# Minimum time delay that can be estimated between two sensors

Consider a 1-dimensional toy problem. I have two sensors at different points along the $x$-axis. Somewhere away from the sensors, a disturbance is created which travels towards the sensors, first reaching sensor 1 and then sensor 2 at a later time. Both sensors produce samples over the entire time window, $W$. Both sensors are sampling at the same rate $f$. Using cross-correlation or other methods, what is the smallest time delay I can resolve?

My hunch is that the answer is given by the Cramer-Rao bound. If so, can someone give a simple intuition on how this bound is derived?

Thanks!

• In the textbook case where the the noiseless signal template is known, and the received signal is a shifted version of the template signal plus additive white Gaussian noise, the CRLB is derived in Kay's estimation theory book Ch. 3. It is given by $\frac{1}{SNR \cdot BW^2}$. You'll have to modify that analysis for your situation where the "template" is not known and you have two noisy signals with a relative shift. – Atul Ingle Jan 23 '18 at 22:20
• A plane wave impinging broadside, to two sensors will have a zero time delay offset. You can register zero delay. I think you're asking what is the accuracy of delay estimate. The CRLB is the high SNR bound on accuracy, assuming an unbiased estimator. It isn't useful at low SNR. It isn't useful in telling you where the threshold SNR is. It is relatively easy to calculate. Intuitively, the CRLB translates the flatness of the log likelihood. (2nd partial derivative) of the estimator at the true value and translates that into a variance. – user28715 Jan 24 '18 at 17:44
• Thanks for your responses - @StanleyPawlukiewicz, I was imagining a different geometry but my original question was not written clearly. I've updated the text to make the setup more explicit. – RedPanda Jan 24 '18 at 23:06