# On the Octonion Cross Wigner Distribution of 3-D Signals
## 2 Basic Definitions
### Octonions  
Octonions \(o\in\mathbb{O}\) are defined as the 8-tuple of real numbers [51], i.e.,  
\[o=o_{0}+o_{1}\mathbf{e}_{1}+o_{2}\mathbf{e}_{2}+o_{3}\mathbf{e}_{3}+o_{4} \mathbf{e}_{4}+o_{5}\mathbf{e}_{5}+o_{6}\mathbf{e}_{6}+o_{7}\mathbf{e}_{7},\]  
where \(o_{0},\ldots,o_{7}\in\mathbb{R}\) and \(\mathbf{e}_{1},\ldots,\mathbf{e}_{7}\) are seven octonion imaginary units, each satisfying the property \(\mathbf{e}_{i}^{2}=\mathbf{e}_{i}\cdot\mathbf{e}_{i}=-1\), \(i=1,\ldots,7\), and other octonion multiplication rules shown in Figure 1. Number \(o_{0}\in\mathbb{R}\) is called the real part of \(o\) (denoted as \(\operatorname{Re}o\)) and the pure imaginary octonion \(o_{1}\mathbf{e}_{1}+o_{2}\mathbf{e}_{2}+\ldots+o_{7}\mathbf{e}_{7}\) is called the imaginary part of \(o\) (and is denoted as \(\operatorname{Im}o\)).  
Octonions form a non-associative and a non-commutative algebra, which means that in general, for \(o_{1}\), \(o_{2}\), \(o_{3}\in\mathbb{O}\)  
\[(o_{1}\cdot o_{2})\cdot o_{3}\
eq o_{1}\cdot(o_{2}\cdot o_{3}),\qquad o_{1}\cdot o _{2}\
eq o_{2}\cdot o_{1}.\]  
As for complex numbers and quaternions, conjugation is also defined for octonions, as  
\[\overline{o}=o_{0}-o_{1}\mathbf{e}_{1}-o_{2}\mathbf{e}_{2}-o_{3}\mathbf{e}_{3} -o_{4}\mathbf{e}_{4}-o_{5}\mathbf{e}_{5}-o_{6}\mathbf{e}_{6}-o_{7}\mathbf{e}_ {7}.\]  
It is then true that for any \(o_{1}\), \(o_{2}\in\mathbb{O}\) we have  
\[\overline{(o_{1}\cdot o_{2})}=\overline{o_{2}}\cdot\overline{o_{1}}.\]  
In the algebra of complex numbers, we have the trigonometric form of a number. In the octonion algebra, we define a similar formula for any nonzero octonion \(o\in\mathbb{O}\):  
\[o=|o|\cdot(\cos\theta+\mathbf{\mu}\cdot\sin\theta),\]  
where \(|o|=\sqrt{o\cdot\overline{o}}\) is the octonion norm, \(\mathbf{\mu}=\frac{\operatorname{Im}o}{|\operatorname{Im}o|}\) is a pure imaginary octonion, and \(\theta\in\mathbb{R}\) is the solution of the system of equations  
\[\cos\theta=\frac{\operatorname{Re}o}{|o|},\qquad\sin\theta=\frac{| \operatorname{Im}o|}{|o|}.\]  
It should be noted that every non-zero octonion is invertible and for every \(o\in\mathbb{O}\setminus\{0\}\) we have \(o^{-1}=\frac{\overline{o}}{|o|}\), which means that for pure unitary octonions \(\mathbf{\mu}\) (\(|\mathbf{\mu}|=1\)) we have \(\mathbf{\mu}^{-1}=-\mathbf{\mu}\).  
To be able to define correctly the octonion Fourier transform, the octonion exponential kernel function has to be introduced. Similarly as for the complex numbers and quaternions [52], for any \(o\in\mathbb{O}\) we write  
\[e^{\rho}=\exp(o):=\sum_{k=0}^{\infty}\frac{o^{k}}{k!}.\]  
It can be shown that if we denote \(\mathbf{o}=\operatorname{Im}o\), then  
\[e^{\rho}=e^{\operatorname{Re}o}\biggl{(}\cos|\mathbf{o}|+\frac{\mathbf{o}}{| \mathbf{o}|}\sin|\mathbf{o}|\biggr{)}.\]  
Figure 1: Multiplication rules in octonion algebra.  
One should keep in mind that the fundamental multiplicative identity of exponentials is in general not valid for all octonions. In particular, for any \(o_{1},o_{2}\in\mathbb{O}\) we have  
\[e^{o_{1}+o_{2}}=e^{o_{1}}\cdot e^{o_{2}}\qquad\text{if and only if}\qquad o_{1} \cdot o_{2}=o_{2}\cdot o_{1}.\]  
It follows from the fact that the octonion multiplication is in general non-commutative.  
From the above considerations it follows immediately that for any \(\alpha\in\mathbb{R}\) we can generalize well-known formulas for trigonometric functions as follows  
\[\cos\alpha=\frac{1}{2}\big{(}e^{\mu\alpha}+e^{-\mu\alpha}\big{)},\qquad\sin \alpha=-\frac{\mu}{2}\big{(}e^{\mu\alpha}-e^{-\mu\alpha}\big{)},\]  
where \(\mu\) is any pure unitary octonion.