On the Octonion Cross Wigner Distribution of 3-D Signals

Abstract

This paper introduces definitions of the octonion cross Wigner distribution (OWD) and the octonion ambiguity function, forming a pair of octonion Fourier transforms. The main part is devoted to the study of the basic properties of the OWD. Among them are the properties concerning its nature (nonlinearity, parity, space support conservation, marginals) and some “geometric” transformations (space shift, space scaling) similar to the case of the complex Wigner distribution. This paper also presents specific forms of the modulation property and an extended discussion about the validity of Moyal’s formula and the uncertainty principle, accompanied by new theorems and examples. The paper is illustrated with examples of 3-D separable Gaussian and Gabor signals. The concept of the application of the OWD for the analysis of multidimensional analytic signals is also proposed. The theoretical results presented in the papers are summarized, and the possibility of further research is discussed.

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 T.A. Ell in 1992 [1]. This concept was applied by T. Bülow 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 Hahn’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 Claasen and Mecklenbräuker [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 Cohen [16], the monograph by Mecklenbräuker and Hlawatsch [17] completely devoted to the Wigner distribution, the Flandrin’s book [18] on time–frequency and time–scale analysis, and numerous co-author papers by Stanković et al. [19,20]. In all of them, the time–frequency analysis has been performed using the complex Fourier approach. The works of Hahn and Snopek generalized the theory of time–frequency distributions into Cohen’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, Bahri introduced the definition of the cross two-sided quaternion Wigner–Ville distribution (CQWVD) [22]. His approach is different compared to the definition proposed by Hahn and Snopek 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 Moyal’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 Fan 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].

Hahn and Snopek are also authors of the definition of the octonion Fourier transformations (OFT) 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 [9,11,31,32,33,34] 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]. The above-cited application-oriented works are accompanied by a growing number of pure theoretical papers concerning octonion algebra [40,41,42,43,44,45].

The octonion Fourier transformation concept has extensively been studied and developed in the papers of Błaszczyk and Snopek [46,47,48,49] where the basic properties of OFT were studied and proved, and continued in the works of other authors [40,41,42,43,44]. The door has been opened to successive generalizations in the field of octonion signal processing, particularly in the multidimensional hypercomplex discrete signal processing. One of the recent articles [50], for example, introduces the definition and presents a study of fundamental properties of the discrete OFT. This paper, devoted to the octonion Wigner distribution and the octonion ambiguity function, is an essential step towards the generalization of the concept of time–frequency analysis in terms of the hypercomplex Cayley–Dickson algebras.

The article is organized as follows. We begin with recalling basic notation and facts concerning the octonion algebra and octonion Fourier transform in Section 2. The main part of this paper, Section 3, is devoted to the octonion Wigner distribution and its most important properties. In Section 4 we illustrate our results with some examples. Section 5 and Section 6 provide comments and thoughts on how to extend OWD theory to other issues such as the octonion ambiguity function and the octonion Wigner–Ville distribution. We conclude with a short résumé of the obtained results and our further research perspectives.

2. Basic Definitions

In this section, we introduce all the tools necessary to present the main results of this work regarding the octonion Wigner distribution and the octonion ambiguity function.

2.1. 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 $o_0,\dots ,o_7\in \mathbb{R}$ and ${\mathbf{e}}_1,\dots ,{\mathbf{e}}_7$ are seven octonion imaginary units, each satisfying the property ${\mathbf{e}}_i^2={\mathbf{e}}_i·{\mathbf{e}}_i=-1$, $i=1,\dots ,7$, and other octonion multiplication rules shown in Figure 1. Number $o_0\in \mathbb{R}$ is called the real part of o (denoted as $\mathrm{Re}o$) and the pure imaginary octonion $o_1{\mathbf{e}}_1+o_2{\mathbf{e}}_2+\dots +o_7{\mathbf{e}}_7$ is called the imaginary part of o (and is denoted as $\mathrm{Im}o$). Octonions form a non-associative and a non-commutative algebra, which means that in general, for $o_1,o_2,o_3\in \mathbb{O}$

$$(o_1·o_2)·o_3\ne o_1·(o_2·o_3),o_1·o_2\ne o_2·o_1.$$

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

$$\overline{(o_1·o_2)}=\overline{o_2}·\overline{o_1}.$$

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 $o\in \mathbb{O}$:

$$o=\left|o\right|·(cos\theta +\mathbf{\mu }·sin\theta ),$$

where $\left|o\right|=\sqrt{o·\overline{o}}$ is the octonion norm, $\mathbf{\mu }=\frac{\mathrm{Im}o}{\left|\mathrm{Im}o\right|}$ is a pure imaginary octonion, and $\theta \in \mathbb{R}$ is the solution of the system of equations

$$cos\theta =\frac{\mathrm{Re}o}{\left|o\right|},sin\theta =\frac{\left|\mathrm{Im}o\right|}{\left|o\right|}.$$

It should be noted that every non-zero octonion is invertible and for every $o\in \mathbb{O}\backslash \left\{0\right\}$ we have $o^{-1}=\frac{\overline{o}}{\left|o\right|}$, which means that for pure unitary octonions $\mathbf{\mu }$ ($\left|\mathbf{\mu }\right|=1$) 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

$$e^o=exp\left(o\right):=\sum _{k=0}^{\infty }\frac{o^k}{k!}.$$

It can be shown that if we denote $\mathbf{o}=\mathrm{Im}o$, then

$$e^o=e^{\mathrm{Re}o}\left(cos\left|\mathbf{o}\right|+\frac{\mathbf{o}}{\left|\mathbf{o}\right|}sin\left|\mathbf{o}\right|\right).$$

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

$$e^{o_1+o_2}=e^{o_1}·e^{o_2}\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}{o}_1·o_2=o_2·o_1.$$

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

$$cos\alpha =\frac{1}{2}\left(e^{\mathbf{\mu }\alpha }+e^{-\mathbf{\mu }\alpha }\right),sin\alpha =-\frac{\mathbf{\mu }}{2}\left(e^{\mathbf{\mu }\alpha }-e^{-\mathbf{\mu }\alpha }\right),$$

where $\mathbf{\mu }$ is any pure unitary octonion.

2.2. Octonion Fourier Transform

Definition of the octonion Fourier transform (OFT) of the real-valued function of three variables was introduced in [53] and used in later publications, as indicated in Section 1. In [49], it was proved that the OFT of real-valued function is well-defined and has some interesting properties. These results were further generalized to the case of octonion-valued functions (see [46,50]). In the next part of this section, we recall the most important results on this topic.

Consider the octonion-valued function of three variables $u:{\mathbb{R}}^3\to \mathbb{O}$, i.e.,

$$u\left(\mathbf{x}\right)=u_0\left(\mathbf{x}\right)+u_1\left(\mathbf{x}\right){\mathbf{e}}_1+\dots +u_7\left(\mathbf{x}\right){\mathbf{e}}_7,$$

where $u_i:{\mathbb{R}}^3\to \mathbb{R}$, $i=0,\dots ,7$, and $\mathbf{x}=(x_1,x_2,x_3)$. The octonion Fourier transform of the integrable (in Lebesgue sense) function u is given by the formula

$$U_{\mathrm{OFT}}\left(\mathbf{f}\right)={\int }_{{\mathbb{R}}^3}u\left(\mathbf{x}\right)e^{-{\mathbf{e}}_{1}2\pi f_1{x}_1}{e}^{-{\mathbf{e}}_{2}2\pi f_2{x}_2}{e}^{-{\mathbf{e}}_{4}2\pi f_3{x}_3}\mathrm{d}\mathbf{x}.$$

Since the octonion algebra is non-associative, we assume that the multiplication in the above integrals is performed from left to right. The conditions for the existence of the OFT and its advantages over the classical approach have been discussed in detail in previous works [49,50].

Theorem 1.

Let $u\in \mathit{S}({\mathbb{R}}^3;\mathbb{O})$. Then for all $\mathbf{x}\in {\mathbb{R}}^3$ we have

$$u\left(\mathbf{x}\right)={\int }_{{\mathbb{R}}^3}{U}_{\mathrm{OFT}}\left(\mathbf{f}\right)e^{{\mathbf{e}}_{4}2\pi f_3{x}_3}{e}^{{\mathbf{e}}_{2}2\pi f_2{x}_2}{e}^{{\mathbf{e}}_{1}2\pi f_1{x}_1}\mathrm{d}\mathbf{f}$$

(where multiplication is performed from left to right).

