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When working on the attached file the preview locks up and shows nothing anymore. This occures for all other files after that and can only be reset by reloading vscode.

EDIT There seems to be much more wrong as I'm getting this after some time when trying to create a new note: connect ECONNREFUSED 127.0.0.1:62760

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File:

---
id: TeQCnGhHwfj2zC1PnZMtV
title: Wigner-Ville-Verteilung
desc: ''
updated: 1643652565548
created: 1643199807029
tags: 
     - msv
---

Sowohl die [[fourier-transformation]], ihre Modifikation, die [[fourier-transformation.kurzzeit-fourier-transformation]], als auch die [[wavelet-transformation]] sind lineare Transformationen, da hier die Funktion $x(t)$ linear in die Transformation eingeht. 
Nachteilig ist dabei, dass zur Bildung eines zweidimensionalen Bildes (der Zeit-Frequenz-Ebene) die Funktion $x(t)$ immer auf einen finiten Bereich eingeschränkt werden muss und es somit aufgrund der Unschärfe im Bildbereich (der [[Zeit-Frequenz-Unschärfe|zeitdauer-bandbreite-produkt]]) zum Leckeffekt kommt. 
Man führt deshalb zur Verhinderung dieser Probleme die *Wigner-Ville-Verteilung*, die eigentlich eine Transformation ist, ein. 
Sie besitzt keine Fensterfunktionen oder Wavelets, wodurch es nicht zum Leckeffekt kommt und sie somit eine optimale Spektralauflösung für alle Abbildungen auf das zweidimensionale Bild (der Zeit-Frequenz-Ebene) besitzt. 
Es handelt sich dabei um eine quadratische Integraltransformation, bei der $x(t)$ quadratisch eingeht. 

## Definition

Die *Wigner-Ville-Verteilung* ist die zweidimensionale Fourier-Transformierte der [[wigner-ville-verteilung.ambiguitätsfunktion]] $A_{xx}(\tau, \vartheta)$ bezüglich der der Verschiebungen $\tau$ und $\vartheta$: 

$$
\begin{aligned}
    W_{xy}(t,f) &= \left\langle A_{xy}(\tau, \vartheta), \exp\left(-\mathrm j 2 \pi \vartheta t\right)\exp\left(-\mathrm j 2 \pi \tau f\right)\right\rangle_{\vartheta, \tau} \\
    &= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} A_{xy}(\tau, \vartheta) \exp\left(-\mathrm j 2 \pi \vartheta t\right)\exp\left(-\mathrm j 2 \pi \tau f\right) \,\mathrm d \vartheta\, \mathrm d \tau \\
\end{aligned}
$$

## Abgeleitete Darstellungen

![[wigner-ville-verteilung.ambiguitätsfunktion#abgeleitete-lokale-korrelation,1:#*]]

Mit Hilfe der Definition der lokalen Autokorrelation und des Satzes von Parseval kann die Wigner-Ville-Verteilung wie folgt dargestellt werden:

$$
\begin{aligned}
    W_{xy}(t,f) &= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} A_{xy}(\tau, \vartheta) \exp\left(-\mathrm j 2 \pi \vartheta t\right)\exp\left(-\mathrm j 2 \pi \tau f\right) \,\mathrm d \vartheta\, \mathrm d \tau \\
    &= \int_{-\infty}^{\infty} x\left(t + \frac{\tau}{2}\right)y^*\left(t - \frac{\tau}{2}\right)\exp\left(-\mathrm j 2 \pi \tau f\right) \, \mathrm d \tau \\
    &= \mathcal F_{\tau} \left\{x\left(t + \frac{\tau}{2}\right)y^*\left(t - \frac{\tau}{2}\right)\right\}
\end{aligned}
$$

bzw.: 

$$
\begin{aligned}
    W_{xy}(t,f) &= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} A_{xy}(\tau, \vartheta) \exp\left(-\mathrm j 2 \pi \vartheta t\right)\exp\left(-\mathrm j 2 \pi \tau f\right) \,\mathrm d \vartheta\, \mathrm d \tau \\
    &= \int_{-\infty}^{\infty} X\left(f - \frac{\vartheta}{2}\right)Y^*\left(f + \frac{\vartheta}{2}\right)\exp\left(-\mathrm j 2 \pi \vartheta t\right) \, \mathrm d \vartheta \\
    &= \mathcal F_{\vartheta} \left\{X\left(f - \frac{\vartheta}{2}\right)Y^*\left(f + \frac{\vartheta}{2}\right)\right\}
\end{aligned}
$$

