How to find the bias gradient for localization problem?

Question

I am doing the work to find the cramer-rao bound when the estimator is biased. The algorithm I am based on is from Rethinking Biased Estimation: Improving Maximum Likelihood and the Cram´er–Rao Bound, and it is also mentioned in Wiki in the "Bound on the variance of biased estimators" section.

According to the algorithm, e.g. a single observation from the binomial probability distribution Bin(m,p), m is known but p is unknown, if the biased estimator is $g(X) = \frac{X+1}{m+2}$, the bias would be $b\{g(X)\} = E\{g(X)\} - p = \frac{mp+1}{m+2} - p = \frac{1-2p}{m+2}$, therefore its bias gradient is $B\{g(X)\} = \frac{\partial b\{g(X)\}}{\partial p} = \frac{-2}{m+2}$.

However, when it comes to localization question, to me it seems I could not follow the above example. Because the observed siganl becomes $r = \alpha s(t - \tau)+ w$ and $\tau = \frac{\sqrt{(x - x_0)^2 + (y - y_0)^2}}{c}$, which $x, y$ are values wanted to be estimated and the others are assumed known, $w$ is zero-mean. The problem is, the observed signal is consisted the desired position values, not a direct function like $g(X) \propto x$ in the example above.

So my question is, with observed signal that is not directly related to the wanted values, how to find its bias gradient? For my idea is like to calculate $b(x,y) = E\{\alpha s(t - \frac{\sqrt{(x - x_0)^2 + (y - y_0)^2}}{c})\} - [x,y]$, however I have no idea how to formula the true $(x,y)$ in the equation.

Thank you.

Answer 1

You're mixing a couple of things up here. Forget about what distribution you're using and just consider what you're actually measuring.

In a typical radar system, you will measure the range, radial velocity (Doppler), and if able the angle(s) of arrival of the target. When you receive the signal and process it, you perform detection where you attempt to measure range, Doppler, and angle.

In your expression, you're taking the expected value of the signal itself, which doesn't yield the information you want. During detection, you measure the range $R$ (delay), Doppler $f_D$, and angles ($az$ and $el$) directly.

It is these measured values that are used for final location estimation and eventually tracking. In your 2D case (let's assume azimuth is the angle you're concerned with and not considering Doppler), you would convert from these radial coordinates $(R, az)$ to $(x, y)$. From there, you can calculate statistics using sequential measurements. As a matter of fact, since you can estimate the SNR of a single return, you have some statistical information about how accurate a single measurement is, which is useful to inform your sample position estimate (e.g., confidence bounds, etc.) and eventually a tracker.