09-EFT.WP.Core.Density v1.0 | Chapter 6 — Spectral Density and Time–Frequency

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Chapter 6 — Spectral Density and Time–Frequency

I. Objectives and Scope

• Establish a unified convention for S_xx(f) (power spectral density, PSD) and time–frequency analysis (STFT / multiresolution), guaranteeing energy consistency and unit coherency: unit( S_xx ) = unit(x)^2 / Hz.
• Fix window normalization and bandwidth metrics U_w, ENBW_Hz; specify single-/two-sided conversions, band-power computation, and uncertainty reporting.
• Provide the engineering workflow Mx-95 and the interface contract I90 6 (I/O shapes), aligned with Core.Sea Chapter 5 and Appendix C (window library).

II. Notation and Conventions

1. Sequence and sampling: x[n], n = 0..N-1; sampling rate fs; segment length L; hop H; number of segments M.
2. DFT frequency grid: f_k = k * ( fs / L ), k = 0..L-1; one-sided indices k = 0..L/2 (for even L).
3. Window and normalization: w[n], with window power U_w and equivalent noise bandwidth ENBW_Hz:
   • S92-9 : U_w = ( 1 / L ) * ∑_{n=0}^{L-1} w[n]^2.
   • S92-8 : ENBW_Hz = fs * ( ∑ w[n]^2 ) / ( ∑ w[n] )^2.
4. Frequency resolution: Delta_f = fs / L. Always state one-/two-sided convention and Delta_f in reports.

III. PSD Definition, One-/Two-Sided Conversion, and Energy Consistency

1. Windowed DFT: X_w[k] = ∑_{n=0}^{L-1} x[n] * w[n] * exp( -j * 2 * pi * k * n / L ).
2. Single-segment periodogram (two-sided convention):
   S92-30 : P_xx[k] = ( 1 / ( fs * L * U_w ) ) * | X_w[k] |^2.
3. Welch PSD (segment averaging with optional overlap):
   • For segment m, x_m[n] = x[n+mH], X_m[k] = ∑ x_m[n] * w[n] * exp( -j * 2 * pi * k * n / L );
   • S92-31 : S_xx[k] = ( 1 / M ) * ∑_{m=0}^{M-1} ( 1 / ( fs * L * U_w ) ) * | X_m[k] |^2 (two-sided).
4. One-/two-sided conversion (real signals):
   • S1_xx[0] = S2_xx[0], and if L even, S1_xx[L/2] = S2_xx[L/2];
   • S1_xx[k] = 2 * S2_xx[k], k = 1..(L/2 - 1).
5. Energy consistency and variance check (zero-mean assumption):
   Continuous view: S92-34 : var(x) ≈ ( ∫_0^{fs/2} S1_xx(f) df ); discrete: var(x) ≈ ∑_{k=0}^{L/2} S1_xx[k] * Delta_f.
6. Band power and amplitude spectral density:
   S92-36 : P_band(B) = ( ∑_{k∈B} S_xx[k] ) * Delta_f; ASD(f) = sqrt( S_xx(f) ).

IV. Window Normalization, Leakage, and Scalloping

• Window normalization revolves around U_w and ENBW_Hz: publishing S_xx must include both.
• Leakage control: choose windows balancing main-lobe width and sidelobe suppression (e.g., hann, blackman); scalloping error in continuous spectra is governed jointly by ENBW_Hz and main-lobe width.
• When fs < 2 * f_max, the manifest must declare the anti-alias filter H(f) and a residual alias estimate (see Core.Sea Chapter 4).

V. Time–Frequency Analysis: STFT and Spectrogram

1. STFT:
   • S92-32 : STFT_x(m,k) = ∑_{n=0}^{L-1} x[n+mH] * w[n] * exp( -j * 2 * pi * k * n / L ).
   • Time tag t_m = ( m * H ) / fs, frequency f_k = k * ( fs / L ).
2. Spectrogram (PSD-normalized):
   S92-33 : S_xx(m,k) = ( 1 / ( fs * L * U_w ) ) * | STFT_x(m,k) |^2 (two-sided; convert to one-sided as above).
3. Energy conservation (frame-average sense):
   ∑_{m,k} S_xx(m,k) * Delta_f * ( H / fs ) ≈ ( 1 / N ) * ∑ x[n]^2 (state the redundancy correction due to window and overlap).

