# Alpha Beta Filter > Use when you must design and run an alpha-beta tracking filter for a constant-velocity target in SI units: predict the target position and velocity at the next sample time, form the residual from a noisy position measurement, apply the alpha and beta gain update to track position and velocity, and select the steady-state alpha and beta gains from the smoothing factor and the target maneuverability index. Produces the predicted and updated position and velocity, the residual sequence, and the tracking error metrics that gate a target tracking assessment. Trigger: alpha beta filter, tracking filter, constant velocity target, steady state gains, smoothing factor, maneuverability index, position tracking, velocity tracking. - Skill: `ashfordeou/alpha-beta-filter` (Agent Skill, multi-file: 3 files) - Install (CLI): `npx skillmds@latest add ashfordeou/alpha-beta-filter` - Raw SKILL.md: https://api.skillmd.com/api/skills/ashfordeou/alpha-beta-filter/raw - Safety review: pending - Works with: Claude Code, Claude.ai, OpenAI Codex - Category: Coding & Dev Tools - License: Apache-2.0 - Author: ashfordeOU (https://skillmd.com/u/ashfordeou) - Updated: 2026-09-17 - Page: https://skillmd.com/skills/ashfordeou/alpha-beta-filter --- # Alpha-Beta Filter (gnc-autonomy/estimation-filtering/alpha-beta-filter) Use when the task is an alpha-beta tracking filter for a constant-velocity target: predicted position and velocity, the alpha-beta gain update from a noisy position measurement, and the steady-state gain selection from the smoothing factor and the maneuverability index. ## Domain quick reference - Target model: constant velocity between samples. State is (position x, velocity v); each sample interval dt in seconds. - Predict step: x_pred = x + dt * v and v_pred = v; the velocity persists and the position advances by dt * v. - Residual: r = z - x_pred, the difference between the noisy position measurement z and the predicted position. - Update step: x_new = x_pred + alpha * r and v_new = v_pred + (beta / dt) * r. alpha weights the position correction, beta weights the velocity correction; both gains are dimensionless constants. - Gain ranges: stable for 0 <= alpha < 2 and 0 <= beta < 4 - 2*alpha. With alpha = 1 and beta = 1 the updated position equals the raw measurement, so the filter tracks the measurement exactly. - Steady-state gains from the smoothing factor (Benedict-Bordner, critical damping): beta = alpha^2 / (2 - alpha) for alpha in (0, 2). Smaller alpha smooths more and lags a maneuvering target; larger alpha tracks faster and passes more measurement noise. - Steady-state gains from the maneuverability index (Kalata tracking index): lambda = sigma_w * dt^2 / sigma_v, formed from the target process noise sigma_w, the sample interval dt, and the measurement noise sigma_v. The critical-damping gains follow the radical closed forms in the logic module; lambda -> 0 gives gains (0, 0) and lambda -> infinity gives gains tending to (1, 2). - Tracking error metrics: root-mean-square error and maximum absolute error between the true and estimated position sequences. - Units: position in the tracked unit (meters), velocity in unit per second (m/s), dt in seconds, gains dimensionless. - ARP4754A (reference-only) frames development assurance for aircraft systems; the alpha-beta filter is common tracking-filter knowledge (Kalata, Benedict and Bordner). ## Workflow 1. Set the sample interval dt and the target model: constant velocity between measurements. 2. Choose the gains. Either pick the smoothing factor alpha and derive beta with steady_state_gains(alpha), or form the maneuverability index lambda = sigma_w * dt^2 / sigma_v and derive both gains with gains_from_tracking_index(lambda). 3. Predict with predict(x, v, dt); confirm the position advanced by dt * v and the velocity held. 4. Compute the residual with residual(z, x_pred) and apply the update with update(x_pred, v_pred, z, dt, alpha, beta), which returns the residual, updated position, and updated velocity. 5. Run the full cycle with step(x, v, z, dt, alpha, beta), or filter a whole measurement batch with run_tracker(measurements, dt, alpha, beta, x0, v0) and inspect the position, velocity, residual, and predicted-position trajectories. 6. Assess the result with tracking_errors(true_positions, estimates); compare the RMSE and maximum error against the measurement-noise level. 7. For stateful use, keep a TrackFilter instance and call its step(z) method per measurement; the filter holds x, v, and the last residual. ## Pitfalls - Using gains outside the stable region; alpha >= 2 or beta >= 4 - 2*alpha makes the filter diverge, so the module raises ValueError. - Confusing the smoothing factor with the maneuverability index; alpha is a dimensionless gain in (0, 2), lambda = sigma_w * dt^2 / sigma_v is a noise ratio, and the two connect through the steady-state gain formulas. - Feeding a non-positive sample interval dt; the module raises ValueError, and a zero dt makes the velocity correction blow up. - Forgetting the dt factor in the velocity update; the residual is divided by dt so the velocity unit stays unit per second. - Expecting the filtered position to equal the measurement; with alpha < 1 the filter lags the measurement by design and trades lag against noise smoothing. - Mixing units in one state; positions and velocities must share one unit set (meters and meters per second). ## Behavior contract (gate 3) The predict, residual, update, batch tracker, steady-state gain selection, tracking error metrics, and the stateful TrackFilter are exercised by the gate 3 contract test: scripts/test_alpha_beta_filter.py against scripts/alpha_beta_filter_logic.py (stdlib unittest, offline). Run: python3 scripts/test_alpha_beta_filter.py ## Compliance - ARP4754A is proprietary (SAE); name and paraphrase only per standards-map.yaml, reference-only: true. - compliance: STANDARDS-REF, gated: false.