What is a distinct feature of an ambiguous result?

[tworitdash]

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

Thanks in advance.

 

[FactChecker]

IMO, you are making a common mistake in your use of the Nyquist limit. Sampling at the Nyquist rate will allow you to precisely determine the amplitude of any lower frequency if the sampling time is infinite. Any finite sampling time leaves uncertainty in the amplitude of lower frequencies. I once saw a formula for the error limit, but I have never been able to find a reference again.

 

[tworitdash]

IMO, you are making a common mistake in your use of the Nyquist limit. Sampling at the Nyquist rate will allow you to precisely determine the amplitude of any lower frequency if the sampling time is infinite. Any finite sampling time leaves uncertainty in the amplitude of lower frequencies. I once saw a formula for the error limit, but I have never been able to find a reference again.

Thank you for the response. When you say uncertainty in the amplitude of lower frequencies, do you by chance refer to the resolution in frequency domain ?

 

[tworitdash]

IMO, you are making a common mistake in your use of the Nyquist limit. Sampling at the Nyquist rate will allow you to precisely determine the amplitude of any lower frequency if the sampling time is infinite. Any finite sampling time leaves uncertainty in the amplitude of lower frequencies. I once saw a formula for the error limit, but I have never been able to find a reference again.

I now understand what you mean.

 

[tworitdash]

I got confused with the terminology. So, I am pasting an edit here.

`` 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. "

The rest of the question remain the same.

 

[FactChecker]

Parts of your question confuse me. Your talk about velocities and Fibonacci numbers and "ambiguity" just confuses me. I recommend that you simplify your question, clarify what your basic question is, and use precise terminology. For instance, by "ambiguity" do you mean "error"?
Maybe I am just unfamiliar with the subject matter that you are referring to.

 

[tworitdash]

Parts of your question confuse me. Your talk about velocities and Fibonacci numbers and "ambiguity" just confuses me. I recommend that you simplify your question, clarify what your basic question is, and use precise terminology. For instance, by "ambiguity" do you mean "error"?
Maybe I am just unfamiliar with the subject matter that you are referring to.

The signal processing terms that I use are usual in practice. For example, "ambiguity" means multiple solutions for the same problem. It is not an error. It can also be said as "grating lobes" in some other fields in science. The paragraph containing the explanation of irregular sampling (for example with Fibonacci numbers) are just an example of how people deal with aliasing in general. This is not part of the question.

All I wanted to ask is if there is a fundamental way to distinguish the ambiguities if the sampling interval (time between consecutive samples) is constant. In short, how to de-alias and pick the real solution.

 

[FactChecker]

All I wanted to ask is if there is a fundamental way to distinguish the ambiguities if the sampling interval (time between consecutive samples) is constant. In short, how to de-alias and pick the real solution.

There is no fundamental mathematical way. Given a finite sample time, there is no basic mathematical way to completely eliminate aliasing. The subject matter determines the limits of the frequencies of interest. The subject matter also determines much accuracy you need for the amplitude of those frequencies. Given those subject matter requirements, the sampling rate must be high enough to satisfy them.
For instance, in aerodynamics, you would have one sampling rate if you were only worried about the flight of a massive airplane. You would have a much faster sampling rate if you were worried about the flutter and vibration of much less massive control surfaces. Those sampling rates are determined by the frequencies of interest, given the physics of the problem.

 

[tworitdash]

There is no fundamental mathematical way. Given a finite sample time, there is no basic mathematical way to completely eliminate aliasing. The subject matter determines the limits of the frequencies of interest. The subject matter also determines much accuracy you need for the amplitude of those frequencies. Given those subject matter requirements, the sampling rate must be high enough to satisfy them.
For instance, in aerodynamics, you would have one sampling rate if you were only worried about the flight of a massive airplane. You would have a much faster sampling rate if you were worried about the flutter and vibration of much less massive control surfaces. Those sampling rates are determined by the frequencies of interest, given the physics of the problem.

Thank you for the reply with the example. I also thought so. That is why I mentioned it as a "silly" question. :D However, I just wanted to ask it here. I have been surprised (especially in the past year in my research) sometimes with fundamentals that completely changed some of my understanding related to mathematics and physics.

