Editing Sensor fusion (section)

== Example calculations ==

Two example sensor fusion calculations are illustrated below.

Let <math>{{x}}_1</math> and <math>{x}_2</math> denote two estimates from two independent sensor measurements, with noise [[variance]]s <math>\scriptstyle\sigma_1^2</math> and <math>\scriptstyle\sigma_2^2</math>
, respectively. One way of obtaining a combined estimate <math>{{x}}_3</math> is to apply [[inverse-variance weighting]], which is also employed within the Fraser-Potter fixed-interval smoother, namely
<ref name="May82">{{cite book | author = Maybeck, S.
 | year = 1982
 | title = Stochastic Models, Estimating, and Control
 | publisher = Academic Press
 | location = River Edge, NJ
 }}</ref>

: <math>{{x}}_3 = \sigma_3^{2} (\sigma_1^{-2}{{x}}_1 + \sigma_2^{-2}{{x}}_2)</math> ,

where <math> \scriptstyle\sigma_3^{2} = (\scriptstyle\sigma_1^{-2} + \scriptstyle\sigma_2^{-2})^{-1}</math> is the variance of the combined estimate. It can be seen that the fused result is simply a linear combination of the two measurements weighted by their respective [[Fisher information|information]]. 

It is worth noting that if <math>{{x}}</math> is a [[random variable]]. The estimates <math>{{x}}_1</math> and <math>{{x}}_2</math> will be correlated through common process noise, which will cause the estimate <math>{{x}}_3</math> to lose conservativeness. <ref>{{Cite book |last=Forsling |first=Robin |url=https://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-199098 |title=The Dark Side of Decentralized Target Tracking : Unknown Correlations and Communication Constraints |date=2023-12-15 |publisher=Linköping University Electronic Press |isbn=978-91-8075-409-5 |series=Linköping Studies in Science and Technology. Dissertations |volume=2359 |location=Linköping |language=en |doi=10.3384/9789180754101}}</ref>

Another (equivalent) method to fuse two measurements is to use the optimal [[Kalman filter]]. Suppose that the data is generated by a first-order system and let <math>{\textbf{P}}_k</math> denote the solution of the filter's [[Riccati equation]]. By applying [[Cramer's rule]] within the gain calculation it can be found that the filter gain is given by:{{Citation needed|date=December 2019|reason=removed citation to predatory publisher content}}

: <math> {\textbf{L}}_k =

\begin{bmatrix}
\tfrac{\scriptstyle\sigma_2^{2}{\textbf{P}}_k}{\scriptstyle\sigma_2^{2}{\textbf{P}}_k + \scriptstyle\sigma_1^{2}{\textbf{P}}_k + \scriptstyle\sigma_1^{2} \scriptstyle\sigma_2^{2}} & \tfrac{\scriptstyle\sigma_1^{2}{\textbf{P}}_k}{\scriptstyle\sigma_2^{2}{\textbf{P}}_k + \scriptstyle\sigma_1^{2}{\textbf{P}}_k + \scriptstyle\sigma_1^{2} \scriptstyle\sigma_2^{2}} \end{bmatrix}.</math>

By inspection, when the first measurement is noise free, the filter ignores the second measurement and vice versa. That is, the combined estimate is weighted by the quality of the measurements.