Optimal estimation of the fusion of two measurements

Question

Suppose I have a sensor measuring a quantity $\text R$. For example the sensor could be a radar estimating the range of a target. We can write: $$R(t)=r(t)+\nu_0(t)$$ where $r(t)$ is the real range and $\nu_0(t)$ the noise that we assume to be gaussian: $\nu_0(t)=\mathcal{N(0,\sigma_0)}$. Another sensor, for example another radar, estimates the same quantity $R(t)$, but the associated noise is now: $\nu_1(t)$ with $\nu_1(t)=\mathcal{N(0,\sigma_1)}$. What is the best estimation I can obtain from the combined information about range? Thanks in advance.

Answer 1

Suppose we have two independent range measurements $R_0,R_1$ of the range $r$ with normal noise $\nu_0(t)=\mathcal{N(0,\sigma_0^2)}$ and $\nu_1(t)=\mathcal{N(0,\sigma_1^2)}$, the Likelihood $\mathcal L(r)$ is: $$\mathcal{L}(r)=p(R_0,R_1|r)=p(R_0|r)p(R_1|r)$$ that means: $$\mathcal{L}(r)\sim\exp\left(\frac{-(R_0-r)^2}{2\sigma_0^2}\right)\times\exp\left(\frac{-(R_1-r)^2}{2\sigma_1^2}\right)$$ The negative Log Likelihood is then: $$-\ln\mathcal{L}(r)=\frac{1}{2}(\sigma_0^{-2}+\sigma_1^{-2})\times\left[r-\dfrac{\sigma_0^{-2}R_0+\sigma_1^{-2}R_1}{\sigma_0^{-2}+\sigma_1^{-2}}\right]$$ which is maximized when: $$r_{MLE}=\left[\dfrac{\sigma_0^{-2}R_0+\sigma_1^{-2}R_1}{\sigma_0^{-2}+\sigma_1^{-2}}\right]$$

Answer 2

First, you will need to detect the signal. For that you have several choices. You have the radar example here, so if you transmit a pulse $p(t)$, then you can apply matched filtering (which maximizes the output SNR). To do this, you need a local template which can be $p(t)$. The output of matched filter is $$x_1=\int_0^TR_1(\tau)p(t-T+\tau)d\tau,$$ which will generally lead to two terms corrsponding to the signal and noise as $$x_1=r_1+n_1.$$If by "optimal" you mean an estimate with minimum mean square error (MMSE), then it is simply the conditional expectation of the joint density of your observations, given their values. If your posterior density is $$f_{Y|X_1,X2}(y|x_1,x_2),$$ then the optimal estimate is $$\hat{y}(x_1,x_2)=E\{Y|X_1=x_1,X_2=x_2\}.$$