Die Zusammenhänge lassen sich über die Fourier-Transformation wie folgt darstellen: 

![](/assets/images/2022-01-26-17-46-12.png){max-width: 45em, display: block, margin 0 auto}

## Eigenschaften

### Kojugation

$$
W_{xy}(t,f) = W_{yx}^*(t,f)
$$

#### Beweis

$$
\begin{aligned}
     W_{yx}^*(t,f) &= \int_{-\infty}^{\infty} x^*\left(t + \frac{\tau}{2}\right)y\left(t - \frac{\tau}{2}\right) \exp\left(\mathrm j 2 \pi \tau f\right) \, \mathrm d \tau && \tau^{\prime}= \tau \\
     &= \int_{-\infty}^{\infty} x^*\left(t - \frac{\tau^{\prime}}{2}\right)y\left(t + \frac{\tau^{\prime}}{2}\right) \exp\left(-\mathrm j 2 \pi \tau^{\prime} f\right) \, \mathrm d \tau^{\prime} && \tau^{\prime}= \tau \\
     &= W_{xy}(t,f)
\end{aligned}
$$

### Reellwertigkeit

Die Auto-Wigner-Ville-Verteilung ist reellwertig: 

$$
W_{xx}(t,f) = \operatorname{Re}\{W_{xx}(t,f)\} = W_{xx}^*(t,f)
$$

#### Beweis

Dies folgt direkt aus der Konjugationseigenschaft. 

<!-- ToDo: Rein mathematische Formulierung -->

### Frequenz-Randbedingung

Die Integration der Auto-Wigner-Ville-Verteilung über die Zeit ergibt das Energiedichtespektrum:

$$
\int_{-\infty}^{\infty} W_{xx}(t,f) \,\mathrm dt = S_{xx}^E(f) = |X(f)|^2
$$

#### Beweis

$$
\begin{aligned}
    \int_{-\infty}^{\infty} W_{xx}(t,f) \,\mathrm dt &= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} x\left(t + \frac{\tau}{2}\right)x^*\left(t - \frac{\tau}{2}\right) \exp\left(-\mathrm j 2 \pi \tau f\right) \, \mathrm d \tau \,\mathrm dt && t^{\prime} = t - \frac{\tau}{2} \\
    &= \int_{-\infty}^{\infty} \left(\int_{-\infty}^{\infty} x\left(t^{\prime} + \tau\right)x^*\left(t^{\prime}\right) \, \mathrm d t^{\prime} \right) \exp\left(-\mathrm j 2 \pi \tau f\right) \,\mathrm d \tau \\
    &= \int_{-\infty}^{\infty} r_{xx}^E(\tau) \exp\left(-\mathrm j 2 \pi \tau f\right) \,\mathrm d \tau \\
    &= S_{xx}^E(f) \\
    &= |X(f)|^2
\end{aligned}
$$

### Zeit-Randbedingung

Die Integration der Auto-Wigner-Ville-Verteilung über die Frequenz ergibt die Energiedichte:

$$
\int_{-\infty}^{\infty} W_{xx}(t,f) \,\mathrm df = s_{xx}^E(t) = |x(t)|^2
$$

#### Beweis

$$
\begin{aligned}
    \int_{-\infty}^{\infty} W_{xx}(t,f) \,\mathrm df &= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} X\left(f - \frac{\vartheta}{2}\right)X^*\left(f + \frac{\vartheta}{2}\right) \exp\left(-\mathrm j 2 \pi \vartheta t\right) \, \mathrm d \vartheta \,\mathrm df && f^{\prime} = f - \frac{\vartheta}{2} \\
    &= \int_{-\infty}^{\infty} \left(\int_{-\infty}^{\infty} X\left(f^{\prime} - \vartheta\right)X^*\left(f^{\prime}\right) \, \mathrm d f^{\prime} \right) \exp\left(-\mathrm j 2 \pi \vartheta t\right) \,\mathrm d \vartheta \\
    &= \int_{-\infty}^{\infty} \rho_{xx}^E(\vartheta) \exp\left(-\mathrm j 2 \pi \vartheta t\right) \,\mathrm d \vartheta \\
    &= s_{xx}^E(t) \\
    &= |x(t)|^2
\end{aligned}
$$

