# On the Octonion Cross Wigner Distribution of 3-D Signals
## 4 Examples
The convention here is to write the OWD as  
\[=\frac{1}{(2\pi)^{3}(\sigma_{1}\sigma_{2}\sigma_{3})^{2}}\cdot \exp\Biggl{\{}-\frac{1}{2}\Bigg{(}\frac{(x_{1}+\frac{y_{1}}{2})^{2}+(x_{1}- \frac{y_{1}}{2})^{2}}{\sigma_{1}^{2}}\]5  
The conclusions that can be drawn from this example are as follows:  
* The shift of the Gaussian function with respect to \(\mathbf{x}\) indicates the decentralization of the set of two Gaussian functions (how far is the midpoint of the segment connecting the centers of those functions from the center of the coordinate system),* Differences between the functions in each direction are visible in the individual imaginary parts--a change in the \(x_{i}\) direction causes the appearance of the imaginary part standing at \(\mathbf{e}_{2^{i-1}}\), a change in the \(x_{i}\) and \(x_{j}\) directions causes the appearance of the imaginary parts standing at \(\mathbf{e}_{2^{i-1}}\), \(\mathbf{e}_{2^{i-1}}\) and \(\mathbf{e}_{2^{i-1}}\cdot\mathbf{e}_{2^{i-1}}\). The relative magnitude of these changes is reflected by the amplitude of the individual imaginary components. The last of the above conclusions shows at the same time the difference (and a significant advantage) of the octonion approach over the classic one. It is worth noting that this is only a theoretical example (which can be directly calculated). Continuing the above example, it is worth considering the OWD of the Gabor signal, i.e., the product of a Gaussian kernel and a sinusoid. Considering only the real Gabor signal, we can use the above considerations and the equivalents of the modulation theorem discussed in Section 3.2. We have two Gabor signals:  
\[=\frac{1}{(2\pi)^{3}(\sigma_{1}\sigma_{2}\sigma_{3})^{2}}\cdot \exp\Biggl{\{}-\frac{1}{2}\Bigg{(}\frac{(x_{1}+\frac{y_{1}}{2})^{2}+(x_{1}- \frac{y_{1}}{2})^{2}}{\sigma_{1}^{2}}\]6 \[=\frac{1}{(2\pi)^{3}(\sigma_{1}\sigma_{2}\sigma_{3})^{2}}\cdot \exp\Biggl{\{}-\frac{1}{2}\Bigg{(}\frac{(x_{1}+\frac{y_{1}}{2})^{2}+(x_{1}- \frac{y_{1}}{2})^{2}}{\sigma_{1}^{2}}\]7  
The choice of variable \(x_{3}\) in the sine carrier is arbitrary. Further considerations can also be made without major changes when selecting other variables. OWD of those two Gabor signals is given by the formula  
\[=\frac{1}{(2\pi)^{3}(\sigma_{1}\sigma_{2}\sigma_{3})^{2}}\cdot \exp\Biggl{\{}-\frac{1}{2}\Bigg{(}\frac{(x_{1}+\frac{y_{1}}{2})^{2}+(x_{1}- \frac{y_{1}}{2})^{2}}{\sigma_{1}^{2}}\]8where \(\mathbf{\omega}_{0}=(0,0,\omega_{0})\) and  
\[=\frac{1}{(2\pi)^{3}(\sigma_{1}\sigma_{2}\sigma_{3})^{2}}\cdot \exp\Biggl{\{}-\frac{1}{2}\Bigg{(}\frac{(x_{1}+\frac{y_{1}}{2})^{2}+(x_{1}- \frac{y_{1}}{2})^{2}}{\sigma_{1}^{2}}\]9 \[\qquad\qquad\qquad+\frac{(x_{2}+\frac{y_{2}}{2})^{2}+(x_{2}- \frac{y_{2}}{2})^{2}}{\sigma_{2}^{2}}+\frac{(x_{3}+\frac{y_{3}}{2})^{2}+(x_{3} -\frac{y_{3}}{2})^{2}}{\sigma_{3}^{2}}\Bigg{)}\Biggr{\}}\]0 \[\qquad\qquad\qquad+\frac{(x_{2}+\frac{y_{2}}{2})^{2}+(x_{2}- \frac{y_{2}}{2})^{2}}{\sigma_{2}^{2}}+\frac{(x_{3}+\frac{y_{3}}{2})^{2}+(x_{3} -\frac{y_{3}}{2})^{2}}{\sigma_{3}^{2}}\Bigg{)}\Biggr{\}}\]1  
In this case, it is also worth paying attention to what the obtained distributions look like when selecting specific parameters of both Gabor signals. Figure 7 shows the plots of the OWD for parameters \(\mathbf{\sigma}_{1}=\left(\frac{1}{2},\frac{1}{5},\frac{1}{2}\right)\), \(\mathbf{\sigma}_{2}=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\) and \(\omega_{0}=\frac{1}{2}\). For comparison, Figure 8 shows plots of the classical Wigner distribution \(W^{C}_{f,g}\) (its real and imaginary parts). There are clear differences in the two approaches to the problem.