Let us recall that the symbol $\mathit{S}({\mathbb{R}}^3;\mathbb{O})$ denotes the Schwartz class of rapidly decreasing functions (generalized to the case of octonion-valued functions). The above claim was proved in one of the previous articles [50] and we omit the details here. It is worth adding, however, that the above theorem is also true under other assumptions, which were also discussed. It should be noted that for the function $u\in L^1({\mathbb{R}}^3;\mathbb{O})$, its octonion Fourier transform is uniformly continuous, and the claim of the Riemann–Lebesgue lemma holds:

$$\underset{\left|\mathbf{f}\right|\to \infty }{lim}{U}_{\mathrm{OFT}}\left(\mathbf{f}\right)=0.$$

In previous studies, we also proved a number of OFT properties [46,47,49,50]. The most important of them (and those that we need in further discussion on the Wigner distribution) are collected in

Table 1. Summary of Octonion Fourier Transform Properties.

	Function 1	Octonion Fourier Transform
1.	$u(\frac{x_1}{a},\frac{x_2}{b},\frac{x_3}{c})$	$\left|abc\right|U(a{f}_1,b{f}_2,c{f}_3)$
2.	$u\left(\mathbf{x}\right)·cos\left(2\pi f_0{x}_1\right)$	$\left(U(f_1+f_0,f_2,f_3)+U(f_1-f_0,f_2,f_3)\right)·\frac{1}{2}$
	$u\left(\mathbf{x}\right)·cos\left(2\pi f_0{x}_2\right)$	$\left(U(f_1,f_2+f_0,f_3)+U(f_1,f_2-f_0,f_3)\right)·\frac{1}{2}$
	$u\left(\mathbf{x}\right)·cos\left(2\pi f_0{x}_3\right)$	$\left(U(f_1,f_2,f_3+f_0)+U(f_1,f_2,f_3-f_0)\right)·\frac{1}{2}$
3.	$u\left(\mathbf{x}\right)·sin\left(2\pi f_0{x}_1\right)$	$\left(U(f_1+f_0,-f_2,-f_3)-U(f_1-f_0,-f_2,-f_3)\right)·\frac{{\mathbf{e}}_1}{2}$
	$u\left(\mathbf{x}\right)·sin\left(2\pi f_0{x}_2\right)$	$\left(U(f_1,f_2+f_0,-f_3)-U(f_1,f_2-f_0,-f_3)\right)·\frac{{\mathbf{e}}_2}{2}$
	$u\left(\mathbf{x}\right)·sin\left(2\pi f_0{x}_3\right)$	$\left(U(f_1,f_2,f_3+f_0)-U(f_1,f_2,f_3-f_0)\right)·\frac{{\mathbf{e}}_4}{2}$
4.	$u\left(\mathbf{x}\right)·exp(-{\mathbf{e}}_{1}2\pi f_0{x}_1)$	$U(f_1+f_0,f_2,f_3)$
	$u\left(\mathbf{x}\right)·exp(-{\mathbf{e}}_{2}2\pi f_0{x}_2)$	$\left(U(f_1,f_2+f_0,f_3)+U(f_1,f_2-f_0,f_3)+U(-f_1,f_2+f_0,f_3)-U(-f_1,f_2-f_0,f_3)\right)·\frac{1}{2}$
	$u\left(\mathbf{x}\right)·exp(-{\mathbf{e}}_{4}2\pi f_0{x}_3)$	$\left(U(f_1,f_2,f_3+f_0)+U(f_1,f_2,f_3-f_0)+U(-f_1,-f_2,f_3+f_0)-U(-f_1,-f_2,f_3-f_0)\right)·\frac{1}{2}$
5.	$u(x_1-\alpha ,x_2,x_3)$	$cos\left(2\pi f_1\alpha \right)U(f_1,f_2,f_3)-sin\left(2\pi f_1\alpha \right)U(f_1,-f_2,-f_3)·{\mathbf{e}}_1$
	$u(x_1,x_2-\beta ,x_3)$	$cos\left(2\pi f_2\beta \right)U(f_1,f_2,f_3)-sin\left(2\pi f_2\beta \right)U(f_1,f_2,-f_3)·{\mathbf{e}}_2$
	$u(x_1,x_2,x_3-\gamma )$	$cos\left(2\pi f_3\gamma \right)U(f_1,f_2,f_3)-sin\left(2\pi f_3\gamma \right)U(f_1,f_2,f_3)·{\mathbf{e}}_4$
6.	$(u\ast v)\left(\mathbf{x}\right)$ 2	$\phantom{+}V(\phantom{-}{f}_1,\phantom{-}{f}_2,\phantom{-}{f}_3)·(\phantom{-}{U}_{eee}\left(\mathbf{f}\right)\phantom{{\mathbf{e}}_1}-U_{eeo}\left(\mathbf{f}\right){\mathbf{e}}_4)+V(\phantom{-}{f}_1,-f_2,-f_3)·(-U_{oee}\left(\mathbf{f}\right){\mathbf{e}}_1+U_{ooe}\left(\mathbf{f}\right){\mathbf{e}}_3)$
		$+V(\phantom{-}{f}_1,\phantom{-}{f}_2,-f_3)·(-U_{eoe}\left(\mathbf{f}\right){\mathbf{e}}_2+U_{oeo}\left(\mathbf{f}\right){\mathbf{e}}_5)+V(-f_1,\phantom{-}{f}_2,-f_3)·(\phantom{-}{U}_{eoo}\left(\mathbf{f}\right){\mathbf{e}}_6-U_{ooo}\left(\mathbf{f}\right){\mathbf{e}}_7)$

¹ Function U denotes the OFT of the function u: ${\mathbb{R}}^3\to \mathbb{O}$. ² Property 6 is valid only for real-valued functions.

. In the further part of the paper, we focus our attention on the most critical signal theory theorems, i.e., the Plancherel’s theorem and the Rayleigh’s theorem proved in our earlier work [49].

Theorem 2.

Let $u,v\in L^2({\mathbb{R}}^3;\mathbb{R})$. Then

$${\left(u,v\right)}_{L^2({\mathbb{R}}^3;\mathbb{O})}={\left(U_{\mathrm{OFT}},V_{\mathrm{OFT}}\right)}_{L^2({\mathbb{R}}^3;\mathbb{O})},$$

where ${\left(·,·\right)}_{L^2({\mathbb{R}}^3;\mathbb{O})}$ denotes the classical scalar product of functions (real- or octonion-valued).

The assumption that the functions u and v are real-valued is relevant [46]. In the case of real-valued functions, Rayleigh’s theorem is a direct corollary of the above-mentioned result, but it is also valid in the general case of octonion-valued functions [44,46].

Theorem 3.

$L^2$-norm of any function $u\in L^2({\mathbb{R}}^3;\mathbb{O})$ is equal to the $L^2$-norm of its octonion Fourier transform, i.e.,

$${∥u∥}_{L^2({\mathbb{R}}^3;\mathbb{O})}={∥U_{\mathrm{OFT}}∥}_{L^2({\mathbb{R}}^3;\mathbb{O})}.$$

On the other hand, we could consider the symmetric real scalar product

$$\begin{array}{cc}{〈f,g〉}_{L^2({\mathbb{R}}^d;\mathbb{O})}& =\frac{1}{2}\left({\left(f,g\right)}_{L^2({\mathbb{R}}^d;\mathbb{O})}+{\left(g,f\right)}_{L^2({\mathbb{R}}^d;\mathbb{O})}\right)={\int }_{{\mathbb{R}}^d}\mathrm{Re}\left(f\left(\mathbf{x}\right)\overline{g\left(\mathbf{x}\right)}\right)\mathrm{d}\mathbf{x}.\end{array}$$

Using exactly the same reasoning as in the proof of the Rayleigh theorem for octonion-valued functions (see [46]), we immediately obtain

$${〈u,v〉}_{L^2({\mathbb{R}}^3;\mathbb{O})}={〈U_{\mathrm{OFT}},V_{\mathrm{OFT}}〉}_{L^2({\mathbb{R}}^3;\mathbb{O})}$$

for all $u,v\in L^2({\mathbb{R}}^3;\mathbb{O})$.

In the context of the signal theory, the uncertainty principle plays a vital role. Let us recall the theorem concerning its form for the OFT, proved by Lian in [44].

Theorem 4.

Let $u\in L^2({\mathbb{R}}^3;\mathbb{O})$. For $s,\beta >0$, there exists a constant $C_{s,\beta }>0$ such that

$$\begin{array}{cc}& {∥{\left|\mathbf{x}\right|}^{s}u\left(\mathbf{x}\right)∥}_{L^2({\mathbb{R}}^3;\mathbb{O})}^{\frac{2\beta }{s+\beta }}{∥{\left|\mathbf{f}\right|}^{\beta }{U}_{\mathrm{OFT}}\left(\mathbf{f}\right)∥}_{L^2({\mathbb{R}}^3;\mathbb{O})}^{\frac{2s}{s+\beta }}\ge C_{s,\beta }{∥u∥}_{L^2({\mathbb{R}}^3;\mathbb{O})}^2.\end{array}$$

3. Octonion Wigner Distribution

We devote this section to the new concept of the octonion Wigner distribution. In addition to the definitions and basic properties (which are analogs of the corresponding properties of the classical Wigner distribution), we also focus on the important (from the point of view of signal theory and applications) theorems concerning Moyal’s formula and the uncertainty principle.