### "Energieerhaltung"

Die Integration der Auto-Wigner-Ville-Verteilung über Zeit und Frequenz ergibt die Signalenergie: 

$$
\int_{-\infty}^{\infty} W_{xx}(t,f) \,\mathrm dt \,\mathrm df = \int_{-\infty}^{\infty} S_{xx}^E(f) \,\mathrm df = \int_{-\infty}^{\infty} s_{xx}^E(t) \,\mathrm dt = E_x
$$

### Translationserhaltend

Die Auto-Wigner-Ville-Verteilung einer verschobenen Funktionen $x^{\prime}(t) = x(t-t_0)$ ist gleich der verschobenen Auto-Wigner-Ville-Verteilung der ursprünglichen Funktion $x(t)$: 

$$
\begin{aligned}
    W_{x^{\prime}x^{\prime}}(t,f) &= \int_{-\infty}^{\infty} x\left(t - t_0 + \frac{\tau}{2}\right)x^*\left(t - t_0- \frac{\tau}{2}\right) \exp\left(-\mathrm j 2 \pi \tau f\right) \, \mathrm d \tau \\
    &= W_{xx}(t-t_0,f)
\end{aligned}
$$

### Modulationserhaltend

Die Auto-Wigner-Ville-Verteilung einer modulierten Funktionen $x^{\prime}(t) = x(t)\exp\left(\mathrm j 2 \pi f_0 t\right)$ ist gleich der modulierten (und damit in $f$-Richtung verschobenen) Auto-Wigner-Ville-Verteilung der ursprünglichen Funktion $x(t)$: 

$$
\begin{aligned}
    W_{x^{\prime}x^{\prime}}(t,f) &= \int_{-\infty}^{\infty} x\left(t + \frac{\tau}{2}\right)\exp\left(\mathrm j 2 \pi f_0 \left(t + \frac{\tau}{2}\right)\right) x^*\left(t - \frac{\tau}{2}\right)\exp\left(\mathrm j 2 \pi f_0 \left(t - \frac{\tau}{2}\right)\right) \exp\left(-\mathrm j 2 \pi \tau f\right) \, \mathrm d \tau \\
    &= \int_{-\infty}^{\infty} x\left(t + \frac{\tau}{2}\right) x^*\left(t - \frac{\tau}{2}\right) \exp\left(-\mathrm j 2 \pi \tau (f - f_0)\right) \, \mathrm d \tau \\
    &= W_{xx}(t,f-f_0)
\end{aligned}
$$

### Symmetrieerhaltend

Die Auto-Wigner-Ville-Verteilung einer reellen symmetrischen Funktionen $\gamma(t) = \gamma^*(t) = \gamma(-t)$ ist gleich der gespiegelten Auto-Wigner-Ville-Verteilung der ursprünglichen Funktion $\gamma(t)$: 

$$
W_{\gamma\gamma}(t,f) = W_{\gamma\gamma}(-t,-f) 
$$

#### Beweis

Der Beweis erfolgt in zwei Teilen: 

$$
\begin{aligned}
    W_{\gamma\gamma}(t,f) &= \int_{-\infty}^{\infty} \gamma\left(t + \frac{\tau}{2}\right)\gamma^*\left(t - \frac{\tau}{2}\right) \exp\left(-\mathrm j 2 \pi \tau f\right) \, \mathrm d \tau && \gamma(t) = \gamma(-t) \\
    &= \int_{-\infty}^{\infty} \gamma\left(-t - \frac{\tau}{2}\right)\gamma^*\left(-t + \frac{\tau}{2}\right) \exp\left(-\mathrm j 2 \pi \tau f\right) \, \mathrm d \tau && \gamma(t) = \gamma^*(t) \\
    &= \int_{-\infty}^{\infty} \gamma\left(-t + \frac{\tau}{2}\right)\gamma^*\left(-t - \frac{\tau}{2}\right) \exp\left(-\mathrm j 2 \pi \tau f\right) \, \mathrm d \tau \\
    &= W_{\gamma\gamma}(-t,f)
\end{aligned}
$$

