# On the Octonion Cross Wigner Distribution of 3-D Signals
## 3 Octonion Wigner Distribution
### Basic Properties  
Many properties of the classical Wigner distribution can be found in the literature [6]. Most often, they result from the properties of the Fourier transformation itself. Most of them can be generalized to the octonion case without much difficulty. We will now present the most important of these properties. From the inversion Formula (2) for \(\mathbf{y}=\mathbf{0}\) we immediately obtain  
\[\int_{\mathbb{R}^{3}}W^{R}_{f,g}(\mathbf{x},\boldsymbol{\omega})\,\mathrm{d} \boldsymbol{\omega}=f(\mathbf{x})\overline{g(\mathbf{x})}, \tag{3}\]  
which can be interpreted as _signal domain marginal_ (or _zero moment_). Again, we need the assumption that for every \(\mathbf{x}\in\mathbb{R}^{3}\) the above integral does exist. In case \(f=g\), it obviously means that the marginal distribution obtained by integrating over frequency is equal to the instantaneous energy \(\left|f\right|^{2}\). Integrating (3) over \(\mathbf{x}\in\mathbb{R}^{3}\) we obtain  
\[\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}W^{R}_{f,g}(\mathbf{x},\boldsymbol{ \omega})\,\mathrm{d}\boldsymbol{\omega}\,\mathrm{d}\mathbf{x}=\int_{\mathbb{R} ^{3}}f(\mathbf{x})\overline{g(\mathbf{x})}\,\mathrm{d}\mathbf{x}\]  
and in the case \(g=f\) we obtain [PERSON]'s equality:  
\[\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}W^{R}_{f}(\mathbf{x},\boldsymbol{ \omega})\,\mathrm{d}\boldsymbol{\omega}\,\mathrm{d}\mathbf{x}=\left\|f\right\| _{L^{2}(\mathbb{R}^{3};\mathbb{O})}^{2}.\]  
Unfortunately, due to the non-associativity of octonion multiplication, the analog of the _frequency marginal_ formula is not possible to be obtained in a general case. We discuss this problem further in Section 6. Few useful properties can be deduced using straightforward calculations (precisely as in proofs of their quaternion or complex analogs):  
* _Space shift_: let \(\mathbf{a}\in\mathbb{R}^{3}\) and \(T_{\mathbf{a}}f(\mathbf{x})=f(\mathbf{x}-\mathbf{a})\), then \[W^{R}_{T_{\mathbf{a}}f,T_{\mathbf{a}}g}(\mathbf{x},\boldsymbol{\omega})=W^{R} _{f,g}(\mathbf{x}-\mathbf{a},\boldsymbol{\omega});\]
* _Space scaling (dilation)_: let \(c>0\) and \(D_{c}f(\mathbf{x})=f\big{(}\frac{\mathbf{x}}{c}\big{)}\), then \[W^{R}_{D_{c}f,D_{c}g}(\mathbf{x},\boldsymbol{\omega})=c^{3}W^{R}_{f,g}\Big{(} \frac{\mathbf{x}}{c},c\boldsymbol{\omega}\Big{)};\]
* _Nonlinearity_: \[W^{R}_{f+g}(\mathbf{x},\boldsymbol{\omega})=W^{R}_{f}(\mathbf{x},\boldsymbol{ \omega})+W^{R}_{g}(\mathbf{x},\boldsymbol{\omega})+W^{R}_{f,g}(\mathbf{x}, \boldsymbol{\omega})+W^{R}_{g,f}(\mathbf{x},\boldsymbol{\omega});\]
* _Space support conservation_: \[\forall_{|\mathbf{x}|>R}f(\mathbf{x})=0\quad\Rightarrow\quad\forall_{| \mathbf{x}|>R,\boldsymbol{\omega}\in\mathbb{R}^{3}}W^{R}_{f}(\mathbf{x}, \boldsymbol{\omega})=0;\]* _Parity_: if \(f\) and \(g\) are \(\mathbb{R}\)-valued, then \[W^{R}_{f,g}(\mathbf{x},\mathbf{\omega})=W^{R}_{g,f}(\mathbf{x},-\mathbf{\omega})\] and \[W^{R}_{f}(\mathbf{x},\mathbf{\omega})=W^{R}_{f}(\mathbf{x},-\mathbf{\omega}).\] We omit the details of the proofs here. Unfortunately, the _modulation property_ and the _convolution theorem_ are not possible to be formulated in their standard forms due to the non-associativity of octonion multiplication. However, we can provide formulas for some specific cases of modulation property.