3.1. Definition and Invertibility

The definition presented below is a natural generalization of the definition of the 4-D Wigner distribution of 2-D quaternion signals (see [6,21]) to the octonion algebra. We introduce it in a general form called the octonion cross Wigner distribution. Throughout the rest of this article, we repeatedly refer to the octonion correlation product of two signals $f,g:{\mathbb{R}}^3\to \mathbb{O}$, i.e.,

$$h(\mathbf{y};\mathbf{x})=f\left(\mathbf{x}+\frac{\mathbf{y}}{2}\right)\overline{g\left(\mathbf{x}-\frac{\mathbf{y}}{2}\right)}.$$

This is obviously a function of the six variables, but we group them into triples $\mathbf{x}$ and $\mathbf{y}$.

Definition 1.

The octonion cross right-sided Wigner distribution (OWD) of the 3-D signals $f,g\in L^2({\mathbb{R}}^3;\mathbb{O})$ is defined as

$$\begin{array}{cc}{W}_{f,g}^R(\mathbf{x},\mathbf{\omega })=& {\int }_{{\mathbb{R}}^3}f\left(\mathbf{x}+\frac{\mathbf{y}}{2}\right)\overline{g\left(\mathbf{x}-\frac{\mathbf{y}}{2}\right)}·e^{-2\pi {\mathbf{e}}_1{\omega }_1{y}_1}{e}^{-2\pi {\mathbf{e}}_2{\omega }_2{y}_2}{e}^{-2\pi {\mathbf{e}}_4{\omega }_3{y}_3}\mathrm{d}\mathbf{y},\end{array}$$

(1)

where $(\mathbf{x},\mathbf{\omega })\in {\mathbb{R}}^3\times {\mathbb{R}}^3$ and multiplication is performed from left to right.

Other definitions of OWD are, of course, possible (left-sided distribution or two different versions of two-sided distributions, etc.), which would be identical for real-valued functions, but we use the one introduced in Definition 1.

It is also worth noting that in the literature, the Wigner distribution is often defined not for two different functions f and g, but a single function f. The definition presented above covers a more general situation, and we denote this particular case as $W_f^R:=W_{f,f}^R$.

It should be noted that if $f,g\in L^2({\mathbb{R}}^3;\mathbb{O})$ then for every $\mathbf{x}\in {\mathbb{R}}^3$ we have (from Cauchy–Schwarz inequality) $h(·;\mathbf{x})\in L^1({\mathbb{R}}^3;\mathbb{O})$, so the OWD of $f,g\in L^2({\mathbb{R}}^3;\mathbb{O})$ is well-defined. The OWD is in fact the OFT of octonion correlation product h (with respect to the first three variables, i.e., $\mathbf{y}$). Applying the properties of OFT proved in previous works [46,50], we can express $h(\mathbf{y};\mathbf{x})$ for every $(\mathbf{x},\mathbf{y})\in {\mathbb{R}}^3\times {\mathbb{R}}^3$ in the form:

$$f\left(\mathbf{x}+\frac{\mathbf{y}}{2}\right)\overline{g\left(\mathbf{x}-\frac{\mathbf{y}}{2}\right)}={\int }_{{\mathbb{R}}^3}{W}_{f,g}^R(\mathbf{x},\mathbf{\omega })e^{2\pi {\mathbf{e}}_4{\omega }_3{y}_3}{e}^{2\pi {\mathbf{e}}_2{\omega }_2{y}_2}{e}^{2\pi {\mathbf{e}}_1{\omega }_1{y}_1}\mathrm{d}\mathbf{\omega },$$

(2)

where multiplication is performed from left to right. Of course this formula is valid under certain assumptions, e.g.,

• $W_{f,g}^R(\mathbf{x},·)\in L^1({\mathbb{R}}^3;\mathbb{O})$ for every $\mathbf{x}\in {\mathbb{R}}^3$,
• $f,g\in \mathit{S}({\mathbb{R}}^3;\mathbb{O})$,
• $h(·;\mathbf{x})\in L^2({\mathbb{R}}^3;\mathbb{O})$ for every $\mathbf{x}\in {\mathbb{R}}^3$,

etc. For the rest of the article (unless stated otherwise) we will assume that f and g are rapidly decreasing functions. Putting $\mathbf{x}=\frac{\mathbf{y}}{2}$ we immediately obtain the formula for recovering the function f from its OWD (as long as $g\left(\mathbf{0}\right)\ne 0$).

Theorem 5.

Let $f,g\in \mathit{S}({\mathbb{R}}^3;\mathbb{O})$ and $g\left(\mathbf{0}\right)\ne 0$. Then for every $\mathbf{y}\in {\mathbb{R}}^3$

$$f\left(\mathbf{y}\right)={\int }_{{\mathbb{R}}^3}{W}_{f,g}^R\left(\frac{\mathbf{y}}{2},\mathbf{\omega }\right)e^{2\pi {\mathbf{e}}_4{\omega }_3{y}_3}{e}^{2\pi {\mathbf{e}}_2{\omega }_2{y}_2}{e}^{2\pi {\mathbf{e}}_1{\omega }_1{y}_1}\mathrm{d}\mathbf{\omega }·\frac{g\left(\mathbf{0}\right)}{\left|g\left(\mathbf{0}\right)\right|},$$

where multiplication is performed from left to right.

3.2. 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

$${\int }_{{\mathbb{R}}^3}{W}_{f,g}^R(\mathbf{x},\mathbf{\omega })\mathrm{d}\mathbf{\omega }=f\left(\mathbf{x}\right)\overline{g\left(\mathbf{x}\right)},$$

(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

$${\int }_{{\mathbb{R}}^3}{\int }_{{\mathbb{R}}^3}{W}_{f,g}^R(\mathbf{x},\mathbf{\omega })\mathrm{d}\mathbf{\omega }\mathrm{d}\mathbf{x}={\int }_{{\mathbb{R}}^3}f\left(\mathbf{x}\right)\overline{g\left(\mathbf{x}\right)}\mathrm{d}\mathbf{x}$$

and in the case $g=f$ we obtain Parseval’s equality:

$${\int }_{{\mathbb{R}}^3}{\int }_{{\mathbb{R}}^3}{W}_f^R(\mathbf{x},\mathbf{\omega })\mathrm{d}\mathbf{\omega }\mathrm{d}\mathbf{x}={∥f∥}_{L^2({\mathbb{R}}^3;\mathbb{O})}^2.$$

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 $T_{\mathbf{a}}f\left(\mathbf{x}\right)=f(\mathbf{x}-\mathbf{a})$, then

  $$W_{T_{\mathbf{a}}f,T_{\mathbf{a}}g}^R(\mathbf{x},\mathbf{\omega })=W_{f,g}^R(\mathbf{x}-\mathbf{a},\mathbf{\omega });$$
• Space scaling (dilation): let $c>0$ and $D_{c}f\left(\mathbf{x}\right)=f\left(\frac{\mathbf{x}}{c}\right)$, then

  $$W_{D_{c}f,D_{c}g}^R(\mathbf{x},\mathbf{\omega })=c^3{W}_{f,g}^R\left(\frac{\mathbf{x}}{c},c\mathbf{\omega }\right);$$
• Nonlinearity:

  $$W_{f+g}^R(\mathbf{x},\mathbf{\omega })=W_f^R(\mathbf{x},\mathbf{\omega })+W_g^R(\mathbf{x},\mathbf{\omega })+W_{f,g}^R(\mathbf{x},\mathbf{\omega })+W_{g,f}^R(\mathbf{x},\mathbf{\omega });$$
• Space support conservation:

  $${\forall }_{\left|\mathbf{x}\right|>R}f\left(\mathbf{x}\right)=0\Rightarrow {\forall }_{\left|\mathbf{x}\right|>R,\mathbf{\omega }\in {\mathbb{R}}^3}{W}_f^R(\mathbf{x},\mathbf{\omega })=0;$$
• Parity: if f and g are $\mathbb{R}$-valued, then

  $$W_{f,g}^R(\mathbf{x},\mathbf{\omega })=W_{g,f}^R(\mathbf{x},-\mathbf{\omega })$$

  and

  $$W_f^R(\mathbf{x},\mathbf{\omega })=W_f^R(\mathbf{x},-\mathbf{\omega }).$$

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. Let ${\mathbf{\omega }}_0>0$ and denote

$$f^{cos,i}\left(\mathbf{x}\right)=f\left(\mathbf{x}\right)·cos\left(2\pi {\omega }_0{x}_i\right),f^{sin,i}\left(\mathbf{x}\right)=f\left(\mathbf{x}\right)·sin\left(2\pi {\omega }_0{x}_i\right).$$

