ) / ( ∑ S_hat_xx[k] ),
  SpecBandwidth = sqrt( ( ∑ ( f[k] - f_centroid )^2 * S_hat_xx[k] ) / ( ∑ S_hat_xx[k] ) ).
• S85-6 Spectral flatness and tonality
  SF = exp( mean( ln S_hat_xx[k] ) ) / mean( S_hat_xx[k] ).
  Tonalness ≈ 1 - SF (heuristic, [0,1]).
• S85-7 Cross-spectrum and coherence
  S_xy(f) = E[ X(f) * Y(f)^* ], Coh^2(f) = | S_xy(f) |^2 / ( S_xx(f) * S_yy(f) ) ∈ [0,1].
  phi_xy(f) = arg S_xy(f). With nu_eff DOF, the null-threshold for Coh^2 at size alpha is approximated by Coh_alpha^2 ≈ 1 - alpha^( 1 / ( K_eff - 1 ) ) (independent segment approx.).

V. Windows and Leakage Management

• Mainlobe vs sidelobes
  Hamming/Hann: lower sidelobes, slightly wider mainlobe; Blackman: even lower sidelobes; Kaiser tunes A_sl_dB vs mainlobe via beta.
• ENBW and resolution
  ENBW = fs * ( ∑ w[n]^2 ) / ( ( ∑ w[n] )^2 ); the effective resolution bandwidth RBW ≈ ENBW.
  Peak resolvability: | f_a - f_b | >= RBW.

VI. Feature Family Definitions (F85-*)

• F85-1 Peaks and harmonics
  f_peak, A_peak via S85-4; estimate fundamental f0 by maximizing ∑_{h=1}^{H} S_hat_xx( h * f0 ).
  HarmonicRatio = ( ∑_{h=2}^{H} S_hat_xx( h * f0 ) ) / S_hat_xx( f0 ).
• F85-2 Band energy and SNR
  SNR_band = 10 * log10( BandPower(signal_band) / BandPower(noise_band) ), with noise band excluding ±RBW around detected peaks.
• F85-3 Spectral centroid and bandwidth
  f_centroid and SpecBandwidth per S85-5; compute per band for fingerprinting.
• F85-4 Flatness and roughness
  SF per S85-6; Roughness ≈ Var( diff( log S_hat_xx[k] ) ) (discrete proxy).
• F85-5 Coherence and phase offset
  Coh^2(f), phi_xy(f); aggregate in-band as
  Coh_band = mean_{f ∈ band}( Coh^2(f) ), Phase_band = atan2( ∑ Im S_xy , ∑ Re S_xy ).
• F85-6 Fault tolerance and missingness
  With missing mask m[n] ∈ {0,1}, use the same window per segment, normalize by effective sample count in U_w, and record missing rate in the manifest.

VII. Time–Frequency Analysis and Tracking

• STFT
  STFT[m,k] = ∑_{n} x[n] * w[n - m*H] * exp( -j * 2 * pi * k * n / N_w ).
  Spectrogram Spec[m,k] = | STFT[m,k] |^2 / ( N_w * fs * U_w ).
• Resolution tradeoffs
  Delta_t ≈ N_w / fs, Delta_f ≈ fs / N_w, with Delta_t * Delta_f ≈ 1; the hop H controls time sampling.
• Trajectories and drift
  Track f_peak[m] via windowed interpolation S85-4; define drift = max | f_peak[m+1] - f_peak[m] | as a rate-of-drift feature, aligned with Chapter 9 drift monitoring.

VIII. Robustness and Uncertainty

• Amplitude bias correction
  For a single tone, divide by coherent gain: A_tone ≈ ( 2 / N ) * | X[k0] | / G_c (one-sided).
• Variance and confidence intervals
  CI_alpha(f) ≈ S_hat_xx(f) * [ nu_eff / χ^2_{1-α/2}(nu_eff) , nu_eff / χ^2_{α/2}(nu_eff) ] (chi-squared approx., independent segments).
• Cross-checks
  Vary N_s/overlap/window (or P for multitaper) to gauge sensitivity of S_hat_xx; if changes exceed thresholds, increase K or switch to multitaper.

IX. Workflow Mx-5 (Spectral Analysis and Features)

• Set objectives: fs / Delta_f / RBW / K or P / one_sided and bands {band_i}; import the Phase compensation plan for H(f) from Chapter 4.
• Choose estimator: method ∈ {"periodogram","welch","multitaper"}, set N or N_s/overlap (or P).
• Window & normalization: choose w[n], compute U_w / G_c / ENBW; declare one-/two-sided convention and units.
• Compute PSD: produce S_hat_xx(f) (and S_hat_yy(f), S_hat_xy(f) as needed); estimate nu_eff and variance.
• Extract features: per band compute BandPower / SNR_band / f_peak / f_centroid / SF / Coh^2 / phi_xy, etc.
• Robustness check: perturb N_s/overlap/window or P; ensure relative feature change <= epsilon_feat.
• Manifest & publish: record parameters and versions; output feature tables with uncertainty columns; for T_arr-related phase features, mark compensation status.

X. Interface Bindings and Returns (I80-4)

• fft(sig:any, window:str="hann") -> complex[]
  Returns X[k]. Caller completes unit normalization and one-sided conversion; return G_c / U_w as helpful metadata.
• psd(sig:any, method:str="welch", seg:int=8, overlap:float=0.5) -> dict
  Return keys: {"f":float[], "Sxx":float[], "one_sided":bool, "U_w":float, "G_c":float, "nu_eff":float, "window":str, "overlap":float}.
• feature_extract(sig:any, feats:list[str]) -> dict
  Supported: "f_peak","A_peak","bandpower:{f1,f2}","snr:{sig,noise}","f_centroid","spec_bw","sf","coh:{y,band}","phi:{y,band}".
  Include {"uncertainty":..., "nu_eff":..., "manifest_ref":...} in results.

XI. Manifest — Minimal Fields

• {method, window, N or N_s, overlap, one_sided, U_w, G_c, ENBW, nu_eff, RBW, bands:[{f1,f2}], H_phase_comp:{"enabled":bool,"ver":str}, ts_range, missing_rate}.
• Feature tables must reference this manifest and record epsilon_feat, CI_alpha parameters, and alpha.

XII. Interlocks and Cross-Volume References

• With Chapter 2: enforce consistency among ENOB/DR/Delta_f; ensure RBW is not below physical resolvability.
• With Chapter 3: keep tau_mono/ts time bases monotone in STFT indexing; combine jitter J (SNR_jitter) impact on high-frequency peaks per Chapters 2 & 4.
• With Chapter 4: any phase-/group-delay-related feature must apply H(f) compensation.
• With Chapter 8: any T_arr frequency-domain estimation must report under the same gamma(ell) and c_ref / n_eff convention.