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what is the relationship between these two things

Perhaps more resolution in a spectrogram is equivalent to knowing more the position of the electron and less resolution is knowing the velocity of the electron.

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    $\begingroup$ Hi! Um, one is fundamental physical theorem (based on a fundamental math theorem), the other is a method of analyzing a measurement. Neither is about electrons in particular (but Heisenberg's uncertainty principle applies to electrons, too): it's not really clear what you're asking for, specifically. Could you edit your question (don't just comment) to specify why you're asking this, and in which context? Without knowing both, this is too unspecific to be answered. $\endgroup$ – Marcus Müller Sep 20 '20 at 10:42
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    $\begingroup$ Also: You're in the very bad habit of asking a question, getting comments and answers, and not reacting at all to them. That's not how this community works. You might want to go through your question history and at least react to the answers (by either accepting them, if they do answer your question, or commenting on them why they don't). Otherwise,I have little trust in the work someone might put in their answer actually benefitting anyone,and that's detrimental to the overall community as it binds resources through low-effort questions with high-effort answers you don't seem to care about. $\endgroup$ – Marcus Müller Sep 20 '20 at 10:47
  • $\begingroup$ Agree with @MarcusMüller here. I was going to explain, but here is link that will require a little studying and a little inference to understand, but I address this connection directly in the "Ideal for a Spectrogram" in dsprelated.com/showarticle/1365.php. Note that "ideal" here is only along a single evaluative criteria, it is not meant to imply a universal ideal. I do not think your observation is applicable. $\endgroup$ – Cedron Dawg Sep 20 '20 at 12:52
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In Quantum Mechanics, state of system specified by a vector (wave function which is a vector in function space), and you could use different basis to represent this vector (imagine one vector in 2 different coordinate system which lead to different components in those coordinate system but they both represent the same vector). For systems composed from moving particles, one basis is the position basis and the second one is momentum basis. The transform between these two basis is the Fourier transform. Considering the wave function give us a probabilistic interpretation, to find the momentum or position of a particle you have to find the average and after that to have a sense about error, you have to find the standard deviation.

The Heisenberg Uncertainty principle, tell us the multiplication of position error and momentum error could not be smaller than some value, for any possible state of system.

Considering this is a property of Fourier transform you could extend that to signals where the signal amplitude and it's spectrum are different representation of same thing in different basis, and say multiplication of effective bandwidth of pulse (as standard deviation of power spectrum around central frequency) and the effective pulse width (as standard deviation of signal's power around it's center in time) could not be smaller than some value for every possible signal.

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  • $\begingroup$ Check out: dsp.stackexchange.com/questions/10429/… $\endgroup$ – Cedron Dawg Sep 22 '20 at 8:05
  • $\begingroup$ @CedronDawg I've read your discussions, and i have to say those are far beyond my knowledge. BTW how electrodynamics is related to Fourier transform eigenfunctions? $\endgroup$ – Mohammad M Sep 22 '20 at 10:22
  • $\begingroup$ MM, I don't know. I'm sort of getting back into Physics in a hobby style after, yikes, almost 40 years. Closest I've come to someone else exploring the connection is this: "Anomaly of the Electromagnetic Duality of Maxwell Theory", Hsieh, et al. $$ $$ <5% Comprehension, but a roadmap. The eigenfunction seems to be the discrete Heisenberg narrowest pulse when viewing that level with a discrete paradigm. $\endgroup$ – Cedron Dawg Sep 22 '20 at 13:13
  • $\begingroup$ My conjecture is this: $$ \text{ There is no such thing as a whirlpool, there is only air and water. } $$ $$ \text{ There is no such thing as a particle, there is only vacuum and ether. } $$ I know they will tell me that 'ether' is an outdated concept. They also said exact frequency formulas weren't possible. I had an insight based on refraction, fluid dynamics, and stable flow patterns. $\endgroup$ – Cedron Dawg Sep 22 '20 at 13:13
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    $\begingroup$ @CedronDawg , very nice. I'm not in a position to suggest scientific topics to you, but when I was an undergraduate student one of my favorite courses was statistical mechanics (a very good book on the subject, statistical physics of particles by Mehran Kardar, it may require some background on thermodynamics). Also another topic was calculus of variations (Calculus of variations by Gelfand, also one chapter of Classical Dynamics of Particles and Systems by Thornton). Personally found these topics very interesting and extremely useful, also related to signal processing. $\endgroup$ – Mohammad M Sep 22 '20 at 14:27
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The standard spectrogram (in the complex and redundant form) is a linear transformation that unfolds a 1D time signal onto a 2D time-frequency space. There, each coefficient represent a "time-times-frequency" square whose dimensions correspond to the standard-deviations or second-order moments of the Weyl-Heisenberg inequality for Fourier analysis.