sowie aufgrund der [[Symmetrie der Spektren bei reellen Funktionen|fourier-transformation#realteil,1:#*]] $\Gamma(f) = \Gamma^*(-f)$

$$
\begin{aligned}
    W_{\gamma\gamma}(t,f) &= \int_{-\infty}^{\infty} \Gamma\left(f - \frac{\vartheta}{2}\right)\Gamma^*\left(f + \frac{\vartheta}{2}\right)\exp\left(-\mathrm j 2 \pi \vartheta t\right) \, \mathrm d \vartheta && \Gamma(f) = \Gamma^*(-f) \\
    &= \int_{-\infty}^{\infty} \Gamma\left(-t - \frac{\vartheta}{2}\right)\Gamma^*\left(-t + \frac{\vartheta}{2}\right)\exp\left(-\mathrm j 2 \pi \vartheta t\right) \, \mathrm d \vartheta 
    &= W_{\gamma\gamma}(t,-f)
\end{aligned}$$

Daraus folgt dann: 

$$
W_{\gamma\gamma}(t,f) = W_{\gamma\gamma}(-t,f) = W_{\gamma\gamma}(t^{\prime},f) = W_{\gamma\gamma}(t^{\prime},-f)
$$

### Skalierungserhaltend

Die Auto-Wigner-Ville-Verteilung einer mit $a \in \mathbb R$ skalierten Funktionen $x_a(t) = \frac{1}{\sqrt{|a|}} x\left(\frac{t}{a}\right)$ ist gleich der skalierten Auto-Wigner-Ville-Verteilung der ursprünglichen Funktion $x(t)$: 

$$
W_{x_ax_a}(t,f) = W_{xx}\left(\frac{t}{a},af\right)
$$

#### Beweis

$$
\begin{aligned}
    W_{x_ax_a}(t,f) &= \frac{1}{a} \int_{-\infty}^{\infty} x\left(\frac{t}{a} + \frac{\tau}{2a}\right)y^*\left(\frac{t}{a} - \frac{\tau}{2a}\right)\exp\left(-\mathrm j 2 \pi \tau f\right) \, \mathrm d \tau && \tau^{\prime} = \frac{\tau}{a}\\
    &= \int_{-\infty}^{\infty} x\left(\frac{t}{a} + \frac{\tau^{\prime}}{2}\right)y^*\left(\frac{t}{a} - \frac{\tau^{\prime}}{2}\right)\exp\left(-\mathrm j 2 \pi a\tau^{\prime} f\right) \, \mathrm d \tau^{\prime} \\
    &= W_{xx}\left(\frac{t}{a},af\right)
\end{aligned}
$$

### Phaseninvarianz

Die Auto-Wigner-Ville-Verteilung einer um eine konstante Phase $\varphi$ Funktion $x^{\prime}(t) = x(t) \cdot \exp(\mathrm j \varphi)$ ist gleich der Auto-Wigner-Ville-Verteilung der ursprünglichen Funktion $x(t)$. 
Die Auto-Wigner-Ville-Verteilung ist invariant gegenüber konstanten Phasenverschiebungen $\varphi_0$: 

$$
\begin{aligned}
    W_{x^{\prime}x^{\prime}}(t,f) &= \int_{-\infty}^{\infty} x\left(t + \frac{\tau}{2}\right)\exp(\mathrm j \varphi)x^*\left(t - \frac{\tau}{2}\right)\exp(-\mathrm j \varphi) \exp\left(-\mathrm j 2 \pi \tau f\right) \, \mathrm d \tau \\
    &= \int_{-\infty}^{\infty} x\left(t + \frac{\tau}{2}\right)x^*\left(t - \frac{\tau}{2}\right) \exp\left(-\mathrm j 2 \pi \tau f\right) \, \mathrm d \tau \\
    &= W_{xx}(t,f)
\end{aligned}
$$

Selbiges gilt auch für phasenverschobene Spektren. 