 

[mgeorge001]

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

Thanks in advance.

Thanks for the good question. Addressing such a question may go beyond applied math or engineering. New types of phenomena, for example, might arise outside the range (both microscopic and macroscopic) of sensors, for example, especially in adversarial games where someone may be trying to deceive you. Therefore, one may require additional sensor data, or advances in basic science, engineering or mathematics. People typically have spent many years setting up the engineering frameworks in which you have experience, and it can be costly or impractical to address the ambiguities. So, strangely, a lot about your question is related to social factors, and how disruptive the adaptations would be: If they are real painful for your employers, better do it yourself on your own time thinking outside the box (which can be tough if we do not even know what the box is.)

 

[FactChecker]

You might be interested in some techniques for combining the information from sensors that are better at different frequency ranges.
The complimentary filter allows you to combine the output of two sensors, one that is accurate at low frequencies and does not drift but is not good for high frequencies with another which is good for high frequencies but which might slowly drift.
Kahlman filters allow you to combine the outputs of several sensors with different strengths and weaknesses.

 

[mgeorge001]

You might be interested in some techniques for combining the information from sensors that are better at different frequency ranges.
The complimentary filter allows you to combine the output of two sensors, one that is accurate at low frequencies and does not drift but is not good for high frequencies with another which is good for high frequencies but which might slowly drift.
Kahlman filters allow you to combine the outputs of several sensors with different strengths and weaknesses.

I like this perspective. Thanks.

 

[tworitdash]

Thank you for the responses. There are many de-aliasing techniques out there that I’m aware of as well. And, as you said, the only way to do it is to use some additional information.

In an image, it is the frequency estimate in each pixel for example. People try to find sharp jumps and try many engineering techniques to address this issue. Sensor fusion is also another way to do it. I call them engineering ways because it’s not very fundamental.

The more scientific way I came across and have used also in practice was the irregular irrational sampling procedures that try to increase the Nyquist limit to a very high value. This is like prevention and not exactly a cure. I love this idea of aperiodic sampling and it gives a lot of philosophical perspective as well.

However, as I already mentioned, I was looking for something more in case of a periodic sampling itself. That is why I called the question very silly. I guess, after the discussion we had, I call it a dead end.

 

[mgeorge001]

I appreciated the question, and actually think it is really a very good question. Also, recognizing what can be done in terms of applied math and engineering is important, whether or not one is able, in the moment to understand the ways in which to go beyond the established knowledge. Certainly, first, one must know what the box is, before one can think outside the box. So I really think your question is great. It is only a dead end if we seek solutions within known math and engineering. In practice, due to social constraints and our ignorance of our times, it may not be practical to get the advances, the needed information, and understand how to go about getting past bottlenecks. I thought the questions and the responses people wrote were edifying.

 

[tworitdash]

I appreciated the question, and actually think it is really a very good question. Also, recognizing what can be done in terms of applied math and engineering is important, whether or not one is able, in the moment to understand the ways in which to go beyond the established knowledge. Certainly, first, one must know what the box is, before one can think outside the box. So I really think your question is great. It is only a dead end if we seek solutions within known math and engineering. In practice, due to social constraints and our ignorance of our times, it may not be practical to get the advances, the needed information, and understand how to go about getting past bottlenecks. I thought the questions and the responses people wrote were edifying.

Thank you for the appreciation. :)

 

[tworitdash]

I know this thread is old, but as we had nice discussions here, I wanted to share that I published a journal paper in Geoscience and Remote Sensing finally regarding this topic. I discuss extensively about the literature that addresses ambiguity / aliasing from different perspectives. In addition, I also discuss theoretical Nyquist limits for aperiodic sequences. Then, I go on and present a sampling procedure that helps in avoiding aliasing, but with a suitable frequency domain conversion approach. The trick is to have more precision in measuring the time instances of the samples in the sequence. I compare my frequency domain conversion approach with the DFT periodogram with aperiodic sequences. I don't want to share all the details otherwise the post will be very long.

If you are looking at this comment and are interested in the topic, you will like it a lot I am sure. After this paper was out, later I compared with other sampling techniques using some 1D Fibonacci or Thue-Morse tiling, but so far the log-periodic one works the best.

Read it here:
Counter-Aliasing Paper