We have

$$\begin{array}{cc}\hfill & cos\left(2\pi {\omega }_0\left(x_i+\frac{y_i}{2}\right)\right)·cos\left(2\pi {\omega }_0\left(x_i-\frac{y_i}{2}\right)\right)=\frac{1}{2}\left(cos\left(2\pi \left(2{\omega }_0\right)x_i\right)+cos\left(2\pi {\omega }_0{y}_i\right)\right),\hfill \\ \hfill & sin\left(2\pi {\omega }_0\left(x_i+\frac{y_i}{2}\right)\right)·sin\left(2\pi {\omega }_0\left(x_i-\frac{y_i}{2}\right)\right)=\frac{1}{2}\left(-cos\left(2\pi \left(2{\omega }_0\right)x_i\right)+cos\left(2\pi {\omega }_0{y}_i\right)\right),\hfill \\ \hfill & cos\left(2\pi {\omega }_0\left(x_i+\frac{y_i}{2}\right)\right)·sin\left(2\pi {\omega }_0\left(x_i-\frac{y_i}{2}\right)\right)=\frac{1}{2}\left(sin\left(2\pi \left(2{\omega }_0\right)x_i\right)-sin\left(2\pi {\omega }_0{y}_i\right)\right),\hfill \\ \hfill & sin\left(2\pi {\omega }_0\left(x_i+\frac{y_i}{2}\right)\right)·cos\left(2\pi {\omega }_0\left(x_i-\frac{y_i}{2}\right)\right)=\frac{1}{2}\left(sin\left(2\pi \left(2{\omega }_0\right)x_i\right)+sin\left(2\pi {\omega }_0{y}_i\right)\right)\hfill \end{array}$$

and using the modulation theorem for the octonion Fourier transform (see [50], Theorem 10 and 11) we immediately deduce that

$$\begin{array}{cc}& W_{f^{cos,i},g^{cos,i}}^R(\mathbf{x},\mathbf{\omega })=\frac{1}{2}{W}_{f,g}^R(\mathbf{x},\mathbf{\omega })·cos\left(2\pi \left(2{\omega }_0\right)x_i\right)\\ & +\frac{1}{4}\left(W_{f,g}^R(\mathbf{x},\mathbf{\omega }+{\mathbf{\omega }}_{0,i})+W_{f,g}^R(\mathbf{x},\mathbf{\omega }-{\mathbf{\omega }}_{0,i})\right),\\ & W_{f^{sin,i},g^{sin,i}}^R(\mathbf{x},\mathbf{\omega })=-\frac{1}{2}{W}_{f,g}^R(\mathbf{x},\mathbf{\omega })·cos\left(2\pi \left(2{\omega }_0\right)x_i\right)\\ & +\frac{1}{4}\left(W_{f,g}^R(\mathbf{x},\mathbf{\omega }+{\mathbf{\omega }}_{0,i})+W_{f,g}^R(\mathbf{x},\mathbf{\omega }-{\mathbf{\omega }}_{0,i})\right),\\ & W_{f^{cos,i},g^{sin,i}}^R(\mathbf{x},\mathbf{\omega })=\frac{1}{2}{W}_{f,g}^R(\mathbf{x},\mathbf{\omega })·sin\left(2\pi \left(2{\omega }_0\right)x_i\right)\\ & -\frac{1}{4}\left(W_{f,g}^R(\mathbf{x},{\mathbf{\omega }}_i+{\mathbf{\omega }}_{0,i})-W_{f,g}^R(\mathbf{x},{\mathbf{\omega }}_i-{\mathbf{\omega }}_{0,i})\right){\mathbf{e}}_{2^{i-1}},\\ & W_{f^{sin,i},g^{cos,i}}^R(\mathbf{x},\mathbf{\omega })=\frac{1}{2}{W}_{f,g}^R(\mathbf{x},\mathbf{\omega })·sin\left(2\pi \left(2{\omega }_0\right)x_i\right)\\ & +\frac{1}{4}\left(W_{f,g}^R(\mathbf{x},{\mathbf{\omega }}_i+{\mathbf{\omega }}_{0,i})-W_{f,g}^R(\mathbf{x},{\mathbf{\omega }}_i-{\mathbf{\omega }}_{0,i})\right){\mathbf{e}}_{2^{i-1}},\end{array}$$

where ${\mathbf{\omega }}_{0,1}=({\omega }_0,0,0)$, ${\mathbf{\omega }}_{0,2}=(0,{\omega }_0,0)$, ${\mathbf{\omega }}_{0,3}=(0,0,{\omega }_0)$ and ${\mathbf{\omega }}_1=({\omega }_1,-{\omega }_2,-{\omega }_3)$, ${\mathbf{\omega }}_2=({\omega }_1,{\omega }_2,-{\omega }_3)$, ${\mathbf{\omega }}_3=\mathbf{\omega }$.

From the fact that OWD is the octonion Fourier transform of the octonion correlation product, it results that:

• for every $(\mathbf{x},\mathbf{\omega })\in {\mathbb{R}}^3$ we have

  $$\left|W_{f,g}^R(\mathbf{x},\mathbf{\omega })\right|\le {\int }_{{\mathbb{R}}^3}\left|f\left(\mathbf{x}+\frac{\mathbf{y}}{2}\right)\right|\left|g\left(\mathbf{x}-\frac{\mathbf{y}}{2}\right)\right|\mathrm{d}\mathbf{y}\le 8{∥f∥}_{L^2({\mathbb{R}}^3;\mathbb{O})}·{∥g∥}_{L^2({\mathbb{R}}^3;\mathbb{O})},$$

  hence $W_{f,g}^R\in L^{\infty }({\mathbb{R}}^3\times {\mathbb{R}}^3;\mathbb{O})$ for all functions $f,g\in L^2({\mathbb{R}}^3;\mathbb{O})$;
• for every $f,g\in L^2({\mathbb{R}}^3;\mathbb{O})$ and $\mathbf{x}\in {\mathbb{R}}^3$, $W_{f,g}^R(\mathbf{x},·)$ is uniformly continuous;
• Riemann–Lebesgue lemma holds with respect to $\mathbf{\omega }$, i.e., for every $f,g\in L^2({\mathbb{R}}^3;\mathbb{O})$ and $\mathbf{x}\in {\mathbb{R}}^3$

  $$\left|W_{f,g}^R(\mathbf{x},\mathbf{\omega })\right|\stackrel{\left|\mathbf{\omega }\right|\to \infty }{\to }0;$$
• for every $f,g\in L^2({\mathbb{R}}^3;\mathbb{O})$ and $\mathbf{x}\in {\mathbb{R}}^3$ such that $h(·;\mathbf{x})\in L^2({\mathbb{R}}^3;\mathbb{O})$ we have $W_{f,g}^R(\mathbf{x},·)\in L^2({\mathbb{R}}^3;\mathbb{O})$.

The last fact characterizes the OWD class for every $\mathbf{x}$. We can also look at the characterization with respect to $\mathbf{\omega }$ using the properties of convolution of $\mathbb{R}$-valued functions. It is known (see [54], Corollary 1.21) that if $F\in L^p({\mathbb{R}}^n;\mathbb{R})$ and $G\in L^1({\mathbb{R}}^n;\mathbb{R})$ for some $p\ge 1$, then $F\ast G\in L^p({\mathbb{R}}^n;\mathbb{R})$, where ∗ denotes classical convolution, and

$${∥F\ast G∥}_{L^p({\mathbb{R}}^n;\mathbb{R})}\le {∥F∥}_{L^p({\mathbb{R}}^n;\mathbb{R})}·{∥G∥}_{L^1({\mathbb{R}}^n;\mathbb{R})}.$$

Assume that $f,g\in L^2({\mathbb{R}}^3;\mathbb{O})\cap L^1({\mathbb{R}}^3;\mathbb{O})$ and $(\mathbf{x},\mathbf{\omega })\in {\mathbb{R}}^3\times {\mathbb{R}}^3$. Then

$$\left|W_{f,g}^R(\mathbf{x},\mathbf{\omega })\right|\le {\int }_{{\mathbb{R}}^3}\left|f\left(\mathbf{x}+\frac{\mathbf{y}}{2}\right)\right|\left|g\left(\mathbf{x}-\frac{\mathbf{y}}{2}\right)\right|\mathrm{d}\mathbf{y}=8{\int }_{{\mathbb{R}}^3}\left|f\left(\mathbf{t}\right)\right|\left|g\left(2\mathbf{x}-\mathbf{t}\right)\right|\mathrm{d}\mathbf{t}.$$

