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This question comes from my experience in radar signal processing. As I am going more deep into the theory of sampling, statistical signal processing and estimation theory in general, I have a very silly but important mathematical question that I want to ask here.

For example, we are estimating a physical quantity after getting some information out of a sensor that doesn't directly measure the quantity. The signal from the sensor is sampled in time domain and based on the sampling interval, the quantity of interest lies within a certain bounds (typically this bound is inversely proportional to the sampling time).

For example, to keep it simple, let's say that the estimated quantity is a Doppler velocity. Due to the sampling, we can only measure for example from $-v$ to $+v$ meters per second in frequency/ velocity domain. If the true velocity of the target lies inside this limit, we can measure the correct velocity.

If we increase our viewing window further, we will see many copies of the same velocity. These are known as ambiguities.

On the other hand, if the true velocity is outside this limit of $-v$ to $+v$, we can still see a copy of it in the limit $-v$ to $+v$. In this case, the solution we have is ambiguous and the true solution is outside our viewing window. In this case, we say it is aliased.

These exist because of the sampling I mentioned above and a quantity like velocity is wrapped and scaled in the signal as an angle and an angular variable has limit from $-\pi$ to $+\pi$ and any solution that is outside this limit is a replica of the real solution. Only one of them is true.

It is like saying $$ \cos(2\pi + \theta) = \cos(\theta) $$

I know that as it is a fundamental limitation and people use irregular sampling intervals (with more fancy patterns like Fibonacci, log normal and so on..) to essentially increase the physical limit of the variable to avoid folding. Irrational sampling intervals also may increase this maximum limit to a very high value. In this case, there is only one solution.

However, I was wondering if these ambiguities can be distinguished somehow in a more fundamental way from the real one in the case of regular sampling. Do these ambiguities have a distinct feature that can make them separable from the real one ? I know it is a very silly question, because they are essentially the multiple solutions to the same question. However, it may happen that my understanding is incomplete.

In short: How to de-alias and get the true value of the quantity in the case of periodic sampling when the true value is outside the Nyquist limit?

Thanks in advance.

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    $\begingroup$ To be honest, I feel your question (explicit in the last paragraph) needs clearance. You must get to the point, what exactly you are asking for. In general, the Nyquist-Shannon sampling theorem (sample at a rate at least twice the highest frequency present in the signal) is the criteria. $\endgroup$ Commented Apr 25, 2023 at 8:24
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    $\begingroup$ I made an edit with the exact point that I wanted to ask. $\endgroup$
    – CfourPiO
    Commented Apr 25, 2023 at 22:41
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    $\begingroup$ Look into what varying the PRF can do, along with the Chinese remainder theorem. $\endgroup$ Commented Apr 25, 2023 at 23:16
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    $\begingroup$ I personally believe that when the true value would be outside the Nyquist limit, and one has merely periodic sampling without any additional information, it is not possible to de-alias and recover the true value, without additional information (to de-entropy). There is an information loss that cannot be recovered. The reason is that aliasing causes higher-frequency constituents of the signal to fold and superposition with the lower-frequency components. Either you must find a way to add information directly or with techniques such as Multiple Signal Classification or ESPRIT. $\endgroup$ Commented Apr 26, 2023 at 7:46

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