# Cramér-Rao lower bound

I have been trying to implement the Cramér-Rao lower bound from the paper - A reference-free time difference of arrival source localization using a passive sensor array (eq. 6 and eq. 7).

\eqalignno{ &{\rm cov}(\hat{d}_{m,n},\hat{d}_{m,n}) \,\,=\,\, {3c^2\over \pi^{2}N\kappa^{2}}{\rho_{m}\rho_{n}\over {\rm SNR}}\left(2+{\rho_{m}\rho_{n}\over{\rm SNR}}\right)&\hbox{(6)}\cr &{\rm cov}(\hat{d}_{i,j},\hat{d}_{k,\ell})\,\,=\,\, {3c^2 \over \pi^{2}N\kappa^{2}}{\rho_{i}\rho_{j}\rho_{k}\rho_{l}\over {\rm SNR}^{2}}\phi_{i,j,k,l}&\hbox{(7)}}

As can be seen the bound is dependent on length of the signal.

Whats the ideal value one must take for the length of the signal?

The bound was derived assuming that $$N$$(length) tends to infinity (reference 8).

So are we supposed to take huge value for $$N$$ like $$10^{21}$$?

• It's a mathematical bound; you can't "implement" it; you could calculate it for a given estimator and a given signal. Could you explain in more detail what you are implementing? Commented Dec 19, 2018 at 17:45

Sounds like you have read reference 8 and understand that for an infinite $$N$$ you can one could theoretically achieve the CRLB. But clearly in practice you need to drop that idea and think about what your constraints are for your problem. That is, do you really need an estimate with such low variance and if you do then how long are you willing to wait? Since increasing $$N$$ means you need more and more samples of your signal to come but if you are doing offline processing then maybe you can afford this and can take a very large $$N$$.