# On the Octonion Cross Wigner Distribution of 3-D Signals
## 3 Octonion Wigner Distribution
### Definition and Invertibility  
The definition presented below is a natural generalization of the definition of the 4-D Wigner distribution of 2-D quaternion signals (see [6; 21]) to the octonion algebra. We introduce it in a general form called the octonion cross Wigner distribution. Throughout the rest of this article, we repeatedly refer to _the octonion correlation product_ of two signals \(f,g\colon\mathbb{R}^{3}\to\mathbb{O}\), i.e.,  
\[h(\mathbf{y};\mathbf{x})=f\Big{(}\mathbf{x}+\frac{\mathbf{y}}{2}\Big{)} \overline{g\Big{(}\mathbf{x}-\frac{\mathbf{y}}{2}\Big{)}}.\]  
This is obviously a function of the six variables, but we group them into triples \(\mathbf{x}\) and \(\mathbf{y}\).  
Definition 1: The octonion cross right-sided Wigner distribution (OWD) of the 3-D signals \(f,g\in L^{2}(\mathbb{R}^{3};\mathbb{O})\) is defined as  
\[W_{f,g}^{R}(\mathbf{x},\mathbf{\omega})=\int_{\mathbb{R}^{3}}f\Big{(}\mathbf{x}+ \frac{\mathbf{y}}{2}\Big{)}\overline{g\Big{(}\mathbf{x}-\frac{\mathbf{y}}{2} \Big{)}}\cdot e^{-2\pi\mathbf{e}_{1}\omega_{1}y_{1}}e^{-2\pi\mathbf{e}_{2} \omega_{2}y_{2}}e^{-2\pi\mathbf{e}_{4}\omega_{3}y_{3}}\,\mathrm{d}\mathbf{y}, \tag{1}\]  
where \((\mathbf{x},\mathbf{\omega})\in\mathbb{R}^{3}\times\mathbb{R}^{3}\) and multiplication is performed from left to right.  
Other definitions of OWD are, of course, possible (left-sided distribution or two different versions of two-sided distributions, etc.), which would be identical for real-valued functions, but we use the one introduced in Definition 1.  
It is also worth noting that in the literature, the Wigner distribution is often defined not for two different functions \(f\) and \(g\), but a single function \(f\). The definition presented above covers a more general situation, and we denote this particular case as \(W_{f}^{R}:=W_{f,f}^{R}\).  
It should be noted that if \(f,g\in L^{2}(\mathbb{R}^{3};\mathbb{O})\) then for every \(\mathbf{x}\in\mathbb{R}^{3}\) we have (from Cauchy-Schwarz inequality) \(h(\cdot;\mathbf{x})\in L^{1}(\mathbb{R}^{3};\mathbb{O})\), so the OWD of \(f,g\in L^{2}(\mathbb{R}^{3};\mathbb{O})\) is well-defined. The OWD is in fact the OFT of octonion correlation product \(h\) (with respect to the first three variables, i.e., \(\mathbf{y}\)). Applying the properties of OFT proved in previous works [46; 50], we can express \(h(\mathbf{y};\mathbf{x})\) for every \((\mathbf{x},\mathbf{y})\in\mathbb{R}^{3}\times\mathbb{R}^{3}\) in the form:  
\[f\Big{(}\mathbf{x}+\frac{\mathbf{y}}{2}\Big{)}\overline{g\Big{(}\mathbf{x}- \frac{\mathbf{y}}{2}\Big{)}}=\int_{\mathbb{R}^{3}}W_{f,g}^{R}(\mathbf{x},\mathbf{ \omega})e^{2\pi\mathbf{e}_{4}\omega_{3}y_{3}}e^{2\pi\mathbf{e}_{2}\omega_{2} y_{2}}e^{2\pi\mathbf{e}_{1}\omega_{1}y_{1}}\,\mathrm{d}\mathbf{\omega}, \tag{2}\]  
where multiplication is performed from left to right. Of course this formula is valid under certain assumptions, e.g.,  
* \(W_{f,g}^{R}(\mathbf{x},\cdot)\in L^{1}(\mathbb{R}^{3};\mathbb{O})\) for every \(\mathbf{x}\in\mathbb{R}^{3}\),
* \(f,g\in\mathcal{S}(\mathbb{R}^{3};\mathbb{O})\),
* \(h(\cdot;\mathbf{x})\in L^{2}(\mathbb{R}^{3};\mathbb{O})\) for every \(\mathbf{x}\in\mathbb{R}^{3}\),etc. For the rest of the article (unless stated otherwise) we will assume that \(f\) and \(g\) are rapidly decreasing functions. Putting \(\mathbf{x}=\frac{\mathbf{y}}{2}\) we immediately obtain the formula for recovering the function \(f\) from its OWD (as long as \(g(\mathbf{0})\
eq 0\)).  
Theorem 3.2: _Let \(f,g\in\mathcal{S}(\mathbb{R}^{3};\mathbb{O})\) and \(g(\mathbf{0})\
eq 0\). Then for every \(\mathbf{y}\in\mathbb{R}^{3}\)_  
\[f(\mathbf{y})=\int_{\mathbb{R}^{3}}W^{R}_{f,g}\Big{(}\frac{\mathbf{y}}{2}, \boldsymbol{\omega}\Big{)}e^{2\pi\mathbf{e}a_{4}\lambda y_{3}y_{5}}e^{2\pi \mathbf{e}a_{2}\lambda y_{2}}e^{2\pi\mathbf{e}a_{1}\lambda y_{1}}\,\mathrm{d} \boldsymbol{\omega}\cdot\frac{g(\mathbf{0})}{\left|g(\mathbf{0})\right|},\]  
_where multiplication is performed from left to right._