# On the Octonion Cross Wigner Distribution of 3-D Signals
## 7 Discussion and Conclusions  
This article introduced a new approach to the Wigner distribution of multivariable signals. Not only the definition of the octonion Wigner distribution was formally introduced, but also a number of its most essential properties were demonstrated. These are generalizations of a considerable number of theorems, known from classical signal theory, for octonion-valued signals of three variables. Among them, the most important are Parseval's equality, Moyal's formula, or the uncertainty principle.  
In the literature, one can already find articles describing the use of octonions for the multispectral signal analysis [31]. The already obtained theoretical results concerning the OWD show the possibility of a wider use of this approach. Moreover, the numerical tools developed in previous research [48] facilitate the theoretical and practical research; however, there are still many limitations to such an approach. Among others, it is the lack of time-frequency symmetry, which may make this variant of the Wigner distribution less popular. Further work will aim to circumvent some of these limitations to define another form of symmetry and extend OWD applications' scope.  
The Wigner distribution approach to time-frequency signal analysis is not the only possibility. Recently, articles (mainly theoretical) have started to appear in which other tools (e.g., linear canonical transform or short-time Fourier transform) are also used based on the octonion algebra [42; 56; 57]; therefore, it is a developmental topic and gives hope for further interesting results, not only theoretical ones. We hope that results of our research will inspire other scientists to look for novel applications in the domain of multidimensional signals.  
It should be noted that the OWD is also related to the concept of the ambiguity function introduced in this paper. The investigation of the properties of this function (including Siebert equality or Price equality) is out of the scope of this paper and can be an interesting subject of future research in this domain.  
It is worth remembering that the presented definitions can be generalized to higher order Cayley-Dickson algebras (e.g., sedenions, which are an algebra of the order 16); however, it should be borne in mind that the higher the order of the Cayley-Dickson algebra, the worse the properties of the multiplication (in particular sedenion multiplication is not alternative and there are zero divisors) [51]. It is therefore necessary to consider the sense of further generalizations.  
**Author Contributions:** Conceptualization, K.S.; methodology, L.B.; software, L.B.; validation, L.B. and K.S; formal analysis, L.B.; investigation, L.B. and K.S.; resources, L.B. and K.S.; data curation, L.B.; writing--original draft preparation, L.B.; writing--review and editing, L.B. and K.S.; visualization, L.B.; supervision, K.S.; project administration, K.S.; funding acquisition, L.B. and K.S. All authors have read and agreed to the published version of the manuscript.  
**Funding:** Research conducted by the first author was funded by POB Cybersecurity and data analysis of Warsaw University of Technology within the Excellence Initiative: Research University (IDUB) programme.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Not applicable.  
**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study, in the writing of the manuscript, or in the decision to publish the results.