Sensor Fusion with Normally Distributed Noise
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.
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.