Alpha Beta Filter — Robotics/Perception Filters

Alpha-Beta Filter

Robotics / Perception Filters

The g-h tracker: takes a noisy position measurement and produces a smoothed position and the velocity implied by it.

• predict – xp = x + v·Ts
• residual – r = z − xp
• correct – x = xp + α·r, v = v + (β/Ts)·r

It is the steady-state Kalman filter for the constant-velocity model – the same structure with the gains frozen rather than propagated. There is no covariance to carry and two scalars to tune, which is the right trade whenever the noise statistics are not known well enough to earn the full recursion.

Ports

• z – the measured position, a scalar, in whatever unit the estimate should carry (metres, radians, counts).
• estimate – a [2,1] column (position, velocity). The velocity is in units per second. Its size is fixed, not inherited.

Parameters

• Alpha (position gain) – how hard the residual pulls the POSITION, in [0, 1] for a sensible filter. 1 snaps to every measurement (no smoothing); 0 ignores measurements entirely and coasts on the model.
• Beta (velocity gain) – how hard the residual pulls the VELOCITY. Larger tracks acceleration faster and amplifies noise more. Stability wants 0 < β < 4 − 2α; a common starting point is β ≈ α²/(2−α), which is the critically damped choice.
• Initial Position / Initial Velocity – the estimate at t = 0. A good initial velocity matters more than a good initial position: the filter corrects position quickly and velocity slowly.
• Sampling Time (s) – zero or less inherits the solver's rate; a positive value runs the block at that period. It must resolve to a positive value, since β/Ts divides by it.

Code export

All ten targets: Python, MATLAB, Java, Rust, C, C++, VHDL, Verilog, SystemVerilog and PLC Structured Text. α, Ts and the folded β/Ts are baked into the generated body at export time rather than exposed as tunable parameters.

The three HDL targets are genuinely synthesizable Q16.16: the per-sample work is four multiplies and four adds, and the one division is folded away at export.

Simulink bridge

None, measured rather than assumed: the tracking filters ship in the Sensor Fusion and Tracking Toolbox, which is not installed on this machine.

Notes

• Stateful and inherently discrete. The state is the (position, velocity) pair.
• The output is the estimate at the START of the step, published before the predict-correct pass, so the first sample is exactly the configured seed and a measurement first appears in the output at k+1.
• It assumes CONSTANT VELOCITY between samples. Under a sustained acceleration a both estimates settle to a constant lag, and the two are not the same size: position lags by a·Ts²·(1−α)/β and velocity by a·Ts·(2α−β)/(2β). Both follow from the steady-state residual r = a·Ts²/β – the value the velocity correction must see each step for the estimate to gain a·Ts. The position lag carries Ts² and is usually the small one; the velocity lag carries Ts and is what a tracker notices first. That is the model's error, not the filter's, and the cure is an alpha-beta-gamma tracker rather than a larger gain.
• β is scaled by Ts inside the block, so the same pair of gains means the same filter at any rate. The exported core carries β/Ts already folded.
• For a full Kalman filter – propagated covariance, time-varying gains – use State Estimation / Kalman Filter.