VI. Cross-Spectrum, Coherence, and Phase

• Cross-spectrum (Welch form):
  S92-37 : S_xy[k] = ( 1 / M ) * ∑_{m} ( 1 / ( fs * L * U_w ) ) * X_m[k] * conj( Y_m[k] ).
• Magnitude-squared coherence:
  S92-38 : Coh_xy[k] = | S_xy[k] |^2 / ( S_xx[k] * S_yy[k] ), in [0,1].
• Phase and group delay: angle( S_xy[k] ) and tau_g(f) = - d( arg H(f) ) / d( 2 * pi * f ) (conditioning chain per Core.Sea Chapter 4).

VII. Uncertainty and Degrees of Freedom (Sketch)

1. Welch nu depends on the number of segments, the window, and the overlap. Recommended reporting:
   • nu ≈ 2 * M_eff, where M_eff ≤ M; estimate M_eff from inter-segment correlation or via library window autocorrelation approximations.
   • Confidence interval (log-normal approximation): 10*log10( S_xx ) ± z * sqrt( var_hat ), and include nu and approximation caveats in the manifest.
2. For cross-spectrum coherence, declare bias-correction method (e.g., randomization / phase scrambling).

VIII. Engineering Workflow Mx-95 (PSD → Energy Verifier)

• Preprocessing. De-mean/de-trend; if needed, band-pass or anti-alias filter (record H(f) and group-delay compensation).
• Segmentation and windowing. Choose L, H, w[n]; compute and record U_w, ENBW_Hz, Delta_f.
• Estimation and convention. Compute S_xx[k] (Welch); state one-/two-sided and DC/Nyquist handling; for coherence, compute S_xy[k], Coh_xy[k] concurrently.
• Energy reconciliation. Compute sigma_hat^2 = ∑ S1_xx[k] * Delta_f and compare to time-domain variance; output rel_err = | sigma_hat^2 - var(x) | / var(x).
• Time–frequency (optional). Compute S_xx(m,k); for a band of interest, emit P_band(t) = ∑_{k∈B} S_xx(m,k) * Delta_f.
• Report and manifest. Publish spec.parquet|nc, writing fs, L, H, window, Delta_f, U_w, ENBW_Hz, nu, side, units, rel_err.
• Alignment and arrival time (multi-source cases). Use tau_mono internally and publish ts; if path timing is applied, record both T_arr forms and delta_form.

IX. Interface Contract (Aligned with I90 6)

• spectral_density(sig:any, method:str="welch", window:str="hann") -> SpecRef
  Expected fields: {"S_xx":"array|tensor", "fs":..., "L":..., "H":..., "window":"...", "U_w":..., "ENBW_Hz":..., "Delta_f":..., "side":"one|two", "nu":..., "qc":{"rel_err":...}}.
• spec_to_energy(spec:SpecRef, band:any) -> float
  Input band = {"f_lo":..., "f_hi":...} or an index set; return P_band preserving units.

X. Implementation Notes and Recommendations

• Window choice. Leakage suppression first: blackman; balanced trade-off: hann; amplitude accuracy (flat-top compensation): flattop (note larger ENBW_Hz).
• Segment length. L sets Delta_f; prefer increasing L (not excessive smoothing) to resolve narrowband components.
• Overlap. Typical 50%–75%; must record H/L. Overlap does not linearly raise degrees of freedom—report nu or M_eff.
• Unit coherency. When publishing S_xx, explicitly state unit(x) and unit( S_xx ) = unit(x)^2/Hz; for ASD, unit(x)/sqrt(Hz).

XI. Cross-Volume / Cross-Chapter Consistency

• Window conventions (ENBW_Hz, U_w) align with Core.Sea Appendix C; arrival-time notation T_arr, path gamma(ell), measure d ell, c_ref, n_eff follow Core.Sea Chapter 8; synchronization and jitter metrics follow Core.Sea Chapter 3.
• This chapter’s S92-8, S92-9, S92-30..S92-38 are part of the global S92-* ledger; workflow Mx-95 and interface I90 6 are the shared anchors for reuse.

XII. Chapter Highlights

• Window normalization via U_w and ENBW_Hz; Welch-form S_xx for stable estimation; explicit one-/two-sided conversion and DC/Nyquist handling.
• STFT spectrograms are PSD-normalized and respect time–frequency energy consistency; cross-spectra and coherence support inter-channel dependency diagnostics.
• The Mx-95 pipeline makes “spectrum → energy” auditable and reconcilable; I90 6 secures implementation binding and cross-system consistency.