Why Cables Matter: Methods & Math Deep Dive

1) Notation & Sampling

• Sampling: audio and LDV at 96 kHz unless stated; time index \( t = n / f_s \).
• Signals: \( v(t) \) velocity at cone (LDV), \( p(t) \) pressure at mic, \( x(t) \) input drive.
• Fourier: uppercase denotes spectra, e.g., \( V(f) = \mathrm{FFT}\{v(t)\} \).

2) LDV Path (Cone Mechanics)

We model the on-axis cone motion locally as a lightly damped second-order system for small signals:

\[ x(t) = 1 - \frac{e^{-\zeta \omega_0 t}}{\sqrt{1-\zeta^2}} \,\sin\!\big(\omega_d t + \phi\big), \qquad \omega_d = \omega_0 \sqrt{1-\zeta^2} \] \[ v(t) = \frac{dx}{dt} \]

We allow small per-condition timing shifts \( \Delta t \) (tens of µs) and gain differences to reflect cable/amp interaction.

3) From Cone to Microphone (Early-Time Window)

The microphone sees the cone after air + room transfer:

\[ p(t) = (r * v)(t), \qquad r(t) \approx \delta(t) + \alpha\,\delta(t-\tau) + e(t) \]

We analyze an early window (e.g., first 6 ms) to emphasize direct sound and the first reflection, reducing late-room confounds.

4) Two-Driver Sum (Woofer + Tweeter)

For a 2-way speaker the mic signal is the complex sum of two paths with distinct delays and directivity:

\[ p(t) = (r_w * v_w)\!\big(t - \tau_w\big) + (r_t * v_t)\!\big(t - \tau_t\big) + \varepsilon(t) \]

In frequency domain (used for fitting):

\[ P(f) = H_w(f)\,V_w(f)\,e^{-j 2\pi f \tau_w} \;+\; H_t(f)\,V_t(f)\,e^{-j 2\pi f \tau_t} \;+\; \varepsilon(f) \]

We estimate \( H_w, H_t \) via least squares in their dominant bands (below/above crossover), then verify prediction vs measured \( P(f) \) and publish residuals.

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5) Time Alignment Metrics

5.1 Cross-Correlation Lag

\[ \hat{\tau} = \arg\max_{\tau \in [-\tau_{\max},\,\tau_{\max}]} \sum_{n} a[n]\, b[n - \tau] \]

We report the best-lag in milliseconds and overlay aligned waveforms.

5.2 Dynamic Time Warping (DTW)

DTW aligns two sequences allowing small local time deformations under constraints (Sakoe–Chiba band):

\[ D(i,j) = |a_i - b_j| + \min\{ D(i\!-\!1,j),\; D(i,j\!-\!1),\; D(i\!-\!1,j\!-\!1) \} \] \[ \mathrm{DTW}(a,b) = \frac{D(N,M)}{Z} \]

\( Z \) normalizes by path length; we also convert the mean path slope to a “ms-equivalent” for readability. DTW settings are pre-registered and never tuned per condition.

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6) Ultrasonic Decay (~30 kHz Band)

1. Band-pass filter around the chosen ultrasonic band to get \( x_{30k}(t) \).
2. Compute amplitude envelope via Hilbert transform: \( a(t) = \big| \mathrm{Hilbert}\{x_{30k}(t)\} \big| \).
3. Fit \( a(t) \approx e^{-t/\tau} \) by linear regression of \( \ln a(t) \) vs \( t \) over a noise-robust window.

We report \( \tau \) with confidence intervals and the noise floor used for truncation.

7) Noise Fingerprints

We capture three baselines on identical scales:

• Electrical (dummy load): amp output noise spectrum with speaker disconnected.
• Acoustic idle: mic in place, system powered, no program material.
• Room baseline: ambient with system off.

\[ \mathrm{PSD}(f) = 10 \log_{10}\!\big( S_{xx}(f) \big) \quad \text{(Welch / windowed \& averaged)} \]

We annotate mains at 50/60 Hz and harmonics, broadband floor, and any RF shelf. Cable routing and grounding notes are included for reproducibility.

8) Listening-Position FR (Why It Can Overlay)

1. Capture log sweep; deconvolve to impulse \( h(t) \).
2. Window early time; FFT to \( H(f) \).
3. Display magnitude with gentle smoothing (e.g., 1/6-oct).

We expect near-overlays at the seat even when time/noise metrics differ—hence complementary analysis.

9) Statistics & Audibility

9.1 Effect Sizes & Confidence

We report mean differences with 95% CIs and standardized effect sizes where appropriate. For null outcomes we run equivalence tests (TOST) against practical JND thresholds.

9.2 Multiple Comparisons

Primary hypotheses are pre-registered; confirmatory p-values are adjusted (Holm–Bonferroni). Exploratory plots are clearly labeled.

9.3 Blind A/B/X Listening

For \( N \) trials with success count \( k \), chance performance is \( \mathrm{Binomial}(N, 0.5) \). We provide exact binomial CIs, Bayes factors, and connect outcomes to measurement deltas.

9.4 Repeatability

We publish test–retest overlays, Bland–Altman plots, and intraclass correlation (ICC) for key metrics.

10) Reporting & Reproducibility

• Release CSV snippets for LDV, mic early windows, decay envelopes, and noise spectra.
• Ship plotting notebook with fixed parameters (gates, bands, DTW constraints).
• Document cable lengths, termination torque, environment logs, and alignment method.

© Princess Pasta Audio Lab. Methods subject to iterative improvement; changes will be versioned.

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