Notice that $H\left(\mathbf{x}\right)={\int }_{{\mathbb{R}}^3}\left|f\left(\mathbf{t}\right)\right|\left|g\left(\mathbf{x}-\mathbf{t}\right)\right|\mathrm{d}\mathbf{t}=(\left|f\right|\ast \left|g\right|)\left(\mathbf{x}\right)$ is the convolution of $\mathbb{R}$-valued functions $\left|f\right|$ and $\left|g\right|$. We have that $H\in L^2({\mathbb{R}}^3;\mathbb{R})\cap L^1({\mathbb{R}}^3;\mathbb{R})$ and

$${∥W_{f,g}^R(·,\mathbf{\omega })∥}_{L^p({\mathbb{R}}^3;\mathbb{O})}\le {∥f∥}_{L^p({\mathbb{R}}^3;\mathbb{O})}·{∥g∥}_{L^1({\mathbb{R}}^3;\mathbb{O})},p=1,2.$$

So we proved that for every $f,g\in L^2({\mathbb{R}}^3;\mathbb{O})\cap L^1({\mathbb{R}}^3;\mathbb{O})$ and $\mathbf{\omega }\in {\mathbb{R}}^3$ we have that $W_{f,g}^R(·,\mathbf{\omega })\in L^2({\mathbb{R}}^3;\mathbb{O})\cap L^1({\mathbb{R}}^3;\mathbb{O})$.

3.3. Moyal’s Formula

One of the important Wigner–Ville distribution properties (in the classical setup) is called Moyal’s formula [17], which states that

$${\left(W_{f_1,g_1},W_{f_2,g_2}\right)}_{L^2({\mathbb{R}}^n\times {\mathbb{R}}^n;\mathbb{C})}={\left(f_1,f_2\right)}_{L^2({\mathbb{R}}^n;\mathbb{C})}·{\left(g_1,g_2\right)}_{L^2({\mathbb{R}}^n;\mathbb{C})}.$$

In the octonion approach, the problem is much more complicated (which is very similar to considerations about the Parseval–Plancherel theorems for the OFT, see [46]).

Assume that $f_1,f_2,g_1,g_2\in L^2({\mathbb{R}}^3;\mathbb{O})$ are real-valued and $W_{f_i,g_i}^R\in L^2({\mathbb{R}}^3\times {\mathbb{R}}^3;\mathbb{O})$. The OWD of $f_i$ and $g_i$ is the octonion Fourier transform (with respect to $\mathbf{y}$) of

$$f_i\left(\mathbf{x}+\frac{\mathbf{y}}{2}\right)\overline{g_i\left(\mathbf{x}-\frac{\mathbf{y}}{2}\right)}.$$

The Plancherel’s theorem is valid for the OFT of real-valued functions, so

$${\int }_{{\mathbb{R}}^3}{W}_{f_1,g_1}^R(\mathbf{x},\mathbf{\omega })\overline{W_{f_2,g_2}^R(\mathbf{x},\mathbf{\omega })}\mathrm{d}\mathbf{\omega }={\int }_{{\mathbb{R}}^3}{f}_1\left(\mathbf{x}+\frac{\mathbf{y}}{2}\right)\overline{g_1\left(\mathbf{x}-\frac{\mathbf{y}}{2}\right)}·\overline{f_2\left(\mathbf{x}+\frac{\mathbf{y}}{2}\right)}{g}_2\left(\mathbf{x}-\frac{\mathbf{y}}{2}\right)\mathrm{d}\mathbf{y}$$

(we could skip the conjugation of real-valued functions). With introducing a new variable $\mathbf{t}=\mathbf{x}+\frac{\mathbf{y}}{2}$ we obtain

$${\int }_{{\mathbb{R}}^3}{W}_{f_1,g_1}^R(\mathbf{x},\mathbf{\omega })\overline{W_{f_2,g_2}^R(\mathbf{x},\mathbf{\omega })}\mathrm{d}\mathbf{\omega }=8{\int }_{{\mathbb{R}}^3}{f}_1\left(\mathbf{t}\right)\overline{f_2\left(\mathbf{t}\right)}·\overline{g_1(2\mathbf{x}-\mathbf{t})}{g}_2(2\mathbf{x}-\mathbf{t})\mathrm{d}\mathbf{t}$$

and by integrating over $\mathbf{x}\in {\mathbb{R}}^3$ and applying the Fubini theorem, we obtain the claim of the following theorem.

Theorem 6.

Let $f_1,f_2,g_1,g_2\in L^2({\mathbb{R}}^3;\mathbb{R})$. Then

$${\left(W_{f_1,g_1}^R,W_{f_2,g_2}^R\right)}_{L^2({\mathbb{R}}^3\times {\mathbb{R}}^3;\mathbb{O})}={\left(f_1,f_2\right)}_{L^2({\mathbb{R}}^3;\mathbb{O})}·{\left(g_1,g_2\right)}_{L^2({\mathbb{R}}^3;\mathbb{O})}.$$

Unfortunately, this formula does not hold for octonion-valued functions, similarly as in the case of the Plancherel theorem for the OFT. Consider

$$\begin{array}{cc}{f}_1\left(\mathbf{x}\right)\hfill & =\frac{{\mathbf{e}}_4}{{\left(2\pi \right)}^{3/2}}exp\left\{-\frac{1}{2}\left(x_1^2+{(x_2-1)}^2+{(x_3-1)}^2\right)\right\},\hfill \\ f_2\left(\mathbf{x}\right)\hfill & =\frac{{\mathbf{e}}_1}{{\left(2\pi \right)}^{3/2}}exp\left\{-\frac{1}{2}\left(x_1^2+{(x_2+1)}^2+{(x_3+1)}^2\right)\right\},\hfill \\ g_1\left(\mathbf{x}\right)\hfill & =g_2\left(\mathbf{x}\right)\hfill \\ & =\frac{{\mathbf{e}}_2}{{\left(2\pi \right)}^{3/2}}exp\left\{-\frac{1}{2}\left(x_1^2+{(x_2-1)}^2+{(x_3+1)}^2\right)\right\}.\hfill \end{array}$$

After many tiring calculations we obtain

$${\left(W_{f_1,g_1}^R,W_{f_2,g_1}^R\right)}_{L^2({\mathbb{R}}^3\times {\mathbb{R}}^3;\mathbb{O})}=-\frac{{\mathbf{e}}_5}{64e^2{\pi }^3}\ne \frac{{\mathbf{e}}_5}{64e^2{\pi }^3}={\left(f_1,f_2\right)}_{L^2({\mathbb{R}}^3;\mathbb{O})}·{∥g_1∥}_{L^2({\mathbb{R}}^3;\mathbb{O})}^2.$$

From Moyal’s formula, we immediately conclude that for $f_1=f_2:=f$ and $g_1=g_2:=g$ we obtain

$${∥W_{f,g}^R∥}_{L^2({\mathbb{R}}^3\times {\mathbb{R}}^3;\mathbb{O})}={∥f∥}_{L^2({\mathbb{R}}^3;\mathbb{R})}·{∥g∥}_{L^2({\mathbb{R}}^3;\mathbb{R})}.$$

Unsurprisingly, this formula holds also for octonion-valued functions. This is due to the fact that Rayleigh’s theorem holds for octonion-valued functions, hence

$${\int }_{{\mathbb{R}}^3}{\left|W_{f,g}^R(\mathbf{x},\mathbf{\omega })\right|}^2\mathrm{d}\mathbf{\omega }={\int }_{R^3}{\left|f\left(\mathbf{x}+\frac{\mathbf{y}}{2}\right)\right|}^2·{\left|g\left(\mathbf{x}-\frac{\mathbf{y}}{2}\right)\right|}^2\mathrm{d}\mathbf{y}$$

and (just like earlier) by introducing a new variable, integrating over $\mathbf{x}\in {\mathbb{R}}^3$ and applying the Fubini theorem, we obtain

$${∥W_{f,g}^R∥}_{L^2({\mathbb{R}}^3\times {\mathbb{R}}^3;\mathbb{O})}^2={∥f∥}_{L^2({\mathbb{R}}^3;\mathbb{O})}^2·{∥g∥}_{L^2({\mathbb{R}}^3;\mathbb{O})}^2,$$

which is a general result.

Similar argument (that proved Moyal’s formula for real-valued functions $f_1,f_2,g_1,g_2\in L^2({\mathbb{R}}^3;\mathbb{R})$) gives us the other Moyal’s formula

$${〈W_{f_1,g_1}^R,W_{f_2,g_2}^R〉}_{L^2({\mathbb{R}}^3\times {\mathbb{R}}^3;\mathbb{O})}={〈f_1,f_2〉}_{L^2({\mathbb{R}}^3;\mathbb{O})}·{\left(g_1,g_2\right)}_{L^2({\mathbb{R}}^3;\mathbb{O})}$$

for $f_1,f_2,g_1,g_2\in L^2({\mathbb{R}}^3;\mathbb{O})$. We are considering a different kind of scalar product here, but this allows us to extend the assumptions of Moyal’s formula to octonion-valued functions.

