# Sensor Fusion with Normally Distributed Noise

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Given unbiased sensor measurements of a scalar quantity, each corrupted by independent, normally distributed noise with variance , the likelihood function for the *true* value (purple, dashed) is also a normal density.

Contributed by: Aaron Becker (June 2015)

University of Houston

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Combining sensor measurements is called *sensor fusion*. If each sensor measurement is corrupted by independent noise, each measurement provides additional information. The *information* of a measurement is equal to the inverse of its variance.

The measurements are combined according to Bayes's theorem. Given two measurements and with variances and , the best estimate for the true value is with variance .

This can be written in recursive form, to update estimate with measurement to form :

,

,

.

These recursive equations are the scalar Kalman filter equations, and is also known as the *optimal Kalman gain. *If only one measurement exists, the combined measurement is identical to the original measurement.

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