### Momente

Angelehnt an die [[Momente|mathematik.stochastik.wahrscheinlichkeitstheorie.moment]] der Wahrscheinlichkeitstheorie können auch von der Wigner-Ville-Verteilung Momente gebildet werden. 
Aufgrund der zweidimensionalen Natur der Wigner-Ville-Verteilung kann dies auf zwei Arten geschehen: 

$$
\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} t^n \cdot W_{xy}(t,f) \,\mathrm d t \,\mathrm d f \;, \qquad \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f^n \cdot W_{xy}(t,f) \,\mathrm d t \,\mathrm d f
$$

Für Auto-Wigner-Ville-Verteilungen kann man über die Randbedingungen vereinfachend schreiben: 

$$
\begin{aligned}
    \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} t^n \cdot W_{xy}(t,f) \,\mathrm d t \,\mathrm d f &= \int_{-\infty}^{\infty} t^n \int_{-\infty}^{\infty} W_{xy}(t,f) \,\mathrm d f \,\mathrm d t \\
    &= \int_{-\infty}^{\infty} t^n |x(t)|^2 \,\mathrm d t \\
\end{aligned}
$$

bzw. 

$$
\begin{aligned}
    \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f^n \cdot W_{xy}(t,f) \,\mathrm d t \,\mathrm d f &= \int_{-\infty}^{\infty} f^n \int_{-\infty}^{\infty} W_{xy}(t,f) \,\mathrm d t \,\mathrm d f \\
    &= \int_{-\infty}^{\infty} f^n |X(f)|^2 \,\mathrm d f \\
\end{aligned}
$$

Für ein [[analytisches Signal|signale.analytisches-signal]] $x_{an}(t)$ entspricht das erste normierte Frequenzmoment der [[signale.momentanfrequenz]]: 

$$
\frac{\int_{-\infty}^{\infty} f \cdot W_{x_{an}x_{an}}(t,f) \,\mathrm d f}{\int_{-\infty}^{\infty} W_{x_{an}x_{an}}(t,f) \,\mathrm d f} = f_{x_{an}}(t)
$$

Für ein [[analytisches Signal|signale.analytisches-signal]] $x_{an}(t)$ entspricht das erste normierte Zeitmoment der [[signale.gruppenlaufzeit]]: 

$$
\frac{\int_{-\infty}^{\infty} t \cdot W_{x_{an}x_{an}}(t,f) \,\mathrm d t}{\int_{-\infty}^{\infty} W_{x_{an}x_{an}}(t,f) \,\mathrm d t} = \tau_{x_{an}}(f)
$$

Beweis der beiden obigen Aussagen siehe [@Kiencke09]

### Multiplikation 

Die Auto-Wigner-Ville-Verteilung eines Produktes zweier Funktionen $y(t) = x(t) \cdot h(t)$ ist gleich der Faltung der beiden Auto-Wigner-Ville-Verteilung der ursprünglichen Funktionen $x(t)$ und $h(t)$ bezüglich $f$: 

$$
W_{yy}(t,f) = W_{xx}(t,f) \underset{f}{\ast} W_{hh}(t,f)
$$

#### Beweis

Der Beweis erfolgt über den [[Faltungssatz|fourier-transformation#faltungssatz,1:#*]] der Fourier-Transformation: 

$$
\begin{aligned}
    W_{yy}(t,f) &= \mathcal F_{\tau} \left\{\left(x\left(t + \frac{\tau}{2}\right)x^*\left(t - \frac{\tau}{2}\right)\right)\cdot\left(h\left(t + \frac{\tau}{2}\right)h^*\left(t - \frac{\tau}{2}\right)\right)\right\} \\
    &= \mathcal F_{\tau} \left\{\left(x\left(t + \frac{\tau}{2}\right)x^*\left(t - \frac{\tau}{2}\right)\right)\right\} \underset{f}{\ast} \mathcal F_{\tau} \left\{\left(h\left(t + \frac{\tau}{2}\right)h^*\left(t - \frac{\tau}{2}\right)\right)\right\} \\
    &= W_{xx}(t,f) \underset{f}{\ast} W_{hh}(t,f)
\end{aligned}
$$

