Cramér–Rao Bound (CRB) calculation when considering NLOS interference w/ multiple targets/clutter

Question

I have a question about the CRB when considering NLOS cultters interference.

Let $x[0] = A + w[0]$ where $w[0]\sim \mathcal N(0,\sigma^2)$, we would first define it's pdf and then calculate the logarithm of it, which is $$\log f_{x[0]}=-\frac{(x[0] - A)^2}{2\sigma^2}.$$

Here, $A$ is the unknwon parameter for a given value of $x[0]$.

However, according to papers I studied, e.g. Power allocation strategies for target localization, it seems the CRB calculation only LOS condition that no clutters exist.

When the model is NLOS, for example there are multiple targets required localization, and the waveform transmitted to target $A$ and reflected to receiver, might also be firstly reflected to target $B$ and then reflected from target $B$ to receiver, how do I consider this kind of interference in the CRB calculation?

For what I think is, the sample $x[0] = A + w[0]$ would be turned into $x[0] = A + w[0+1]$ where $w[0]\sim \mathcal N(0,\sigma ^2+ \Sigma^2)$, where $\Sigma^2$ is caused by the NLOS clutters.

Therefore the CRB calculation would be $$\log f_{x[0]}=-\frac{(x[0] - A)^2}{2(\sigma^2+\Sigma^2)}.$$

But I am not sure if this idea is correct for the CRB calculation when considering NLOS clutters.

Answer 1

It doesn't sound the right thing to me – because multiple reflections are fundamentally different from noise, in that their contribution is not independent from your transmission.

For example, when you set $A=0$, then you should only get noise, no matter what the clutter/multipath environment looks like. Thus, adding a constant to your white noise variance must be wrong.

You simply need a different channel model. What you're describing isn't inherently NLOS – you only imply there's an NLOS component. So, you'd model this as Rician channel instead! That would mean that your reception would contain two independent random variables: a random variable describing your noise, as before, with a variance that is constant, and another one describing the random distribution of the power you receive.