Sensor Filtering: Alpha-Beta, Kalman, and Complementary

Visual: predict-correct filtering loop comparing alpha-beta, Kalman covariance gain, and complementary high/low-frequency split.

Simple filters are often the right tool when the state, noise, and latency requirements are clear. Alpha-beta filters, Kalman filters, and complementary filters all combine prediction with measurement correction, but they make different assumptions about dynamics, uncertainty, and frequency content.

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Related docs

• Bayesian Filtering and Error-State Kalman Filters
• Information Filters and Smoothers
• Sampling, FFT, Windowing, and Filtering
• Sensor Likelihoods, Noise, and Error Budgets
• Time Sync, PTP, Timestamping, and Latency Models

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Why it matters for AV, perception, SLAM, and mapping

Not every signal needs a full nonlinear smoother. Object range rates, wheel speed, low-level radar tracks, IMU attitude priors, lane boundary smoothing, and diagnostic residuals often need low-latency filtering with predictable behavior. The wrong simple filter, however, can add delay, suppress real maneuvers, or hide faults.

The practical question is: what states are being estimated, what dynamics are assumed, and how much lag can the autonomy stack tolerate?

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Core math and algorithm steps

Alpha-beta filter

The alpha-beta filter is a fixed-gain position-velocity tracker for roughly constant velocity motion. State:

x = position
v = velocity
dt = sample interval

Predict:

x_pred = x + v * dt
v_pred = v

Residual:

r = z - x_pred

Update:

x = x_pred + alpha * r
v = v_pred + (beta / dt) * r

Larger alpha and beta follow measurements more aggressively. Smaller gains smooth more but lag and under-react to maneuvers. The alpha-beta filter can be viewed as a steady-state Kalman-like tracker with fixed gains.

Kalman filter

For linear-Gaussian dynamics:

x_k = F x_{k-1} + B u_k + w_k,   w_k ~ N(0, Q)
z_k = H x_k + v_k,               v_k ~ N(0, R)

Prediction:

x_pred = F x + B u
P_pred = F P F^T + Q

Update:

y = z - H x_pred
S = H P_pred H^T + R
K = P_pred H^T S^-1
x = x_pred + K y
P = (I - K H) P_pred

Unlike alpha-beta filtering, Kalman gain adapts from covariance, process noise, and measurement noise.

Complementary filter

A complementary filter fuses sensors by frequency content. For attitude, gyro integration is good at short time scales but drifts; accelerometer gravity is noisy under vehicle acceleration but stable long term for roll/pitch.

One simple scalar form:

angle_gyro = angle_prev + gyro_rate * dt
angle = alpha * angle_gyro + (1 - alpha) * angle_accel

where:

alpha = tau / (tau + dt)

This is equivalent to high-pass behavior on the gyro-integrated estimate and low-pass behavior on the absolute reference. Nonlinear complementary filters on SO(3) generalize this idea to rotations without treating orientation as a flat Euclidean vector.

Choosing the filter

Filter	Use when	Avoid when
Moving average	visualization or simple denoising	control or tracking needs low lag
Alpha-beta	constant-velocity scalar/vector tracking	noise varies strongly or maneuvers dominate
Kalman	linear state with meaningful Q and R	non-Gaussian or multi-modal belief dominates
EKF/ESKF	nonlinear geometry with local Gaussian errors	poor initialization or severe nonlinearity
Complementary	sensors separate by frequency band	biases and dynamics are not observable

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Implementation notes

• Always compute filters at measurement timestamps. Variable dt must be part of the update.
• Track filter delay and expose it in the timestamp or latency budget.
• Initialize state and covariance explicitly; startup transients are often mistaken for sensor faults.
• Use innovation gating for Kalman-style updates.
• Tune alpha-beta gains against maneuver response and noise rejection, not just a smooth-looking plot.
• Use complementary filters for attitude only with clear assumptions about acceleration and magnetic disturbances.
• Publish health signals: innovation, gain, covariance, saturation, dropped samples, and reset reason.

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Failure modes and diagnostics

Failure mode	Symptom	Diagnostic
Excessive lag	Filtered object or speed trails true motion.	Step response and phase delay test.
Under-modeled acceleration	Position residual grows during maneuvers.	Residual correlates with acceleration or yaw rate.
Wrong Q/R ratio	Kalman filter either jitters or ignores measurements.	NIS distribution and gain trace.
Complementary filter drift	Heading or roll slowly biases.	Compare against static and dynamic reference intervals.
Magnetometer corruption	Yaw jumps near steel, motors, aircraft, or chargers.	Magnetic norm and innovation monitoring.
Reset discontinuity	Downstream planner sees state jump.	Reset reason and covariance inflation audit.
Hidden dropout	Filter coasts smoothly through missing data.	Expose measurement age and update count.

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Sources

• Welch and Bishop, "An Introduction to the Kalman Filter": https://www.cs.unc.edu/~welch/media/pdf/kalman_intro.pdf
• Kalman, "A New Approach to Linear Filtering and Prediction Problems": https://www.cs.unc.edu/~welch/kalman/media/pdf/Kalman1960.pdf
• MathWorks alpha-beta filter documentation: https://www.mathworks.com/help/fusion/ref/trackingabf.html
• NIST, "Equivalency of LMSF and Alpha-Beta-Gamma Filters": https://www.nist.gov/publications/equivalency-lmsf-and-alpha-beta-gamma-filters
• Mahony et al., "Nonlinear Complementary Filters on the Special Orthogonal Group": https://core.ac.uk/download/pdf/52787176.pdf
• Madgwick, "An efficient orientation filter for inertial and inertial/magnetic sensor arrays": https://www.x-io.co.uk/res/doc/madgwick_internal_report.pdf
• SciPy signal filtering documentation: https://docs.scipy.org/doc/scipy/reference/signal.html