### Faltung

Die Auto-Wigner-Ville-Verteilung einer Faltung zweier Funktionen $y(t) = x(t) \ast h(t)$ ist gleich der Faltung der beiden Auto-Wigner-Ville-Verteilung der ursprünglichen Funktionen $x(t)$ und $h(t)$ bezüglich $t$: 

$$
W_{yy}(t,f) = W_{xx}(t,f) \underset{t}{\ast} W_{hh}(t,f)
$$

#### Beweis

Der Beweis erfolgt über den [[Faltungssatz|fourier-transformation#faltungssatz,1:#*]] der Fourier-Transformation: 

$$
\begin{aligned}
    W_{yy}(t,f) &= \mathcal F_{\vartheta} \left\{X\left(f - \frac{\vartheta}{2}\right)X^*\left(f + \frac{\vartheta}{2}\right) \cdot H\left(f - \frac{\vartheta}{2}\right)H^*\left(f + \frac{\vartheta}{2}\right)\right\} \\
    &= \mathcal F_{\vartheta} \left\{X\left(f - \frac{\vartheta}{2}\right)X^*\left(f + \frac{\vartheta}{2}\right)\right\} \underset{t}{\ast} \mathcal F_{\vartheta} \left\{H\left(f - \frac{\vartheta}{2}\right)H^*\left(f + \frac{\vartheta}{2}\right)\right\} \\
    &= W_{xx}(t,f) \underset{t}{\ast} W_{hh}(t,f)
\end{aligned}
$$

### Moyals Formel

Moyals Formel stellt das Pendant zum [[fourier-transformation.satz-von-plancherel]] für Wigner-Ville-Verteilungen dar. 
Das Produkt des Innenprodukts der Funktionen $x_1(t)$ und $y_1(t)$ sowie des komplex konjugierten Innenproduktes $x_2(t)$ und $y_2(t)$ im Urbildbereich ist gleich dem Innenprodukt der Wigner-Ville-Verteilungen: 

$$
\langle x_1(t), y_1(t) \rangle_t \cdot \langle x_2(t), y_2(t) \rangle_t^* = \langle W_{x_1x_2}(t,f), W_{y_1y_2}(t,f) \rangle_{t,f} = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} W_{x_1x_2}(t,f) \cdot W_{y_1y_2}^*(t,f) \,\mathrm d t \,\mathrm d f
$$

Für den Sonderfall $x(t) = x_1(t) = x_2(t)$ und $y(t) = y_1(t) = y_2(t)$ wird dies zu: 

$$
|\langle x(t), y(t) \rangle_t|^2 = \langle W_{xx}(t,f), W_{yy}(t,f) \rangle_{t,f}
$$

## Zusammenhang zwischen dem Spektrogramm der Kurzzeit-Fourier-Transormation und der Wigner-Ville-Verteilung

<!-- ToDo: Einheitliche Bezeichnung für tau, t, f und f_0 einführen. -->

![[fourier-transformation.kurzzeit-fourier-transformation#definition:#folgerungen-aus-den-interpretationen]]

![[fourier-transformation.kurzzeit-fourier-transformation.spektrogramm#definition]]

Nach Moyals Formel und der [[translationserhaltenden|wigner-ville-verteilung#translationserhaltend]] bzw. [[modulationserhaltenden|wigner-ville-verteilung#modulationserhaltend]] Eigenschaft der Wigner-Ville-Verteilung gilt somit: 

$$
S_x^{\gamma} (\tau, f_0) = |\mathcal F^{\gamma} x (\tau,f_0)|^2 = |\left\langle x(t), \gamma(t-\tau) \mathrm e^{\mathrm j 2\pi f_0t}\right\rangle_t|^2 = \langle W_{xx}(t,f), W_{yy}(t-\tau,f-f_0) \rangle_{t,f}
$$