Higher dimensional versions exist.

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The spectogram is a specific member of a class of tools used for time-freq analysis of signals.

The time-freq analysis of a signal may be performed by a number of tools, but most common is to use a time dependent Fourier transform (or a wavelet transform) which is a Fourier transform applied on a sliding window accross the whole signal length.

Such a time-freq analysis provides not only a spectral view of the signal (inside the window) but also its time information of the events in it.

An event is something a like a pulse, a transient, a time-varying frequency of sinusoid (a modulated signal). These signals carry information both in their frequencies and in their timings.

The time-freq analysis has a frequency-resolution determined by the window length & shape, and a time-resolution determined by the window jumps.

A shorter window will provide a better time localisation (resolution) of a particular event, whereas a longer window will provide a better frequency resolution. The dilemma between the short and long windows for better time and frequency resolutions is reminiscent of a well known dilemma from quantum physics which was stated as Heisenberg's uncertainty principle.

Therefore, it's analogously named as the uncertainty principle of signal processing. But there's no physical link in between the two. It says that you cannot increase time and frequency resolutions simultaneously. If one is increasing, then the other must decrease. And their product, which is constant for a given observation time, is called the capacity of analysis.

Unless you increase the total observation interval (per sampling frequency) and thus increase the capacity of analysis, your frequency & time resolutions will be inversely dependent on the other.

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  • $\begingroup$ This is the erroneous 'Jacobsen' interpretation of the uncertainty principle. You are simply talking about bin widths with that, nothing more. The Heisenberg/Gabor uncertainty has to do with a pulse width going through the FT. A gaussian is the eigenfunction of the CTFT. Make it wider, and the output is narrower and vice versa. For some parameter it is the same. The analogous (should that be discretous?) situation for the DFT is the eigenvectors found in my latest article. $\endgroup$ – Cedron Dawg Sep 23 '20 at 12:36
  • $\begingroup$ @CedronDawg Jacobsen interpretation ??? In signal processing the time-freq localisation uncertainty is about simultaneously resolving the frequency and timing of signals. And that's all about observation window length. Not much interested in your articles. Unfortunately, your lack of experience & literature in the field of (electrical) engineering somehow limits your ability to clearly describe yourself... You are lacking technical clarity. It's hard to distinguish the novel from gimmicks in them. And no one would spend time to do so. It's your duty to use the standard language. $\endgroup$ – Fat32 Sep 23 '20 at 13:24
  • $\begingroup$ Tongue in cheek reference to this commonly held misperception. The term "bin width" is standard, sometimes also called "frequency resolution" which is misleading in itself. I have no such duty. Nor does EE "own" this domain. Jacobsen used it explicitly to explain to me quite clearly that an exact frequency formula is "impossible". Horse/water/drink. Ain't my duty to make the horse drink either. It's there if you want to look at it. $\endgroup$ – Cedron Dawg Sep 23 '20 at 13:31
  • $\begingroup$ @CedronDawg Unfortunately, it's observed increasingly in the latest decades that due to an excessive competition among collegues, unbreakable prejudice among profs, and a lack of sufficient control on the publishers, several hazardous mechanism have emerged, that can publish trash into web and even into the offical papers. So many papers claim doing so many things but they r either fake (gimmick) or they achieve something that's not verifiable by the claims they propose... Day by day it's getting harder to find true authentic fidelity in the scientific field accross the world. $\endgroup$ – Fat32 Sep 23 '20 at 13:33
  • $\begingroup$ That might be true. But you will find fidelity to math in my articles and derivations without reliance on Calculus. And the best frequency formulas in the world, independently tested and verified. To drink, or not to drink, that is the question for you here. tsdconseil.fr/log/scriptscilab/festim/index-en.html Funny thing, when my work is finally recognized as it should have been long ago, august professors are going to have to cite a dog. :-) $\endgroup$ – Cedron Dawg Sep 23 '20 at 13:36
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Maybe this answers your question:

time frequency resolution for TF-Transformations

source: (older thread: Which time-frequency coefficients does the Wavelet transform compute?)

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