<p>Disclaimer: This may be considered off-topic. I apologise if this is the case, but I really could not find a better place to post this.</p>
<p>In general, a signal can be built from a combination of parameters. A simple example is a Gaussian distribution of the form-</p>
<p><span class="math-container">$f(x) = \frac{1}{\sigma \sqrt{2\pi}}e^{(\frac{x - \mu}{\sigma})^2}$</span></p>
<p>which can be always represented by <span class="math-container">$\mu$</span> and <span class="math-container">$\sigma$</span>.</p>
<p>In the specific problem I am working on, I have a differential equation that describes the propagation of a signal starting from a Gaussian pulse. But I don't really care about the signal itself, but care only about the pulse width.</p>
<p>The pulse width of a signal carries &quot;less&quot; information about the signal than the signal itself. By this, I mean the value of the pulse width is hidden within the signal, but the signal cannot be described from the knowledge of the pulse width alone.</p>
<p>This makes me think that it would be justified if an algorithm could be developed that can track the propagation of the pulse width alone which is computationally inexpensive as compared to track the propagation of the pulse itself and then calculate the pulse width from it.</p>
<p>Am I right in saying this? My initial guess is no. But I do not know why it is wrong to say so.</p>
<p>As for tags, I am not sure what this question could be tagged under. Going with a generic &quot;information&quot; tag for now. Please let me know if I have to add any others.</p>
<p>Edit:</p>
<p>My input is a Gaussian signal. I would like to see how the pulse width of the signal changes when it propagates through a system described by a partial differential equation. My current method is to see how the signal itself propagates and then calculate the pulse width from the signal. I would rather like to see how only the pulse width propagates without actually seeing how the signal propagates. Intuitively, I feel that it should be easier to compute the pulse width as a function instead of the entire signal as the pulse width alone has lesser information than the entire signal.</p>
<p>The differential equation I had in mind is the Nonlinear Schrodinger Equation (NLSE).</p>
<p>Although an algorithm would be great, that is <strong>not</strong> what I am asking. <strong>I am asking if it is possible to prove that such an algorithm exists or doesn't exist in the general case where the propagation of a single parameter of a signal can be tracked with lesser computation than the computation required for propagating the entire signal.</strong></p>

CC BY-SA 4.0
2020-11-10T18:35:37.167Z
Can you make your question more concrete?  What is the computational task you want to solve?  What does it mean to "track the propagation of .." as a computational task - what are the inputs and what is the desired output?  I suspect the answer will depend on what specific differential equation you are working with so I am not sure you are getting to get an answer in general.  Also, what exactly is your question?  Are you asking for such an algorithm?  Right now your question appears to be asking whether such an algorithm would be desirable, for which it seems you already know the answer.
D.W.
CC BY-SA 4.0
2020-11-11T07:04:07.957Z
@D.W., I hope the edits clarify my question.
Paddy
<information>
Computing specific information of a signal propagation instead of entire signal
Paddy
# Computing specific information of a signal propagation instead of entire signal

(Asked by: username_0 on 2020-11-10)

Disclaimer: This may be considered off-topic. I apologise if this is the case, but I really could not find a better place to post this.

In general, a signal can be built from a combination of parameters. A simple example is a Gaussian distribution of the form-

$f(x) = \frac{1}{\sigma \sqrt{2\pi}}e^{(\frac{x - \mu}{\sigma})^2}$

which can be always represented by $\mu$ and $\sigma$.

In the specific problem I am working on, I have a differential equation that describes the propagation of a signal starting from a Gaussian pulse. But I don't really care about the signal itself, but care only about the pulse width.

The pulse width of a signal carries "less" information about the signal than the signal itself. By this, I mean the value of the pulse width is hidden within the signal, but the signal cannot be described from the knowledge of the pulse width alone.

This makes me think that it would be justified if an algorithm could be developed that can track the propagation of the pulse width alone which is computationally inexpensive as compared to track the propagation of the pulse itself and then calculate the pulse width from it.

Am I right in saying this? My initial guess is no. But I do not know why it is wrong to say so.

As for tags, I am not sure what this question could be tagged under. Going with a generic "information" tag for now. Please let me know if I have to add any others.

Edit:

My input is a Gaussian signal. I would like to see how the pulse width of the signal changes when it propagates through a system described by a partial differential equation. My current method is to see how the signal itself propagates and then calculate the pulse width from the signal. I would rather like to see how only the pulse width propagates without actually seeing how the signal propagates. Intuitively, I feel that it should be easier to compute the pulse width as a function instead of the entire signal as the pulse width alone has lesser information than the entire signal.

The differential equation I had in mind is the Nonlinear Schrodinger Equation (NLSE).

Although an algorithm would be great, that is **not** what I am asking. **I am asking if it is possible to prove that such an algorithm exists or doesn't exist in the general case where the propagation of a single parameter of a signal can be tracked with lesser computation than the computation required for propagating the entire signal.**

**Tags:** information
> **Comments:**
> **username_1**: Can you make your question more concrete?  What is the computational task you want to solve?  What does it mean to "track the propagation of .." as a computational task - what are the inputs and what is the desired output?  I suspect the answer will depend on what specific differential equation you are working with so I am not sure you are getting to get an answer in general.  Also, what exactly is your question?  Are you asking for such an algorithm?  Right now your question appears to be asking whether such an algorithm would be desirable, for which it seems you already know the answer.  
> **username_0**: @D.W., I hope the edits clarify my question.  

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