# On the Octonion Cross Wigner Distribution of 3-D Signals
## 6 OWD and Analytic Signals
### Time-Frequency Symmetry  
One of the most important results in the literature on the properties of the Wigner distribution is the symmetry between the time and frequency definitions [14]. In classical setup, the cross Wigner distribution of two functions \(f\) and \(g\) is the Fourier transform of their correlation product, i.e.,  
\[W_{f,g}(\mathbf{x},\mathbf{\omega})=\int_{\mathbb{R}^{3}}\underbrace{f\Big{(} \mathbf{x}+\frac{\mathbf{y}}{2}\Big{)}\overline{g\Big{(}\mathbf{x}-\frac{ \mathbf{y}}{2}\Big{)}}}_{=h(\mathbf{y};\mathbf{x})}e^{-2\pi\mathrm{i}\mathbf{ \omega}\cdot\mathbf{y}}\,\mathrm{d}\mathbf{y}.\]  
If we consider the correlation product of Fourier transforms of two functions, i.e.,  
\[H(\mathbf{\zeta};\mathbf{\omega})=F\bigg{(}\mathbf{\omega}+\frac{\mathbf{\zeta}}{2}\bigg{)} \overline{G\bigg{(}\mathbf{\omega}-\frac{\mathbf{\zeta}}{2}\bigg{)}},\]  
where \(F\) and \(G\) are (classical) Fourier transforms of \(f\) and \(g\), respectively, and compute the cross Wigner distribution of \(F\) and \(G\) (as the classical Fourier transform of \(H(\cdot;\mathbf{\omega})\)), we obtain  
\[W_{F,G}(\mathbf{\omega},\mathbf{x})=\int_{\mathbb{R}^{3}}F\bigg{(}\mathbf{\omega}+ \frac{\mathbf{\zeta}}{2}\bigg{)}\overline{G\bigg{(}\mathbf{\omega}-\frac{\mathbf{\zeta}} {2}\bigg{)}}e^{-2\pi\mathrm{i}\mathbf{\kappa}\cdot\mathbf{\zeta}}\,\mathrm{d}\mathbf{\zeta}. \tag{4}\]  
Then the following formula is true  
\[W_{f,g}(\mathbf{x},\mathbf{\omega})=W_{F,G}(\mathbf{\omega},-\mathbf{x}).\]  
This equality can be interpreted as follows: the Wigner distribution of the spectra of two signals can be obtained from that of those signals in time domain by simply interchanging frequency and time variables (and also reversing the sign of time variable) [14]. So we have a very characteristic property of symmetry.  
It would seem that an attempt could be made to demonstrate a similar property in the case of the octonion Wigner distribution. Consider the octonion correlation product of the OFT of two signals \(f,g\colon\mathbb{R}^{3}\to\mathbb{O}\), i.e.,  
\[H(\mathbf{\zeta};\mathbf{\omega})=F_{\text{OFF}}\bigg{(}\mathbf{\omega}+\frac{\mathbf{\zeta}} {2}\bigg{)}\overline{G_{\text{OFF}}\bigg{(}\mathbf{\omega}-\frac{\mathbf{\zeta}}{2} \bigg{)}}.\]  
The cross right-sided OWD of those OFTs is  
\[W_{F_{\text{OFF}},G_{\text{OFF}}}^{R}(\mathbf{\omega},\mathbf{x})=\int_{\mathbb{R }^{3}}F_{\text{OFF}}\bigg{(}\mathbf{\omega}+\frac{\mathbf{\zeta}}{2}\bigg{)} \overline{G_{\text{OFF}}\bigg{(}\mathbf{\omega}-\frac{\mathbf{\zeta}}{2}\bigg{)}}\cdot e ^{-2\pi\mathrm{e}_{1}x_{1}\xi_{1}}e^{-2\pi\mathrm{e}_{2}x_{2}\xi_{2}}e^{-2\pi \mathrm{e}_{4}x_{3}\xi_{3}}\,\mathrm{d}\mathbf{\zeta},\]  
where the multiplication is performed from left to right. Unfortunately, the expected property of symmetry, i.e.,  
\[W_{f,g}^{R}(\mathbf{x},\mathbf{\omega})=W_{F_{\text{OFF}},G_{\text{OFF}}}^{R}(\bm {\omega},-\mathbf{x}) \tag{5}\]  
is satisfied in just a few specific cases, e.g., for separable real-valued functions even with respect to every coordinate, such as Gaussian functions.  
One could try a slightly different approach. Note that \(W_{F,G}(\mathbf{\omega},-\mathbf{x})\) defined as in Equation (4) can also be treated as the inverse Fourier transform of \(H(\cdot;\mathbf{\omega})\) computed at \(\mathbf{x}\). This suggests the use of the inverse octonion Fourier transform as well, i.e., considering the expression  
\[\int_{\mathbb{R}^{3}}F_{\text{OFF}}\bigg{(}\mathbf{\omega}+\frac{\mathbf{\zeta}}{2} \bigg{)}\overline{G_{\text{OFF}}\bigg{(}\mathbf{\omega}-\frac{\mathbf{\zeta}}{2}\bigg{)} }\cdot e^{2\pi\mathrm{e}_{4}x_{3}\xi_{3}}e^{2\pi\mathrm{e}_{2}x_{2}\xi_{2}}e^{ 2\pi\mathrm{e}_{1}x_{1}\xi_{1}}\,\mathrm{d}\mathbf{\zeta}.\]As before, this expression is only equal to \(W_{f,g}^{R}(\mathbf{x},\mathbf{\omega})\) in very specific cases. Unfortunately, the above observations have an impact on another (important from the point of view of applications) issue.