3.4. Uncertainty Principle

The general (Heisenberg) uncertainty principle has been formulated for the octonion Fourier transform in [44] (see Section 2). Since OWD is the OFT of $h(·;\mathbf{x})$, we can write that if $h(·;\mathbf{x})\in L^2({\mathbb{R}}^3;\mathbb{O})$ and $s,\beta >0$, then there exists the constant $C_{s,\beta }>0$ such that

$${\left({\int }_{{\mathbb{R}}^3}{\left|\mathbf{y}\right|}^{2s}{\left|h(\mathbf{y};\mathbf{x})\right|}^2\mathrm{d}\mathbf{y}\right)}^{\frac{\beta }{s+\beta }}·{\left({\int }_{{\mathbb{R}}^3}{\left|\mathbf{\omega }\right|}^{2\beta }{\left|W_{f,g}^R(\mathbf{x};\mathbf{\omega })\right|}^2\mathrm{d}\mathbf{\omega }\right)}^{\frac{s}{s+\beta }}\ge C_{s,\beta }{\int }_{{\mathbb{R}}^3}{\left|h(\mathbf{y};\mathbf{x})\right|}^2\mathrm{d}\mathbf{y}.$$

Integrating both sides with respect to $\mathbf{x}\in {\mathbb{R}}^3$ and using the Hölder inequality with $p=\frac{s+\beta }{\beta }$ and $q=\frac{s+\beta }{s}$, we conclude immediately with the following theorem.

Theorem 7.

Let $f,g:{\mathbb{R}}^3\to \mathbb{O}$ be such that for every $\mathbf{x}\in {\mathbb{R}}^3$ we have $h(·;\mathbf{x})\in L^2({\mathbb{R}}^3;\mathbb{O})$. If $s,\beta >0$, then there exists constant $C_{s,\beta }>0$ such that

$${∥\phantom{{\left|\mathbf{\omega }\right|}^{\beta }}{\left|\mathbf{y}\right|}^{s}h(\mathbf{y};\mathbf{x})∥}_{L^2({\mathbb{R}}^3\times {\mathbb{R}}^3;\mathbb{O})}^{\frac{2\beta }{s+\beta }}{∥{\left|\mathbf{\omega }\right|}^{\beta }{W}_{f,g}^R(\mathbf{x},\mathbf{\omega })∥}_{L^2({\mathbb{R}}^3\times {\mathbb{R}}^3;\mathbb{O})}^{\frac{2s}{s+\beta }}\ge C_{s,\beta }{∥f∥}_{L^2({\mathbb{R}}^3;\mathbb{O})}^2{∥g∥}_{L^2({\mathbb{R}}^3;\mathbb{O})}^2.$$

The exact values of the $C_{s,\beta }$ constants can be found in the literature (see [44]). In the above theorem, they remain the same.

4. Examples

Let us illustate the theoretical results presented above with the examples of a separable Gaussian function:

$$f(x_1,x_2,x_3)=\frac{1}{{\left(2\pi \right)}^{3/2}{\sigma }_1{\sigma }_2{\sigma }_3}·exp\left\{-\frac{1}{2}\left(\frac{x_1^2}{{\sigma }_1^2}+\frac{x_2^2}{{\sigma }_2^2}+\frac{x_3^2}{{\sigma }_3^2}\right)\right\}.$$

Due to the fact that, in addition to illustrating the previous statements, we would like to compare the obtained results with the classical theory (derived for complex-valued signals); we limit ourselves only to examining the OWD of real-valued signals. Notice that

$$\begin{array}{cc}h(\mathbf{y};\mathbf{x})\hfill & =f\left(\mathbf{x}+\frac{\mathbf{y}}{2}\right)\overline{f\left(\mathbf{x}-\frac{\mathbf{y}}{2}\right)}\hfill \\ \hfill & =\frac{1}{{\left(2\pi \right)}^3{\left({\sigma }_1{\sigma }_2{\sigma }_3\right)}^2}·exp\left\{-\frac{1}{2}\left(\frac{{(x_1+\frac{y_1}{2})}^2+{(x_1-\frac{y_1}{2})}^2}{{\sigma }_1^2}\right.\right.\hfill \\ \hfill & +\frac{{(x_2+\frac{y_2}{2})}^2+{(x_2-\frac{y_2}{2})}^2}{{\sigma }_2^2}\left.\left.+\frac{{(x_3+\frac{y_3}{2})}^2+{(x_3-\frac{y_3}{2})}^2}{{\sigma }_3^2}\right)\right\}\hfill \\ \hfill & =\frac{1}{{\left(2\pi \right)}^3{\left({\sigma }_1{\sigma }_2{\sigma }_3\right)}^2}·exp\left\{-\frac{1}{2}\left(\frac{2x_1^2}{{\sigma }_1^2}+\frac{2x_2^2}{{\sigma }_2^2}+\frac{2x_3^2}{{\sigma }_3^2}\right.\right.\left.\left.+\frac{y_1^2}{2{\sigma }_1^2}+\frac{y_2^2}{2{\sigma }_2^2}+\frac{y_3^2}{2{\sigma }_3^2}\right)\right\}\hfill \\ \hfill & =\frac{1}{{\left(2\pi \right)}^{3/2}\frac{{\sigma }_1}{\sqrt{2}}\frac{{\sigma }_2}{\sqrt{2}}\frac{{\sigma }_3}{\sqrt{2}}}exp\left\{-\frac{1}{2}\left(\frac{x_1^2}{{({\sigma }_1/\sqrt{2})}^2}\right.\right.\left.\left.+\frac{x_2^2}{{({\sigma }_2/\sqrt{2})}^2}+\frac{x_3^2}{{({\sigma }_3/\sqrt{2})}^2}\right)\right\}\hfill \\ \hfill & ·\frac{1}{{\left(2\pi \right)}^{3/2}\sqrt{2}{\sigma }_1\sqrt{2}{\sigma }_2\sqrt{2}{\sigma }_3}exp\left\{-\frac{1}{2}\left(\frac{y_1^2}{{\left(\sqrt{2}{\sigma }_1\right)}^2}\right.\right.\left.\left.+\frac{y_2^2}{{\left(\sqrt{2}{\sigma }_2\right)}^2}+\frac{y_3^2}{{\left(\sqrt{2}{\sigma }_3\right)}^2}\right)\right\}\hfill \end{array}$$

which is the product of two independent separable Gaussian functions. Since OWD is actually the OFT of $h(·;\mathbf{x})$, then (repeating the calculations for the 3-D Gaussian signal presented in ([49], Example 3) we obtain

$$\begin{array}{cc}{W}_f^R(\mathbf{x},\mathbf{\omega })\hfill & =\frac{1}{{\left(2\pi \right)}^{3/2}\frac{{\sigma }_1}{\sqrt{2}}\frac{{\sigma }_2}{\sqrt{2}}\frac{{\sigma }_3}{\sqrt{2}}}·exp\left\{-\frac{1}{2}\left(\frac{x_1^2}{{({\sigma }_1/\sqrt{2})}^2}\right.\right.\left.\left.+\frac{x_2^2}{{({\sigma }_2/\sqrt{2})}^2}+\frac{x_3^2}{{({\sigma }_3/\sqrt{2})}^2}\right)\right\}\hfill \\ \hfill & ·exp\left\{-2{\pi }^2\left(2{\sigma }_1^2{\omega }_1^2+2{\sigma }_2^2{\omega }_2^2+2{\sigma }_3^2{\omega }_3^2\right)\right\},\hfill \end{array}$$

which (again) is the product of two independent separable Gaussian functions. This result should come as no surprise—in the case of separable real-valued even (with respect to each coordinate) functions, the octonion Fourier transform coincides with the classical one [49]. The same is true for the Wigner distribution of such functions.

