# On the Octonion Cross Wigner Distribution of 3-D Signals
## 4 Examples  
Let us illustrate the theoretical results presented above with the examples of a separable Gaussian function:  
\[f(x_{1},x_{2},x_{3})=\frac{1}{(2\pi)^{3/2}\sigma_{1}\sigma_{2}\sigma_{3}}\cdot \exp\left\{-\frac{1}{2}\left(\frac{x_{1}^{2}}{\sigma_{1}^{2}}+\frac{x_{2}^{2} }{\sigma_{2}^{2}}+\frac{x_{3}^{2}}{\sigma_{3}^{2}}\right)\right\}.\]  
Due to the fact that, in addition to illustrating the previous statements, we would like to compare the obtained results with the classical theory (derived for complex-valued signals); we limit ourselves only to examining the OWD of real-valued signals. Notice that  
\[h(\mathbf{y};\mathbf{x}) =f\Big{(}\mathbf{x}+\frac{\mathbf{y}}{2}\Big{)}\overline{f\Big{(} \mathbf{x}-\frac{\mathbf{y}}{2}\Big{)}}\] \[=\frac{1}{(2\pi)^{3}(\sigma_{1}\sigma_{2}\sigma_{3})^{2}}\cdot \exp\Biggl{\{}-\frac{1}{2}\Bigg{(}\frac{(x_{1}+\frac{y_{1}}{2})^{2}+(x_{1}- \frac{y_{1}}{2})^{2}}{\sigma_{1}^{2}}\] \[\qquad\qquad\qquad+\frac{(x_{2}+\frac{y_{2}}{2})^{2}+(x_{2}- \frac{y_{2}}{2})^{2}}{\sigma_{2}^{2}}+\frac{(x_{3}+\frac{y_{3}}{2})^{2}+(x_{3} -\frac{y_{3}}{2})^{2}}{\sigma_{3}^{2}}\Bigg{)}\Biggr{\}}\] \[=\frac{1}{(2\pi)^{3/2}\sqrt{2}\sigma_{1}\sqrt{2}\sigma_{2}\sqrt{2 }\sigma_{3}}\exp\Biggl{\{}-\frac{1}{2}\Bigg{(}\frac{x_{1}^{2}}{(\sigma_{1}/ \sqrt{2})^{2}}\ +\frac{x_{2}^{2}}{(\sigma_{2}/\sqrt{2})^{2}}+\frac{x_{3}^{2}}{( \sigma_{3}/\sqrt{2})^{2}}\Bigg{)}\Biggr{\}}\] \[\quad\cdot\frac{1}{(2\pi)^{3/2}\sqrt{2}\sigma_{1}\sqrt{2}\sigma_{ 2}\sqrt{2}\sigma_{3}}\exp\Biggl{\{}-\frac{1}{2}\Bigg{(}\frac{y_{1}^{2}}{( \sqrt{2}\sigma_{1})^{2}}\ +\frac{y_{2}^{2}}{(\sqrt{2}\sigma_{2})^{2}}+\frac{y_{3}^{2}}{( \sqrt{2}\sigma_{3})^{2}}\Bigg{)}\Biggr{\}}\]  
which is the product of two independent separable Gaussian functions. Since OWD is actually the OFT of \(h(\cdot;\mathbf{x})\), then (repeating the calculations for the 3-D Gaussian signal presented in ([49], Example 3) we obtain  
\[W_{f}^{R}(\mathbf{x},\boldsymbol{\omega}) =\frac{1}{(2\pi)^{3/2}\frac{\sigma_{1}}{\sqrt{2}}\frac{\sigma_{2}} {\sqrt{2}}\frac{\sigma_{3}}{\sqrt{2}}}\cdot\exp\Biggl{\{}-\frac{1}{2}\Bigg{(} \frac{x_{1}^{2}}{(\sigma_{1}/\sqrt{2})^{2}}\ +\frac{x_{2}^{2}}{(\sigma_{2}/\sqrt{2})^{2}}+\frac{x_{3}^{2}}{( \sigma_{3}/\sqrt{2})^{2}}\Bigg{)}\Biggr{\}}\] \[\quad\cdot\exp\Bigl{\{}-2\pi^{2}\Big{(}2\sigma_{1}^{2}\omega_{1}^ {2}+2\sigma_{2}^{2}\omega_{2}^{2}+2\sigma_{3}^{2}\omega_{3}^{2}\Big{)}\Bigr{\}},\]  
which (again) is the product of two independent separable Gaussian functions. This result should come as no surprise--in the case of separable real-valued even (with respect to each coordinate) functions, the octonion Fourier transform coincides with the classical one [49]. The same is true for the Wigner distribution of such functions. Consider now the cross right-sided OWD of two separable Gaussian functions:  
\[f(x_{1},x_{2},x_{3}) =\prod_{i=1}^{3}\frac{1}{\sqrt{2\pi}\sigma_{1,i}}\exp\left\{-\frac{1 }{2}\cdot\frac{(x_{i}-\mu_{1,i})^{2}}{\sigma_{1,i}^{2}}\right\},\] \[g(x_{1},x_{2},x_{3}) =\prod_{i=1}^{3}\frac{1}{\sqrt{2\pi}\sigma_{2,i}}\exp\left\{- \frac{1}{2}\cdot\frac{(x_{i}-\mu_{2,i})^{2}}{\sigma_{2,i}^{2}}\right\}.\]  
Sample graphs of the above functions are shown in \[h(\mathbf{y};\mathbf{x}) =f\Big{(}\mathbf{x}+\frac{\mathbf{y}}{2}\Big{)}\overline{f\Big{(} \mathbf{x}-\frac{\mathbf{y}}{2}\Big{)}}\]0 Consider the midpoint of the line connecting the centers of these two Gaussian functions, i.e., \(\tilde{\mu}=\frac{1}{2}(\mu_{1}+\mu_{2})\)\(=(\tilde{\mu}_{1},\tilde{\mu}_{2},\tilde{\mu}_{3})\) (where \(\mathbf{\mu}_{i}=(\mu_{i,1},\mu_{i,2},\mu_{i,3})\)), and denote \(\mathbf{\alpha}=\frac{1}{2}(\mathbf{\mu}_{1}-\mathbf{\mu}_{2})\), \(\mathbf{\alpha}=(\alpha_{1},\alpha_{2},\alpha_{3})\).