Question

Two sensors have measurement errors that are Gaussian distributed with zero means and variances ${\sigma }_1^{2}and{\sigma }_2^2,$ respectively. The two sensor measurements $x_{1}and{x}_2$ are combined to form the weighted average $x=\alpha x_1+(1-\alpha )x_2,0\le \alpha \le 1.$ Assuming that the measurement errors of the two sensors are uncorrelated, the weighting factor $\alpha$ that yields the smallest error variance of x is

• A. $\frac{{\sigma }_2^2}{{\sigma }_1^2+{\sigma }_2^2}$
• B. $\frac{{\sigma }_1^2}{{\sigma }_1^2+{\sigma }_2^2}$
• C. $\frac{{\sigma }^2}{{\sigma }_1+{\sigma }_2}$
• D. 0.5

Answer 1

The correct option is A $\frac{{\sigma }_2^2}{{\sigma }_1^2+{\sigma }_2^2}$

$\frac{{\sigma }_2^2}{{\sigma }_1^2+{\sigma }_2^2}$