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 $10^{21}$?

  • 1
    $\begingroup$ 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? $\endgroup$ Dec 19 '18 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 $N$.

  • $\begingroup$ My problem is that the mean square error is going below the CRLB. $\endgroup$
    – qwerty
    Dec 19 '18 at 19:46
  • $\begingroup$ Basically I want to implement the first simulation given in the paper. However, the Mean square error line is coming below the CRLB. I do the following 1. Compute the range differences 2. Add Gaussian noise to it with covariance matrix given by equation 6 and 7. 3. Estimate the position and find the error 4. Repeat it for 1000 iterations .. $\endgroup$
    – qwerty
    Dec 19 '18 at 19:55
  • $\begingroup$ If your estimator's variance is better than the CRLB, then it's either not an unbiased estimator (which might be fine!) or your method of finding the variance isn't reliable. $\endgroup$ Dec 19 '18 at 20:18

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