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{{WikiProject Statistics|importance=low}}
{{WikiProject Mathematics|importance=low}}
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== example ==
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"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. <!-- Template:Unsigned IP --><small class="autosigned">— Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[Special:Contributions/2A01:C22:8842:E400:15C:A911:1061:A3B3|2A01:C22:8842:E400:15C:A911:1061:A3B3]] ([[User talk:2A01:C22:8842:E400:15C:A911:1061:A3B3#top|talk]]) 16:53, 25 April 2020 (UTC)</small> <!--Autosigned by SineBot-->
: 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. [[User:Cuzkatzimhut|Cuzkatzimhut]] ([[User talk:Cuzkatzimhut|talk]]) 17:59, 25 April 2020 (UTC)
::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? [[User:Sdwehe|Sdwehe]] ([[User talk:Sdwehe|talk]]) 15:27, 28 July 2022 (UTC)
== "masking" ==
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 [[Special:Contributions/132.147.144.113|132.147.144.113]] ([[User talk:132.147.144.113|talk]]) 22:23, 12 May 2024 (UTC)
== Issues with definitions of the projection property section ==
Currently it is said: "
; Projection property
: <math>\begin{align}
|x(t)|^2 &= \int_{-\infty}^\infty W_x(t,f)\,df \\
|X(f)|^2 &= \int_{-\infty}^\infty W_x(t,f)\,dt
\end{align}</math>
"
But this could be misinterpreted, since both integral are the same it is saying that
: <math>|x(t)|^2 = |X(f)|^2</math>
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
: <math>\begin{align}
\|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{align}</math>
"
where will show without ambiguity that
: <math>\|x(t)\|_1^2 = \|X(f)\|_1^2</math>
But following [[Plancherel theorem]] I think the real norms it should be used are:
"
; Projection property
: <math>\begin{align}
\|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{align}</math>
"
where will show the true statement and without ambiguity that
: <math>\|x(t)\|_2^2 = \|X(f)\|_2^2</math>
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. [[Special:Contributions/191.115.171.3|191.115.171.3]] ([[User talk:191.115.171.3|talk]]) 19:22, 25 October 2024 (UTC)
: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.
:[[User:David, but not Hilbert|David, but not Hilbert]] ([[User talk:David, but not Hilbert|talk]]) 14:27, 27 October 2024 (UTC)
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