# On the Octonion Cross Wigner Distribution of 3-D Signals
## 3 Octonion Wigner Distribution
### [PERSON]'s Formula  
One of the important Wigner-Ville distribution properties (in the classical setup) is called _[PERSON]'s formula_[17], which states that  
\[\left(W_{f_{1}g1},W_{f_{2}g2}\right)_{L^{2}(\mathbb{R}^{n}\times\mathbb{R}^{ n};\mathbb{C})}=(f_{1},f_{2})_{L^{2}(\mathbb{R}^{n};\mathbb{C})}\cdot(g_{1},g_{ 2})_{L^{2}(\mathbb{R}^{n};\mathbb{C})}.\]  
In the octonion approach, the problem is much more complicated (which is very similar to considerations about the Parseval-Plancherel theorems for the OFT, see [46]). Assume that \(f_{1},f_{2},g_{1},g_{2}\in L^{2}(\mathbb{R}^{3};\mathbb{O})\) are real-valued and \(W^{R}_{f_{1},g_{1}}\in L^{2}(\mathbb{R}^{3}\times\mathbb{R}^{3};\mathbb{O})\). The OWD of \(f_{i}\) and \(g_{i}\) is the octonion Fourier transform (with respect to \(\mathbf{y}\)) of  
\[f_{i}\Big{(}\mathbf{x}+\frac{\mathbf{y}}{2}\Big{)}\overline{g_{i}\Big{(} \mathbf{x}-\frac{\mathbf{y}}{2}\Big{)}}.\]  
The [PERSON]'s theorem is valid for the OFT of real-valued functions, so  
\[\int_{\mathbb{R}^{3}}W^{R}_{f_{1},g_{1}}(\mathbf{x},\boldsymbol{\omega}) \overline{W^{R}_{f_{2},g_{2}}(\mathbf{x},\boldsymbol{\omega})}\,\mathrm{d} \boldsymbol{\omega}=\int_{\mathbb{R}^{3}}f_{1}\Big{(}\mathbf{x}+\frac{\mathbf{y} }{2}\Big{)}\overline{g_{1}\Big{(}\mathbf{x}-\frac{\mathbf{y}}{2}\Big{)}} \cdot\overline{f_{2}\Big{(}\mathbf{x}+\frac{\mathbf{y}}{2}\Big{)}}g_{2}\Big{(} \mathbf{x}-\frac{\mathbf{y}}{2}\Big{)}\,\mathrm{d}\mathbf{y}\]  
(we could skip the conjugation of real-valued functions). With introducing a new variable \(\mathbf{t}=\mathbf{x}+\frac{\mathbf{y}}{2}\) we obtain  
\[\int_{\mathbb{R}^{3}}W^{R}_{f_{1},g_{1}}(\mathbf{x},\boldsymbol{\omega}) \overline{W^{R}_{f_{2},g_{2}}(\mathbf{x},\boldsymbol{\omega})}\,\mathrm{d} \boldsymbol{\omega}=8\int_{\mathbb{R}^{3}}f_{1}(\mathbf{t})\overline{f_{2}( \mathbf{t})}\cdot\overline{g_{1}(2\mathbf{x}-\mathbf{t})}g_{2}(2\mathbf{x}- \mathbf{t})\,\mathrm{d}\mathbf{t}\]  
and by integrating over \(\mathbf{x}\in\mathbb{R}^{3}\) and applying the Fubini theorem, we obtain the claim of the following theorem. **Theorem 6**. : _Let \(f_{1},f_{2},g_{1},g_{2}\in L^{2}(\mathbb{R}^{3};\mathbb{R})\). Then_  
\[\left(W^{R}_{f_{1},g_{1}},W^{R}_{f_{2},g_{2}}\right)_{L^{2}(\mathbb{R}^{3}\times \mathbb{R}^{3};\mathbb{O})}=(f_{1},f_{2})_{L^{2}(\mathbb{R}^{3};\mathbb{O})} \cdot(g_{1},g_{2})_{L^{2}(\mathbb{R}^{3};\mathbb{O})}.\]  
Unfortunately, this formula does not hold for octonion-valued functions, similarly as in the case of the Plancherel theorem for the OFT. Consider  
\[f_{1}(\mathbf{x}) =\frac{\mathbf{e}_{4}}{(2\pi)^{3/2}}\exp\biggl{\{}-\frac{1}{2} \Bigl{(}x_{1}^{2}+(x_{2}-1)^{2}+(x_{3}-1)^{2}\Bigr{)}\biggr{\}},\] \[f_{2}(\mathbf{x}) =\frac{\mathbf{e}_{1}}{(2\pi)^{3/2}}\exp\biggl{\{}-\frac{1}{2} \Bigl{(}x_{1}^{2}+(x_{2}+1)^{2}+(x_{3}+1)^{2}\Bigr{)}\biggr{\}},\] \[g_{1}(\mathbf{x}) =g_{2}(\mathbf{x})\] \[=\frac{\mathbf{e}_{2}}{(2\pi)^{3/2}}\exp\biggl{\{}-\frac{1}{2} \Bigl{(}x_{1}^{2}+(x_{2}-1)^{2}+(x_{3}+1)^{2}\Bigr{)}\biggr{\}}.\]  
After many tiring calculations we obtain  
\[\left(W^{R}_{f_{1},g_{1}},W^{R}_{f_{2},g_{1}}\right)_{L^{2}(\mathbb{R}^{3} \times\mathbb{R}^{3};\mathbb{O})}=-\frac{\mathbf{e}_{5}}{64e^{2}\pi^{3}}\
eq \frac{\mathbf{e}_{5}}{64e^{2}\pi^{3}}=(f_{1},f_{2})_{L^{2}(\mathbb{R}^{3}; \mathbb{O})}\cdot\left\|g_{1}\right\|_{L^{2}(\mathbb{R}^{3};\mathbb{O})}^{2}.\]  
From [PERSON]'s formula, we immediately conclude that for \(f_{1}=f_{2}:=f\) and \(g_{1}=g_{2}:=g\) we obtain  
\[f_{i}\Big{(}\mathbf{x}+\frac{\mathbf{y}}{2}\Big{)}\overline{g_{i}\Big{(} \mathbf{x}-\frac{\mathbf{y}}{2}\Big{)}}.\]0  
Unsurprisingly, this formula holds also for octonion-valued functions.