Consider now the cross right-sided OWD of two separable Gaussian functions:

$$\begin{array}{cc}f(x_1,x_2,x_3)& ={\displaystyle \prod _{i=1}^3}\frac{1}{\sqrt{2\pi }{\sigma }_{1,i}}exp\left\{-\frac{1}{2}·\frac{{(x_i-{\mu }_{1,i})}^2}{{\sigma }_{1,i}^2}\right\},\\ g(x_1,x_2,x_3)& ={\displaystyle \prod _{i=1}^3}\frac{1}{\sqrt{2\pi }{\sigma }_{2,i}}exp\left\{-\frac{1}{2}·\frac{{(x_i-{\mu }_{2,i})}^2}{{\sigma }_{2,i}^2}\right\}.\end{array}$$

Sample graphs of the above functions are shown in Figure 2. Consider the midpoint of the line connecting the centers of these two Gaussian functions, i.e., $\tilde{\mathbf{\mu }}=\frac{1}{2}({\mathbf{\mu }}_1+{\mathbf{\mu }}_2)=({\tilde{\mu }}_1,{\tilde{\mu }}_2,{\tilde{\mu }}_3)$ (where ${\mathbf{\mu }}_i=({\mu }_{i,1},{\mu }_{i,2},{\mu }_{i,3})$), and denote $\mathbf{\alpha }=\frac{1}{2}({\mathbf{\mu }}_1-{\mathbf{\mu }}_2)$, $\mathbf{\alpha }=({\alpha }_1,{\alpha }_2,{\alpha }_3)$. Notice that

$$\begin{array}{cc}F\left(\mathbf{x}\right)& =f(x_1+{\tilde{\mu }}_1,x_2+{\tilde{\mu }}_2,x_3+{\tilde{\mu }}_3)={\displaystyle \prod _{i=1}^3}\frac{1}{\sqrt{2\pi }{\sigma }_{1,i}}exp\left\{-\frac{1}{2}·\frac{{(x_i-{\alpha }_i)}^2}{{\sigma }_{1,i}^2}\right\},\\ G\left(\mathbf{x}\right)& =g(x_1+{\tilde{\mu }}_1,x_2+{\tilde{\mu }}_2,x_3+{\tilde{\mu }}_3)={\displaystyle \prod _{i=1}^3}\frac{1}{\sqrt{2\pi }{\sigma }_{2,i}}exp\left\{-\frac{1}{2}·\frac{{(x_i+{\alpha }_i)}^2}{{\sigma }_{2,i}^2}\right\}\end{array}$$

and it is enough to consider the OWD of functions F and G and then use the shift property of the OWD. We now need to investigate the behavior of the function $h(\mathbf{y};\mathbf{x})$. Note that

$$\frac{{(x_i+\frac{y_i}{2}-{\alpha }_i)}^2}{{\sigma }_{1,i}^2}+\frac{{(x_i-\frac{y_i}{2}+{\alpha }_i)}^2}{{\sigma }_{2,i}^2}=\frac{x_i^2}{{\left(\frac{\sqrt{{\sigma }_{1,i}^2+{\sigma }_{2,i}^2}}{2}\right)}^2}+\frac{{\left(y_i-2\left({\alpha }_i-\frac{{\sigma }_{2,i}^2-{\sigma }_{1,i}^2}{{\sigma }_{1,i}^2+{\sigma }_{2,i}^2}{x}_i\right)\right)}^2}{{\left(\frac{2{\sigma }_{1,i}{\sigma }_{2,i}}{\sqrt{{\sigma }_{1,i}^2+{\sigma }_{2,i}^2}}\right)}^2},$$

which means that (once again) h is the product of two separable Gaussian functions:

$$\begin{array}{cc}h(\mathbf{y};\mathbf{x})\hfill & =F\left(\mathbf{x}+\frac{\mathbf{y}}{2}\right)\overline{G\left(\mathbf{x}-\frac{\mathbf{y}}{2}\right)}\hfill \\ \hfill & ={\displaystyle \prod _{i=1}^3}\left(\frac{2}{\sqrt{2\pi }\sqrt{{\sigma }_{1,i}^2+{\sigma }_{2,i}^2}}exp\left\{-2·\frac{x_i^2}{{\sigma }_{1,i}^2+{\sigma }_{2,i}^2}\right\}\right)\hfill \\ \hfill & ·{\displaystyle \prod _{i=1}^3}\left(\frac{1}{\sqrt{2\pi }·\frac{2{\sigma }_{1,i}{\sigma }_{2,i}}{\sqrt{{\sigma }_{1,i}^2+{\sigma }_{2,i}^2}}}exp\left\{-\frac{1}{2}·\frac{{\left(y_i-2\left({\alpha }_i-\frac{{\sigma }_{2,i}^2-{\sigma }_{1,i}^2}{{\sigma }_{1,i}^2+{\sigma }_{2,i}^2}{x}_i\right)\right)}^2}{{\left(\frac{2{\sigma }_{1,i}{\sigma }_{2,i}}{\sqrt{{\sigma }_{1,i}^2+{\sigma }_{2,i}^2}}\right)}^2}\right\}\right).\hfill \end{array}$$

What is more important is the fact that it is a shifted Gaussian function with respect to the variable $\mathbf{y}$. Using the shift theorem (for the OFT, and having in mind that the OFT of Gaussian function is even with respect to all variables) and calculations from ([49], Example 3), we have that

$$\begin{array}{cc}{W}_{F,G}^R(\mathbf{x},\mathbf{\omega })\hfill & =\left(\right({\displaystyle \prod _{i=1}^3}\left(\frac{2}{\sqrt{2\pi }\sqrt{{\sigma }_{1,i}^2+{\sigma }_{2,i}^2}}·exp\left\{-2·\frac{x_i^2}{{\sigma }_{1,i}^2+{\sigma }_{2,i}^2}\right\}\right)\hfill \\ \hfill & ·{\displaystyle \prod _{i=1}^3}exp\left\{-8{\pi }^2{\sigma }_{1,i}^2{\sigma }_{2,i}^2·\frac{{\omega }_i^2}{{\sigma }_{1,i}^2+{\sigma }_{2,i}^2}\right\}\hfill \\ \hfill & ·exp\left\{-4\pi {\mathbf{e}}_1\left({\alpha }_1-\frac{{\sigma }_{2,1}^2-{\sigma }_{1,1}^2}{{\sigma }_{1,1}^2+{\sigma }_{2,1}^2}{x}_1\right){\omega }_1\right\})\hfill \\ \hfill & ·exp\left\{-4\pi {\mathbf{e}}_2\left({\alpha }_2-\frac{{\sigma }_{2,2}^2-{\sigma }_{1,2}^2}{{\sigma }_{1,2}^2+{\sigma }_{2,2}^2}{x}_2\right){\omega }_2\right\})\hfill \\ \hfill & ·exp\left\{-4\pi {\mathbf{e}}_4\left({\alpha }_3-\frac{{\sigma }_{2,3}^2-{\sigma }_{1,3}^2}{{\sigma }_{1,3}^2+{\sigma }_{2,3}^2}{x}_3\right){\omega }_3\right\}.\hfill \end{array}$$

Going back to the original functions f and g and shifts ${\mathbf{\mu }}_1$ and ${\mathbf{\mu }}_2$, using the shift theorem we obtain that

$$\begin{array}{cc}{W}_{f,g}^R(\mathbf{x},\mathbf{\omega })\hfill & =\left(\right({\displaystyle \prod _{i=1}^3}\left(\frac{2}{\sqrt{2\pi }\sqrt{{\sigma }_{1,i}^2+{\sigma }_{2,i}^2}}·exp\left\{-2·\frac{{\left(x_i-\frac{1}{2}({\mu }_{1,i}+{\mu }_{2,i})\right)}^2}{{\sigma }_{1,i}^2+{\sigma }_{2,i}^2}\right\}\right)\hfill \\ \hfill & ·{\displaystyle \prod _{i=1}^3}exp\left\{-8{\pi }^2{\sigma }_{1,i}^2{\sigma }_{2,i}^2·\frac{{\omega }_i^2}{{\sigma }_{1,i}^2+{\sigma }_{2,i}^2}\right\}\hfill \\ \hfill & ·exp\left\{-4\pi {\mathbf{e}}_1\left(\frac{1}{2}({\mu }_{1,1}-{\mu }_{2,1})-\frac{{\sigma }_{2,1}^2-{\sigma }_{1,1}^2}{{\sigma }_{1,1}^2+{\sigma }_{2,1}^2}{x}_1\right){\omega }_1\right\})\hfill \\ \hfill & ·exp\left\{-4\pi {\mathbf{e}}_2\left(\frac{1}{2}({\mu }_{1,2}-{\mu }_{2,2})-\frac{{\sigma }_{2,2}^2-{\sigma }_{1,2}^2}{{\sigma }_{1,2}^2+{\sigma }_{2,2}^2}{x}_2\right){\omega }_2\right\})\hfill \\ \hfill & ·exp\left\{-4\pi {\mathbf{e}}_4\left(\frac{1}{2}({\mu }_{1,3}-{\mu }_{2,3})-\frac{{\sigma }_{2,3}^2-{\sigma }_{1,3}^2}{{\sigma }_{1,3}^2+{\sigma }_{2,3}^2}{x}_3\right){\omega }_3\right\}.\hfill \end{array}$$

