NOTES¶

ITD — Ibrahim Time Domain: AR Models on Channel Pairs

Build autoregressive (AR) models on each pair of sensor channels, extract candidate modal parameters, then cluster and vote. Catches weakly-excited modes that SVD-based methods miss.

Intuition¶

Why ITD Finds Weak Modes¶

SVD-based methods (FDD, ERA) find the "highest energy" components. If a mode is energetically weak (e.g., a high-order local mode), SVD may discard it as noise.

ITD processes channel pairs: even if a mode is globally weak, as long as it has detectable correlation between a channel pair, the AR model will catch it. Per-pair + voting is the secret.

How It Works¶

For each pair (a,b):
  compute correlation matrices R₀, R₁
  build 4×4 system matrix A
  eig(A) → candidate freq/damping
     ↓
Pool all candidates (10ch → 45 pairs × 4 = 180 candidates)
     ↓
Cluster: 10% tolerance, ≥ 2 votes → accept

Design Deep Dive¶

1. Why sn=4?¶

AR order sn=4 constructs a 4×4 system matrix per channel pair, capturing up to 2 complex mode pairs. sn=2 can only capture 1 mode; sn=8 causes overfitting on small samples.

2. Pairwise vs Global¶

	Pairwise (ITD)	Global (all ch)
Matrix size	4×4	20×20
Weak modes	Detected	Drowned
Candidates	40	4

ITD's unique strength in weak-mode detection comes from pairwise decomposition.

3. Damping Overestimation¶

AR models overestimate damping 2-10× because noise energy is forcibly fit as AR process variance. Damping from ITD is used only for frequency pairing, never as final output.

4. Edge WDT Handling¶

ITD is the most time-consuming OMA method (2-5s on ESP32 for 5ch×3000). The Watchdog Timer must be temporarily disabled.

Characteristics¶

Strength: best weak-mode detection
Weakness: damping overestimated by 2-10×, unreliable
Best for: complementary cross-validation to SVD methods

When to Use ITD¶

Suspect SVD methods are missing modes
Short data (≥ 500 samples = 10s)
Need independent cross-validation method