# Cross-Correlation Signal Delay Estimation Variance

I am working on a project which intends to use Time Difference of Arrival (TDoA) for localization.

Firstly, my understanding is that a matched filter is the most common method for estimating signal-delay.

To get performance metrics for TDoA, I need the variance of the time-delay estimate. Thus, what I'm looking for is a way to calculate the variance in the signal-delay estimate using a matched filter as an estimator.

This seems like it would be a very common problem; however, I have consulted three books and searched online, and I have not found anything which addresses this issue.

One warning first: this is going to be a bit literature-heavy, because I think that's what you're more or less explicitly ask for. You want something that you can read, understand and cite for your publication.

Firstly, my understanding is that a matched filter is the most common method for estimating signal-delay.

No. A matched filter can be used to effectively correlate a receive signal against a transmit signal shape. It's can but be a puzzle piece in a TDOA system.

Thus, what I'm looking for is how to calculate the variance in the signal-delay estimate using a matched filter as an estimator.

Again, the matched filter is not the delay estimator; think of it as a signal preconditioner!

Now, what you probably mean is that you use matched filters for the TX pulse shape and the RX pulse shape filter, and then, based on the post-filter signal, do a delay estimate.

I'd like to cite a paper,[1] :

which doesn't actually deal with matched filtering directly, as the TDoA system isn't able to define the pulse shape the transmitter uses; instead, it's about cross-correlation of the received signals at different positions.

The interesting point is that the defining entity (aside from the obvious $\frac{c}{f_\text{Nyquist}}$ limit) is the SNR of the received signal -- that's pretty obvious, if you think about it: The "certainty" with which the correlation of RX and TX signal has a maximum at the "right" position is reduced with increased SNR.

In previous work of the author, she uses in [2] that with AWGN assumption and cross-correlation (i.e. matched filter with a known TX signal)

\begin{align*} \sigma^2_{\hat\tau}&=\frac{3}{8\pi^2}\,\frac{1+2\, \text{SNR}}{\text{SNR}^2}\, \frac {1}{T_o B^3}\text,\,\text{with}\\ \sigma^2_{\hat\tau}&: \text{Variance of the timing estimate}\\ T_o&: \text{Duration of observation, and}\\ B&: \text{bandwidth of the system.} \end{align*}

Interesting here, aside from the $B^{-3}$ relationship of variance and bandwidth, is the cited paper (if you find it in fulltext available online for free; I have IEEExplore access, but I don't know about you...) [9], which pretty much should give you plenty of background to work with.

[1]Noha El Gemayel, Holger Jäkel, Friedrich K. Jondral: Error Analysis of a Low Cost TDoA Sensor Network, IEEE/ION Position, Location and Navigation Symposium 2014, Monterey, CA, USA, May 5-8, 2014

[2]Noha El Gemayel, Sebastian Koslowski, Friedrich K. Jondral: A low cost TDOA Localization System: Setup, Challenges and Results, , 10th Workshop on Positioning, Navigation and Communication 2013 (WPNC'13), Dresden, Germany, March 20-21, 2013

[9] G. Carter, Coherence and time delay estimation, Proceedings of the IEEE, vol. 75, no. 2, pp. 236 – 255, Feb. 1987

• Thank you for your thorough comment. I understand that a matched filter is not, in and of itself a signal-delay estimator. To be more precise, the signal-delay estimator is the time at which the maximum output from a matched filter is found. But I think you understood that's what I'm after. I'd have to look at your citations in more depth; however, since I posted this I've been doing some Monte Carlo Simulations, and discovered that the distribution of the positions of matched filter maximums is not Gaussian. Furthermore, the distribution is a function of the signal type (squre vs chirp). – Stephen Hartzell Feb 8 '16 at 21:49
• "function of the signal type": probably result of smaller SNR for the square? – Marcus Müller Feb 8 '16 at 21:53