03. Signal Processing & 3D Kalman Filter

Diátaxis Type: Explanation & Reference | Status: Canonical | Review: Continuous

This document covers the Digital Signal Processing (DSP) tier in Bio-Quant: 3D Kalman state estimation, sensor jitter attenuation, velocity spike rejection, and Kovatchev risk-space transformations.

---

1. 3D Kalman Filter Mathematical Formulation

The core signal estimator models physiological glucose kinetics as a 3-dimensional continuous-time kinematic system tracking position (\(g\)), velocity (\(v = \dot{g}\)), and acceleration (\(a = \ddot{g}\)):

\[\mathbf{x}_k = \begin{bmatrix} g_k \\ v_k \\ a_k \end{bmatrix} \in \mathbb{R}^3\]

State Transition Matrix (\(\mathbf{F}\)) & Process Noise Covariance (\(\mathbf{Q}\))

For a discrete sampling interval \(\Delta t = 5 \text{ minutes}\):

\[\mathbf{F} = \begin{bmatrix} 1 & \Delta t & \frac{1}{2}\Delta t^2 \\ 0 & 1 & \Delta t \\ 0 & 0 & 1 \end{bmatrix}\]

\[\mathbf{Q} = q \cdot \begin{bmatrix} \frac{\Delta t^5}{20} & \frac{\Delta t^4}{8} & \frac{\Delta t^3}{6} \\ \frac{\Delta t^4}{8} & \frac{\Delta t^3}{3} & \frac{\Delta t^2}{2} \\ \frac{\Delta t^3}{6} & \frac{\Delta t^2}{2} & \Delta t \end{bmatrix}\]

Measurement Model & Innovation

The continuous glucose monitor only measures blood glucose concentration (\(g\)):

\[\mathbf{H} = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix}, \quad R = \sigma_{sensor}^2\]

\[\tilde{\mathbf{y}}_k = z_k - \mathbf{H} \hat{\mathbf{x}}_{k|k-1}\]

\[\mathbf{S}_k = \mathbf{H} \mathbf{P}_{k|k-1} \mathbf{H}^T + R\]

\[\mathbf{K}_k = \mathbf{P}_{k|k-1} \mathbf{H}^T \mathbf{S}_k^{-1}\]

\[\hat{\mathbf{x}}_{k|k} = \hat{\mathbf{x}}_{k|k-1} + \mathbf{K}_k \tilde{\mathbf{y}}_k\]

\[\mathbf{P}_{k|k} = (\mathbf{I} - \mathbf{K}_k \mathbf{H}) \mathbf{P}_{k|k-1}\]

Ponytail Resilience Invariant: If \(\mathbf{S}_k\) approaches singularity (\(\det(\mathbf{S}_k) < 10^{-12}\)), the filter falls back to Moore-Penrose pseudo-inversion (np.linalg.pinv) to prevent LinAlgError crashes during sensor connection transients.

---

2. Biological Velocity Spike & Jitter Rejection

Continuous glucose sensors occasionally suffer from compression lows, electrostatic discharge, or rapid interstitial fluid pressure shifts. The DSP pipeline applies biological acceleration bounds:

1. Maximum Biological Velocity: \(|v| \le 0.35 \text{ mmol/L/min}\) (\(6.3 \text{ mg/dL/min}\)).
2. Maximum Biological Acceleration: \(|a| \le 0.05 \text{ mmol/L/min}^2\).
3. Artifact Flagging: Readings exceeding physiological bounds trigger an increased sensor measurement variance (\(R \leftarrow 10 \cdot R\)), forcing the filter to rely on metabolic inertia until clean signals resume.

---

3. Kovatchev Risk-Space Transformation

Raw blood glucose measurements are logarithmically asymmetric: a drop from \(5.0\) to \(3.0 \text{ mmol/L}\) presents extreme immediate mortality risk, whereas an equivalent numerical rise from \(5.0\) to \(7.0 \text{ mmol/L}\) is clinically benign.

The Kovatchev transform projects glucose into a symmetric risk space:

\[f(g) = 1.509 \cdot \left( \ln(g)^{1.084} - 5.381 \right)\]

\[r(g) = 10 \cdot f(g)^2\]

\[\text{LBGI} = \begin{cases} r(g) & \text{if } f(g) < 0 \\ 0 & \text{otherwise} \end{cases}, \quad \text{HBGI} = \begin{cases} r(g) & \text{if } f(g) > 0 \\ 0 & \text{otherwise} \end{cases}\]

\[\text{Risk Index (RI)} = \text{LBGI} + \text{HBGI}\]

flowchart LR
    RAW["Raw Glucose (g)<br/>[1.0 - 25.0 mmol/L]"] --> LOG_TRANS["Kovatchev Transform<br/>f(g)"]
    LOG_TRANS --> LBGI_BOX["Low Blood Glucose Index (LBGI)<br/>Hypoglycemia Risk"]
    LOG_TRANS --> HBGI_BOX["High Blood Glucose Index (HBGI)<br/>Hyperglycemia Risk"]
    LBGI_BOX & HBGI_BOX --> RI["Composite Risk Index (RI)"]