08-EFT.WP.Core.Sea v1.0 | Appendix D — Spectral & Statistical Formula Sheet

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Appendix D — Spectral & Statistical Formula Sheet

I. Frequency Axis and Indexing

• Discrete frequency vs. sampling
• f_k = k * ( fs / N ), k ∈ {0,1,...,N-1}
• Two-sided vs. one-sided spectra
• two-sided: k ∈ {0,...,N-1} ; one-sided: k ∈ {0,...,N/2}
• Angular frequency
• omega_k = 2 * pi * f_k / fs
• Frequency resolution
• Delta_f = fs / N

II. DFT/IDFT and Scaling

• Definitions (volume-wide convention)
• X[k] = ∑_{n=0}^{N-1} x[n] * exp( -j * 2 * pi * k * n / N )
• x[n] = (1/N) * ∑_{k=0}^{N-1} X[k] * exp( j * 2 * pi * k * n / N )
• Parseval identity
• (1/N) * ∑_{n=0}^{N-1} |x[n]|^2 = (1/N^2) * ∑_{k=0}^{N-1} |X[k]|^2

III. Windowing, Coherent Gain, and ENBW

• Windowed DFT
• X_w[k] = ∑_{n=0}^{N-1} ( x[n] * w[n] ) * exp( -j * 2 * pi * k * n / N )
• Coherent gain and amplitude correction
• CG = ( ∑ w[n] ) / N , C_amp = 1 / CG
• Energy normalization (window power)
• U_w = (1/N) * ∑ w[n]^2
• Equivalent noise bandwidth
• ENBW_bins = N * ( ∑ w[n]^2 ) / ( ∑ w[n] )^2
• ENBW_Hz = ENBW_bins * ( fs / N )

IV. Periodogram and PSD (One-Sided Convention)

• Single-segment periodogram (power spectral density)
• S_xx[k] = ( 2 / ( fs * N * U_w ) ) * | X_w[k] |^2 , k = 1,...,N/2-1
• Endpoints
• S_xx[0] = ( 1 / ( fs * N * U_w ) ) * | X_w[0] |^2 ; if N even, S_xx[N/2] likewise
• Welch averaging (K segments, optional overlap)
• S_xx^Welch[k] = (1/K) * ∑_{i=1}^{K} S_{xx}^{(i)}[k]
• nu_eff approx 2 * K (no overlap) ; nu_eff approx 1.5 * K (50% overlap + Hann)
• Line-spectrum amplitude (single tone on bin k0)
• A_hat approx ( 2 / N ) * | X_w[k0] | * C_amp

V. Cross-Spectrum, Coherence, and Transfer Estimates

• Cross power spectral density
• S_xy[k] = ( 2 / ( fs * N * U_w ) ) * X_w[k] * conj( Y_w[k] )
• Magnitude-squared coherence
• gamma_xy^2[k] = | S_xy[k] |^2 / ( S_xx[k] * S_yy[k] ), gamma_xy^2[k] ∈ [0,1]
• Transfer function estimates (input x, output y)
• H1[k] = S_yx[k] / S_xx[k] (output-noise robust)
• H2[k] = S_yy[k] / S_xy[k] (input-noise robust)
• Hv[k] = sqrt( H1[k] * H2[k] ) (hybrid)
• Phase and group delay
• phi[k] = angle( H[k] )
• tau_g[k] = - d phi / d omega |_{omega=omega_k} approx - ( phi[k+1] - phi[k-1] ) / ( 2 * Delta_omega )
• Delta_omega = 2 * pi * Delta_f / fs

VI. Autocorrelation, Cross-Correlation, and Wiener–Khinchin

• Autocorrelation
• r_xx[m] = ∑_{n} x[n] * conj( x[n-m] )
• Cross-correlation
• r_xy[m] = ∑_{n} x[n] * conj( y[n-m] )
• Wiener–Khinchin
• S_xx[k] = FFT( r_xx[m] ) , r_xx[m] = IFFT( S_xx[k] )

VII. Peak Location and Sub-Bin Interpolation

• Parabolic interpolation (power spectrum P[k] at peak k0)
• delta = ( P[k0+1] - P[k0-1] ) / ( 2 * ( 2 * P[k0] - P[k0-1] - P[k0+1] ) )
• k_hat = k0 + delta , f_hat = k_hat * ( fs / N )

VIII. Quantization Noise, ENOB, and Dynamic Range

• Uniform quantization noise power (step Delta)
• P_q = Delta^2 / 12
• Full-scale sine quantization SNR
• SNR_q_dB = 6.02 * ADC_bits + 1.76
• Effective number of bits
• ENOB = ( SNR_meas_dB - 1.76 ) / 6.02
• Noise spectral density (one-sided, near-white)
• S_q ≈ ( 2 * P_q ) / fs

IX. Sampling Jitter and Phase-Noise Approximations

• Jitter-limited SNR (single-tone f_in, time jitter sigma_t)
• SNR_j_dB approx -20 * log10( 2 * pi * f_in * sigma_t )
• Equivalent phase jitter
• sigma_phi = 2 * pi * f_in * sigma_t

