Measuring an Event — Question 9

Question 9

Radar time and range have different propagated uncertainties

For an immediate echo from positive x, $t_E=(t_++t_-)/2$ and $x_E=c(t_+-t_-)/2$. The two timestamp errors are independent, each with standard deviation $\sigma=20\,\mathrm{ns}$. Use $c=3.00\times10^8\,\mathrm{m/s}$.

1. Find the standard uncertainties in event time and range.

2. Calculate their covariance and contrast a common clock-offset error with independent timestamp noise.

Question 9 - Solution

Radar time and range have different propagated uncertainties

(a) Method and calculation

$$\sigma_t^2=\tfrac14(\sigma^2+\sigma^2),\quad\boxed{\sigma_t=14.14\,\mathrm{ns}}.$$$$\sigma_x^2=\tfrac{c^2}4(\sigma^2+\sigma^2),\quad\boxed{\sigma_x=4.24\,\mathrm m}.$$

(b) Interpretation and further calculation

$$\operatorname{Cov}(t_E,x_E)=\tfrac c4(\sigma_+^2-\sigma_-^2)=\boxed{0}.$$$$t_\pm\to t_\pm+b:\quad\boxed{t_E\to t_E+b,\quad x_E\to x_E}.$$

Physical check

Standard deviations combine in quadrature under independence; a shared systematic offset cancels from the range.