Ist die Funktion $\gamma(t)$ reellwertig und symmetrisch ist die zugehörige Wigner-Ville-Verteilung ebenso [[symmetrisch|wigner-ville-verteilung#symmetrieerhaltend]], wodurch sich 

$$
S_x^{\gamma} (\tau, f_0) = \langle W_{xx}(t,f), W_{yy}(t-\tau,f-f_0) \rangle_{t,f} = \langle W_{xx}(t,f), W_{yy}(\tau-t,f_0-f) \rangle_{t,f} = W_{xx}(\tau,f_0) \underset{\tau, f_0}{\ast} W_{yy}(\tau,f_0)
$$

ergibt. 
Durch die Faltung der Wigner-Ville-Verteilungen kommt es zu einer Verbreiterung. 
Man sieht daran, dass das Spektrogramm eine grundsätzlich schlechtere Auflösung als die Wigner-Ville-Verteilung besitzt. 

## Zusammenhang zwischen dem Skalogramm der Wavelet-Transformation und der Wigner-Ville-Verteilung

![[wavelet-transformation#definition:#interpretation-als-inverse-fourier-transformation]]

![[wavelet-transformation.skalogramm#definition,2]]

Nach Moyals Formel, der [[translationserhaltenden|wigner-ville-verteilung#translationserhaltend]] und der [[skalierungserhaltenden|wigner-ville-verteilung#skalierungserhaltend]] Eigenschaft entspricht das Skalogramm gerade dem Innenprodukt der Wigner-Ville-Verteilungen: 

$$
|W^{\psi}x(a,b)|^2 = |\langle x(t), \psi_{a,b}(t) \rangle_t|^2 = \left\langle W_{xx}(t,f), W_{\psi\psi}\left(\frac{t-b}{a},af\right) \right\rangle_{t,f}
$$

Man sieht daran, dass das Skalogramm eine grundsätzlich schlechtere Auflösung als die Wigner-Ville-Verteilung besitzt. 

## Rekonstruktion aus der Auto-Wigner-Ville-Verteilung

Man setzt zur Rekonstruktion die abgeleitete Darstellungen an:

![[wigner-ville-verteilung#abgeleitete-darstellungen,2]] 

### Rekonstruktion in den Zeitbereich

$$
x\left(t + \frac{\tau}{2}\right)x^*\left(t - \frac{\tau}{2}\right) = \mathcal F_{f}^{-1} \left\{W_{xx}(t,f)\right\} = \int_{-\infty}^{\infty} W_{xx}(t,f) \exp\left(\mathrm j 2 \pi f \tau \right) \,\mathrm d f
$$

Setzt man nun $t = \frac{\tau}{2}$ erhält man daraus: 

$$
\tilde x(\tau) = x(\tau)x^*(0) = \int_{-\infty}^{\infty} W_{xx}(\frac{\tau}{2},f) \exp\left(\mathrm j 2 \pi f \tau \right) \,\mathrm d f
$$

Die Funktion $\tilde x(\tau)$ stimmt also mit der ursprünglichen Funktion $x(t)$ bis auf einen konstanten Faktor $x^*(0)$ überein. 
Für $\tau = 0$ folgt daraus ein Spezialfall der [[Zeit-Randbedingung|wigner-ville-verteilung#zeit-randbedingung]]: 

$$
\tilde x(0) = x(0)x^*(0) = |x(0)|^2 = \int_{-\infty}^{\infty} W_{xx}(0,f) \,\mathrm d f
$$

Bringt man beide Gleichungen zusammen und substituiert für eine einheitliche Darstellung $\tau$ durch $t$, folgt daraus: 

$$
\hat x(t) = \frac{\tilde x(t)}{|x(0)|} = \frac{\int_{-\infty}^{\infty} W_{xx}(\frac{t}{2},f) \exp\left(\mathrm j 2 \pi f t \right) \,\mathrm d f}{\sqrt{\int_{-\infty}^{\infty} W_{xy}(0,f) \,\mathrm d f}}
$$

Aufgrund der [[Phaseninvarianz|wigner-ville-verteilung#phaseninvarianz]] der Wigner-Ville-Verteilung kann aus dieser die Funktion lediglich bis auf einen konstanten Phasenfaktor rekonstruiert werden. 
Dies wird auch von obiger Gleichung, bei der durch den Betrag geteilt wird, bestätigt. 

