Talk:Wigner distribution function

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I just changed the caption of the image so that it does not advertises an unknown method. SinPantuflas (talk) 09:39, 7 November 2017 (UTC)[reply]

If it is indeed an autocorrelation that is meant, then it is not a time-reversed copy. Auto correlation uses the cross-correlation (shifted), not the convolution (time-reversed, shifted). The current article text essentially suggests one operation, but explains the other. I am not so experienced with integral transforms, but it appears to indeed be a time-reversed, shifted copy based on the definition given. That means that is more like an autoconvolution. This term exists, but doesn't have a Wikipedia page. Any other mathematicians care to weigh in?--75.80.43.80 (talk) 23:39, 18 April 2011 (UTC)[reply]

I do not see how it can be that "The WDF is essentially the Fourier transform of the input signal’s autocorrelation function". The link to autocorrelation shows that the autocorrelaiton itself is an integral. None of it appears in the definition of the WDF. What about removing this sentence? — Preceding unsigned comment added by 130.246.132.178 (talk) 14:34, 12 September 2013 (UTC)[reply]

I fixed this part of the article. Lots of stuff could be added to this article, like references to Ville and sections on negativity and the uncertainty principle. But I don't agree with merging this with the QM article, because the wave function and a time series are two very different things with very different interpretations... even though there will necessarily be many redundant/analogous relations. CHF (talk) 18:45, 19 September 2013 (UTC)[reply]

The article claims "the Wigner distribution function provides the highest possible temporal vs frequency resolution which is mathematically possible within the limitations of uncertainty in quantum wave theory. " without giving a source. I do not understand how a mathematical operation can be bound by some physical theory. I doubt this claim. I guess there is some lower bound one can show for a class of transformations which is achieved by the wigner distribution function and that this bound also occurs somewhere in the quantum wave theory. — Preceding unsigned comment added by 2A01:C22:8842:E400:15C:A911:1061:A3B3 (talk) 16:53, 25 April 2020 (UTC)[reply]

I suspect your are misreading the admittedly loose statement. There is no physics involved, and the allusion to "quantum wave theory" is just shorthand for the uncertainty principle, a property of Fourier analysis, so pure math. It' just that most readers recognize the inequality as a quantum mechanical principle. Mere language. Cuzkatzimhut (talk) 17:59, 25 April 2020 (UTC)[reply]

Perhaps you could quantify this idea (or point to a source) for the mathematically challenged among us. I can intuit that the resolution of the spectral content vs. time for the Wigner Transform would be much better than say the straightforward Short Time Fourier Transform method but could you help quantify that? Sdwehe (talk) 15:27, 28 July 2022 (UTC)[reply]

Integrating from negative infinity to positive infinity is not difficult at all, its easier than the finite case actually. The article should avoid saying something is difficult because that is subjective 132.147.144.113 (talk) 22:23, 12 May 2024 (UTC)[reply]

Currently it is said: "

Projection property

| x ( t ) | 2 = ∫ − ∞ ∞ W x ( t , f ) d f | X ( f ) | 2 = ∫ − ∞ ∞ W x ( t , f ) d t {\displaystyle {\begin{aligned}|x(t)|^{2}&=\int _{-\infty }^{\infty }W_{x}(t,f)\,df\\|X(f)|^{2}&=\int _{-\infty }^{\infty }W_{x}(t,f)\,dt\end{aligned}}}

" But this could be misinterpreted, since both integral are the same it is saying that

| x ( t ) | 2 = | X ( f ) | 2 {\displaystyle |x(t)|^{2}=|X(f)|^{2}}

which makes sense only thinking about the absolute values as norms comparing two values, since the directly interpretation as absolute values gives two functions that aren't the same at all.

Consider using instead: "

Projection property

‖ x ( t ) ‖ 1 2 = ∫ − ∞ ∞ W x ( t , f ) d f ‖ X ( f ) ‖ 1 2 = ∫ − ∞ ∞ W x ( t , f ) d t {\displaystyle {\begin{aligned}\|x(t)\|_{1}^{2}&=\int _{-\infty }^{\infty }W_{x}(t,f)\,df\\\|X(f)\|_{1}^{2}&=\int _{-\infty }^{\infty }W_{x}(t,f)\,dt\end{aligned}}}

" where will show without ambiguity that

‖ x ( t ) ‖ 1 2 = ‖ X ( f ) ‖ 1 2 {\displaystyle \|x(t)\|_{1}^{2}=\|X(f)\|_{1}^{2}}

But following Plancherel theorem I think the real norms it should be used are:

"

Projection property

‖ x ( t ) ‖ 2 2 = ∫ − ∞ ∞ W x ( t , f ) d f ‖ X ( f ) ‖ 2 2 = ∫ − ∞ ∞ W x ( t , f ) d t {\displaystyle {\begin{aligned}\|x(t)\|_{2}^{2}&=\int _{-\infty }^{\infty }W_{x}(t,f)\,df\\\|X(f)\|_{2}^{2}&=\int _{-\infty }^{\infty }W_{x}(t,f)\,dt\end{aligned}}}

" where will show the true statement and without ambiguity that

‖ x ( t ) ‖ 2 2 = ‖ X ( f ) ‖ 2 2 {\displaystyle \|x(t)\|_{2}^{2}=\|X(f)\|_{2}^{2}}

But I am not familiarized with the Winger transform, so I left to you the validity of the results, but should consider to be specific about the norm you are using. 191.115.171.3 (talk) 19:22, 25 October 2024 (UTC)[reply]

I am not familiar with it either, but maybe you did not notice the two integrals of the projection property are about two different variables? The two integrals are not necessarily equal.

David, but not Hilbert (talk) 14:27, 27 October 2024 (UTC)[reply]