Often we need to estimate the time difference of arrival between two signals to find the location of a target. Many algorithms gives the time delay corresponding to a sample number or time delay is a function of the sampling frequency. To get away with this problem subsample time delay estimation procedure is used. One can see that if the sampling frequency is very high the subsample may not be necessary. Can anyone help to find when subsample is necessary and when it become irrelevant? I am trying to understand the other associated factor.
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Even though the signals are sampled you can get accuracy which is well above the accuracy offered by the samples as long as you sample using Nyquist.
Actually, Using the Matched Filter you can achieve the CRLB (Cramer Rao Lower Bound) for Delay Estimation (Easy to derive for white noise).
If you calculate the CRLB for Time Delay Estimation you'll see it depends on the Signal BW and the SNR and not the sampling frequency (Given Nyquist sampling).
How can that be achieved?
You can interpolate the the cross correlation signal and by simple math infer the time delay in resolutions well beyond the sampling resolution.
For instance, with signals with BW of ~ 15 MHz we shall sample by 60 Mhz.
The sampling resolution (Assuming RADAR) is ~ 2.5 [Meter] yet the CRLB at decent SNR will give you a bound of few Centi Meters which can be achieved using the Matched Filter + Interpolation.
Enjoy.
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$\begingroup$ Could you clarify more? Using prior parametric model of the signal you can estimate the delay without any interpolation of the whole signal. $\endgroup$– RoyiCommented Jul 13, 2015 at 15:21
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$\begingroup$ The problem is related to the link below. Question is at what sampling frequency these methods are not necessary. Matched filter is one method which uses the original signal. mathworks.com/matlabcentral/fileexchange/… $\endgroup$– CreatorCommented Jul 13, 2015 at 20:24
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$\begingroup$ The methods in the link are the same ones I talked about. Namely you can assume the Cross Correlation (the MF) is symmetric around its peak and hence you can use a prior model to locally interpolate it and have a sub sample accuracy. Such prior can be Gaussian, Parabolic or any symmetric curve. $\endgroup$– RoyiCommented Jul 14, 2015 at 14:11
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$\begingroup$ Question is when this is not required? At sampling frequency? $\endgroup$– CreatorCommented Jul 14, 2015 at 17:46
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$\begingroup$ There is no way having the perfect solution (Namely if the signals were continuous and no noise is there and you have the perfect replicate of the echo signal) with no interpolation (Locally or globally) besides the zero probability case where the delay matches integer factor of the sampling interval. $\endgroup$– RoyiCommented Jul 14, 2015 at 20:20