# On the Octonion Cross Wigner Distribution of 3-D Signals
## 1 Introduction  
This paper is focused on spatial-frequency distributions of hypercomplex multidimensional signals defined by means of Cayley-Dickson algebras. In the last thirty years, the hypercomplex Cayley-Dickson algebras interfered with the multidimensional signal theory. The first step was the definition of the quaternion Fourier transformation introduced by [PERSON] in 1992 [1]. This concept was applied by [PERSON] in 1999 in his quaternion analytic signal defined as the inverse quaternion Fourier transform of the single-quadrant quaternion spectrum [2]. In parallel, the theory of [PERSON]'s multidimensional analytic signals [3] has been developing, and finally, it was combined for the 2-D case with the quaternion Fourier approach [4]. In recent years, more and more works have started to appear, in which authors use in their research hypercomplex algebras [5; 6; 7]. The area of application is focused mainly on the analysis of color images [8; 9; 10; 11; 12], as well as the biomedical signals processing [13]. From the practical point of view, applying the complex Fourier transformation for frequency analysis of nonstationary signals is not useful. This problem was solved in the works of [PERSON] and [PERSON] [14] who adapted the Wigner distribution [15] as a tool of time-frequency analysis of 1-D complex signals. In the 1990s, the theory of different bilinear time-frequency distributions has been intensively developing and resulted in a multitude of monographs and scientific papers. Among them, the most significant are: the review paper by [PERSON] [16], the monograph by [PERSON] and [PERSON] [17] completely devoted to the Wigner distribution, the [PERSON]'s book [18] on time-frequency and time-scale analysis, and numerous co-author papers by [PERSON] et al. [19; 20]. In all of them, the time-frequency analysis has been performed using the complex Fourier approach. The works of [PERSON] and [PERSON] generalized the theory of time-frequency distributions into [PERSON]'s class of multidimensional space-frequency distributions, and the final result of the long-term research was the monograph published in 2016 [6]. One of its chapters is devoted to the quaternion Wigner distribution and the quaternion Woodward's ambiguity function defined by the same authors a decade earlier [21]. Independently in 2014, [PERSON] introduced the definition of the cross two-sided quaternion Wigner-Ville distribution (CQWVD) [22]. His approach is different compared to the definition proposed by [PERSON] and [PERSON] in 2005, who applied the right-sided quaternion Fourier transform of the auto-correlation product. The CQWVD was introduced in two forms: as the two-sided quaternion Fourier transform of the cross-correlation product and the right-sided one. Their basic properties were presented with proofs, namely the shift, modulation, and convolution theorems, as well as [PERSON]'s formula. In [22], the cross two-sided quaternion ambiguity function was also defined, and some of its properties were presented. This concept has been generalized further by [PERSON] et al. [23] in association with the linear canonical transform presented in detail in [24]. In the literature, one can also find considerations on the Fourier-Wigner transformations using hypercomplex commutative algebras, e.g., bicomplex numbers [25]. [PERSON] and [PERSON] are also authors of the definition of the octonion Fourier transformations (DFT) proposed in 2011 and based on the octonion algebra [26]. It should be noted that at the beginning, octonion algebra attracted the great interest of physicists and was mainly applied in mechanics, electrodynamics, and gravity theory [27; 28; 29; 30]; however, its potential has also been recognized in completely different research fields such as color and multispectral image processing [9; 31; 32; 33; 34] or biomedical acoustic signal analysis via the artificial neural networks [35]. The idea of the octonion-based neural networks also appeared a few years ago [36; 37; 38; 39].