### Rekonstruktion in den Frequenzbereich

Die Rekonstruktion in den Frequenzbereich erfolgt analog zur Rekonstruktion in den Zeitbereich. 

$$
X\left(f - \frac{\vartheta}{2}\right)X^*\left(f + \frac{\vartheta}{2}\right) = \mathcal F_{t}^{-1} \left\{W_{xx}(t,f)\right\} = \int_{-\infty}^{\infty} W_{xx}(t,f) \exp\left(\mathrm j 2 \pi \vartheta t \right) \,\mathrm d t
$$

Setzt man nun $f = \frac{\vartheta}{2}$ erhält man daraus: 

$$
\tilde X^*(\vartheta) = X(0)X^*(\vartheta) = \int_{-\infty}^{\infty} W_{xx}(t,\frac{\vartheta}{2}) \exp\left(\mathrm j 2 \pi \vartheta t \right) \,\mathrm d f
$$

Da die Auto-Wigner-Ville-Verteilung [[reell|wigner-ville-verteilung#reellwertigkeit]] ist, gilt damit auch: 

$$
\tilde X(\vartheta) = X^*(0)X(\vartheta) = \int_{-\infty}^{\infty} W_{xx}(t,\frac{\vartheta}{2}) \exp\left(- \mathrm j 2 \pi \vartheta t \right) \,\mathrm d f
$$

Die Funktion $\tilde X(\vartheta)$ stimmt also mit der ursprünglichen Funktion $X(t)$ bis auf einen konstanten Faktor $X^*(0)$ überein. 
Für $\vartheta = 0$ folgt daraus ein Spezialfall der [[Frequenz-Randbedingung|wigner-ville-verteilung#frequenz-randbedingung]]: 

$$
\tilde X(0) = X^*(0)X(0) = |X(0)|^2 = \int_{-\infty}^{\infty} W_{xx}(t,0) \,\mathrm d f
$$

Bringt man beide Gleichungen zusammen und substituiert für eine einheitliche Darstellung $\vartheta$ durch $f$, folgt daraus: 

$$
\hat X(f) = \frac{\tilde X(f)}{|X(0)|} = \frac{\int_{-\infty}^{\infty} W_{xx}(t,\frac{f}{2}) \exp\left(-\mathrm j 2 \pi f t \right) \,\mathrm d t}{\sqrt{\int_{-\infty}^{\infty} W_{xy}(t,0) \,\mathrm d t}}
$$

Aufgrund der [[Phaseninvarianz|wigner-ville-verteilung#phaseninvarianz]] der Wigner-Ville-Verteilung kann aus dieser die Funktion lediglich bis auf einen konstanten Phasenfaktor rekonstruiert werden. 
Dies wird auch von obiger Gleichung, bei der durch den Betrag geteilt wird, bestätigt. 

## Wigner-Ville-Verteilung von Mehrkomponentensignalen

Wendet man die Wigner-Ville-Verteilung auf [[Mehrkomponentensignale|signale.mehrkomponentensignal]] an, so ergeben sich aufgrund der quadratischen Transformation neue Probleme in Form von Kreuztermen: 

![[signale.mehrkomponentensignal]]

Daraus erhält man dann 

$$
\begin{aligned}
    W_{xx}(t,f) &= \int_{-\infty}^{\infty} x\left(t + \frac{\tau}{2}\right)x^*\left(t - \frac{\tau}{2}\right)\exp\left(-\mathrm j 2 \pi \tau f\right) \, \mathrm d \tau \\
    &= 
\end{aligned}
$$

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[Harshita-mindfire]

Thanks @funnym0nk3y for reporting the issue. We'll keep track of this and investigate how to fix this issue.

[Harshita-mindfire]

On a second note @funnym0nk3y , does this solution by @kazephil works for you?https://github.com/dendronhq/dendron/issues/2214#issuecomment-1029025000

[funnym0nk3y]

@Harshita-mindfire The hosts file does resolve to localhost. I'm having this issue just with the note mentioned in my original post. Most (all I've used recently) notes work just fine, though they are shorter and don't include as many references. Also when including the note above as a reference view into other notes the same thing happens.

[Harshita-mindfire]

Thank you for the clarification 🙏 . Keeping this issue open for further tracking.