X. Confidence Intervals and Significance Tests

• PSD (1 - alpha) confidence interval (chi-square)
• [ S_low, S_high ] = [ ( nu_eff * S_hat ) / chi2_{1 - alpha/2}(nu_eff),
• ( nu_eff * S_hat ) / chi2_{alpha/2}(nu_eff) ]
• Mean (known sample variance estimate sigma_hat)
• mu_CI = mu_hat ± z_{1 - alpha/2} * ( sigma_hat / sqrt(N_eff) )
• Variance
• [ sigma2_low, sigma2_high ] = [ ( (N_eff - 1) * sigma_hat^2 ) / chi2_{1 - alpha/2}(N_eff - 1),
• ( (N_eff - 1) * sigma_hat^2 ) / chi2_{alpha/2}(N_eff - 1) ]
• Coherence significance threshold (K-segment average)
• gamma_crit^2 = 1 - alpha^{ 1 / ( K - 1 ) }

XI. Missingness and Weights (m ∈ {0,1})

• Effective sample count
• N_eff = ∑ m[n]
• Weighted mean and variance
• mu_hat = ( ∑ m[n] * x[n] ) / N_eff
• sigma_hat^2 = ( ∑ m[n] * ( x[n] - mu_hat )^2 ) / ( N_eff - 1 )
• Weighted, windowed DFT
• X_{wm}[k] = ∑ ( m[n] * w[n] * x[n] ) * exp( -j * 2 * pi * k * n / N )
• U_{wm} = (1/N) * ∑ ( m[n]^2 * w[n]^2 )

XII. Robust Statistics and Drift Metrics

• Median absolute deviation
• MAD = median( | x - median(x) | )
• sigma_robust approx 1.4826 * MAD
• IQR scale
• sigma_IQR approx 0.7413 * ( Q3 - Q1 )
• EWMA
• z_t = lambda * x_t + (1 - lambda) * z_{t-1}
• CUSUM (upper-sided)
• C_t^+ = max( 0, C_{t-1}^+ + x_t - mu0 - k )
• Simplified drift score
• drift_score = | mu_window - mu_ref | / sigma_robust

XIII. Arrival Time and Path (Cross-Volume Anchors)

• Cross-correlation TOA estimate
• r_xy[tau] = ∑ x[n] * y[n - tau] , tau_hat = argmax_tau r_xy[tau]
• Sub-sample parabolic refinement
• tau_hat_frac = tau0 + ( r[tau0+1] - r[tau0-1] ) / ( 2 * ( 2 * r[tau0] - r[tau0-1] - r[tau0+1] ) ) / fs
• Two canonical arrival-time forms (consistency)
• T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
• T_arr = ( ∫ ( n_eff / c_ref ) d ell )
• delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |

XIV. Frequency-Domain Filtering and Group Delay (Reminders)

• FIR linear-phase group delay
• tau_g(H) = ( N - 1 ) / ( 2 * fs )
• IIR nonlinear phase
• tau_g[k] via - d phi / d omega numerical approximation (see Chapter 4)

XV. Features and Spectral Moments

• Spectral centroid
• f_centroid = ( ∑ f_k * S_xx[k] ) / ( ∑ S_xx[k] )
• Bandwidth (2nd moment)
• BW_rms = sqrt( ( ∑ ( f_k - f_centroid )^2 * S_xx[k] ) / ( ∑ S_xx[k] ) )
• Skewness and kurtosis (power or amplitude domain)
• skew = E[ ( x - mu )^3 ] / sigma^3
• kurt = E[ ( x - mu )^4 ] / sigma^4

XVI. Anti-Aliasing and Leakage (Checks)

• Aliasing image
• f_alias = | f_true - r * fs | , r ∈ Z with f_alias ∈ [0, fs/2]
• Leakage vs. main lobe
• Delta_f_main approx K_main * ( fs / N ) (K_main from Appendix C)

XVII. I80 Mapping (Field Cross-Check)

• fft(sig, window) → X_w[k]
• psd(sig, method, seg, overlap) → S_xx[k] and nu_eff
• feature_extract(sig, feats) → e.g., f_centroid, BW_rms, gamma_xy^2
• estimate_toa(sig, "xcorr") → tau_hat

XVIII. Reporting Fields and Units (Unified Convention)

• Required basics
• fs, N, window.name, U_w, CG, ENBW_Hz, seg, overlap, nu_eff,
• PSD_unit = power/Hz (or amp^2/Hz)
• Endpoint note
• For one-sided spectra, bins k=0 and k=N/2 are NOT doubled; all others are ×2 (energy conservation).

XIX. Reference Consistency Checklist

• Timebase: Evaluate delay, jitter, and consistency on tau_mono; publish and audit with ts.
• Environment: Any corr_env(x; RefCond) must explicitly record RefCond.
• Arrival time: Any T_arr reference must also provide gamma(ell), d ell, c_ref, n_eff, and report delta_form.

XX. Closing

This sheet standardizes formulas and scaling for spectral estimation, cross-spectral coherence, robust statistics, and confidence intervals, ensuring that S_xx(f), H(f), tau_g(H), and tau_hat remain reproducible, auditable, and comparable across scenarios.