ACR
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Determination of signal amplitude distribution
You are 100% right! It stems from Shannon's work of 1940s. And I have contacted a dozen of mathematicians all over the world without much success. It is a difficult problem, not in communications but chemistry rather, but doable, hopefully one day.
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Determination of signal amplitude distribution
Let me suggest a problem which I am trying to tackle (certainly not as a student!). Suppose you have a noisy signal that contains a Gaussian peak or it can have multiple peaks from a instrument. It is deterministic in the sense, that we have that data and it is reproducible. The objective is to minimize the information entropy = denoising of this signal. Image processing folks use the histograms (of amplitudes etc.) to calculate entropies of images signals etc. This is where I was trying to go.
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Determination of signal amplitude distribution
Marcus, see my comment to Hilmar, which mentions the entire purpose of this effort.
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Determination of signal amplitude distribution
Hilmar, You are right. I think we have to focus on a time interval otherwise instead of infinite time on both ends. As you stated, we will have a near delta function from far away at 0. My eventual goal of this entire exercise was eventually on how to calculate the (information) entropy, which requires a knowledge of probability amplitudes, of a signal that contains a Gaussian peak with some noise within a given time frame.
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Determination of signal amplitude distribution
Yes, you are right. It is deterministic. If I create a histogram of a Gaussian, it is also like a asymmetric U, with more values near zero, and a small branch on the right near the maximum amplitude values. I am wondering, how to estimate it mathematically.
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Determination of signal amplitude distribution
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Determination of signal amplitude distribution
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Power spectrum scaling conventions
Okay, right. Welch is another story compared to the simple power spectrum.
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Power spectrum scaling conventions
What did you mean by overlap in a power spectrum?
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Power spectrum scaling conventions
Thank you Baddioes, I was interested in scaling power spectrum not power spectrum density, so in power spectrum 1/$N^2$, scaling is correct, provided there is no window.
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Power spectrum scaling conventions
@Jdip Thanks for the useful chapter.
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Calculating Shannon-like entropy function of a 1D signal with random noise
In chemical literature, sadly, optimization functions which are not even close to this entropy definition.
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Calculating Shannon-like entropy function of a 1D signal with random noise
I have copied a large portion of the text from one of the papers here: mathoverflow.net/questions/475196/…
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Calculating Shannon-like entropy function of a 1D signal with random noise
These papers are behind paywall, but they do not use probabilities. This is clear for sure. They simply use the raw data. Their $x$ is apparently nothing but raw data. However, it begs big questions, how did they deal with negative values if they were adding random Gaussian news. My best bet is that they used absolute values because the Shannon-like formula with absolute signal value shows the same trend as the histogram method (ignoring the blips) as I showed in the figure. Since this is an optimization, it may not care if it were true Shannon's entropy.