It may seem that the mathematical formulas we have obtained do not contribute much; however, it is worth observing what are the consequences of these formulas on the examples of functions f and g with specific numerical parameters. Figure 3, Figure 4, Figure 5 and Figure 6 show OWD plots (real part and subsequent imaginary parts) for different values of ${\mathbf{\mu }}_1,{\mathbf{\mu }}_2,{\mathbf{\sigma }}_1=({\sigma }_{1,1},{\sigma }_{1,2},{\sigma }_{1,3}),{\mathbf{\sigma }}_2=({\sigma }_{2,1},{\sigma }_{2,2},{\sigma }_{2,3})$ parameters of f and g functions. The convention here is to write the OWD as

$$W_{f,g}^R=W_{f,g}^{R,0}+W_{f,g}^{R,1}{\mathbf{e}}_1+W_{f,g}^{R,2}{\mathbf{e}}_2+W_{f,g}^{R,3}{\mathbf{e}}_3+W_{f,g}^{R,4}{\mathbf{e}}_4+W_{f,g}^{R,5}{\mathbf{e}}_5+W_{f,g}^{R,6}{\mathbf{e}}_6+W_{f,g}^{R,7}{\mathbf{e}}_7.$$

The conclusions that can be drawn from this example are as follows:

• The shift of the Gaussian function with respect to $\mathbf{x}$ indicates the decentralization of the set of two Gaussian functions (how far is the midpoint of the segment connecting the centers of those functions from the center of the coordinate system),
• Differences between the functions in each direction are visible in the individual imaginary parts—a change in the $x_i$ direction causes the appearance of the imaginary part standing at ${\mathbf{e}}_{2^{i-1}}$, a change in the $x_i$ and $x_j$ directions causes the appearance of the imaginary parts standing at ${\mathbf{e}}_{2^{i-1}}$, ${\mathbf{e}}_{2^{j-1}}$ and ${\mathbf{e}}_{2^{i-1}}·{\mathbf{e}}_{2^{j-1}}$. The relative magnitude of these changes is reflected by the amplitude of the individual imaginary components.

The last of the above conclusions shows at the same time the difference (and a significant advantage) of the octonion approach over the classic one. It is worth noting that this is only a theoretical example (which can be directly calculated).

Continuing the above example, it is worth considering the OWD of the Gabor signal, i.e., the product of a Gaussian kernel and a sinusoid. Considering only the real Gabor signal, we can use the above considerations and the equivalents of the modulation theorem discussed in Section 3.2. We have two Gabor signals:

$$\begin{array}{cc}f(x_1,x_2,x_3)& ={\displaystyle \prod _{i=1}^3}\left(\frac{1}{\sqrt{2\pi }{\sigma }_{1,i}}exp\left\{-\frac{1}{2}·\frac{x_i^2}{{\sigma }_{1,i}^2}\right\}\right)·sin\left(2\pi {\omega }_0{x}_3\right),\\ g(x_1,x_2,x_3)& ={\displaystyle \prod _{i=1}^3}\left(\frac{1}{\sqrt{2\pi }{\sigma }_{2,i}}exp\left\{-\frac{1}{2}·\frac{x_i^2}{{\sigma }_{2,i}^2}\right\}\right)·cos\left(2\pi {\omega }_0{x}_3\right).\end{array}$$

The choice of variable $x_3$ in the sine carrier is arbitrary. Further considerations can also be made without major changes when selecting other variables. OWD of those two Gabor signals is given by the formula

$$W_{f,g}^R(\mathbf{x},\mathbf{\omega })=\frac{1}{2}W(\mathbf{x},\mathbf{\omega })·sin\left(2\pi \left(2{\omega }_0\right)x_3\right)+\frac{1}{4}\left(W(\mathbf{x},\mathbf{\omega }+{\mathbf{\omega }}_0)-W(\mathbf{x},\mathbf{\omega }-{\mathbf{\omega }}_0)\right){\mathbf{e}}_4,$$

where ${\mathbf{\omega }}_0=(0,0,{\omega }_0)$ and

$$\begin{array}{cc}{W}_{f,g}^R(\mathbf{x},\mathbf{\omega })\hfill & =\left(\right({\displaystyle \prod _{i=1}^3}\left(\frac{2}{\sqrt{2\pi }\sqrt{{\sigma }_{1,i}^2+{\sigma }_{2,i}^2}}·exp\left\{-2·\frac{x_i^2}{{\sigma }_{1,i}^2+{\sigma }_{2,i}^2}\right\}\right)\hfill \\ \hfill & ·{\displaystyle \prod _{i=1}^3}exp\left\{-8{\pi }^2{\sigma }_{1,i}^2{\sigma }_{2,i}^2·\frac{{\omega }_i^2}{{\sigma }_{1,i}^2+{\sigma }_{2,i}^2}\right\}·exp\left\{-4\pi {\mathbf{e}}_1\frac{{\sigma }_{1,1}^2-{\sigma }_{2,1}^2}{{\sigma }_{1,1}^2+{\sigma }_{2,1}^2}{x}_1{\omega }_1\right\})\hfill \\ \hfill & ·exp\left\{-4\pi {\mathbf{e}}_2\frac{{\sigma }_{1,2}^2-{\sigma }_{2,2}^2}{{\sigma }_{1,2}^2+{\sigma }_{2,2}^2}{x}_2{\omega }_2\right\})·exp\left\{-4\pi {\mathbf{e}}_4\frac{{\sigma }_{1,3}^2-{\sigma }_{2,3}^2}{{\sigma }_{1,3}^2+{\sigma }_{2,3}^2}{x}_3{\omega }_3\right\}.\hfill \end{array}$$

In this case, it is also worth paying attention to what the obtained distributions look like when selecting specific parameters of both Gabor signals. Figure 7 shows the plots of the OWD for parameters ${\mathbf{\sigma }}_1=\left(\frac{1}{2},\frac{1}{5},\frac{1}{2}\right)$, ${\mathbf{\sigma }}_2=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)$ and ${\omega }_0=\frac{1}{2}$. For comparison, Figure 8 shows plots of the classical Wigner distribution $W_{f,g}^{\mathbb{C}}$ (its real and imaginary parts). There are clear differences in the two approaches to the problem.

Choosing other parameters of Gabor’s signals, one can draw similar conclusions as in the previous experiment—differences in functions in each direction result in the appearance of non-zero values in the corresponding imaginary parts (changes in the $x_i$ direction cause the appearance of the ${\mathbf{e}}_{2^i}$ component, changes in the two directions $x_i$ and $x_j$ cause the appearance of imaginary components ${\mathbf{e}}_{2^i}$, ${\mathbf{e}}_{2^j}$, and ${\mathbf{e}}_{2^i}·{\mathbf{e}}_{2^j}$). Changes in all directions result in the appearance of all imaginary components. Moreover, the variable $x_k$ used in the sine carrier also influences the appearance of additional imaginary components (${\mathbf{e}}_{2^k}$, ${\mathbf{e}}_{2^i}·{\mathbf{e}}_{2^k}$, ${\mathbf{e}}_{2^j}·{\mathbf{e}}_{2^k}$, and ${\mathbf{e}}_{2^i}·{\mathbf{e}}_{2^j}·{\mathbf{e}}_{2^k}$).

There is a clear difference between this approach and the classical one, where changes result in the appearance of the imaginary part, but since there is only one imaginary part, it is not possible to immediately extract information about the direction of change from the analysis of the Wigner distribution alone.

5. Octonion Ambiguity Function

The concept of the Wigner distribution (WD) is directly related to the definition of the (Woodward) ambiguity function (AF) [21]. It can be defined in many different ways, but it seems that in the case of an octonion approach, each of them will have considerable limitations (in the context of analogous relations between WD and AF in the classical case).

Definition 2.

The octonion cross right-sided ambiguity function (OAF) of the 3-D signals $f,g\in L^2({\mathbb{R}}^3;\mathbb{O})$ is defined as

$$A_{f,g}^R(\mathbf{x},\mathbf{\omega })={\int }_{{\mathbb{R}}^3}f\left(\mathbf{y}+\frac{\mathbf{x}}{2}\right)\overline{g\left(\mathbf{y}-\frac{\mathbf{x}}{2}\right)}·e^{-2\pi {\mathbf{e}}_1{\omega }_1{y}_1}{e}^{-2\pi {\mathbf{e}}_2{\omega }_2{y}_2}{e}^{-2\pi {\mathbf{e}}_4{\omega }_3{y}_3}\mathrm{d}\mathbf{y},$$

where $(\mathbf{x},\mathbf{\omega })\in {\mathbb{R}}^3\times {\mathbb{R}}^3$ and multiplication is performed from left to right.

Analogous properties of this function can be proved as in the case of OWD, but from our point of view, the mutual dependencies between these functions will be more critical. Notice that

$$\begin{array}{cc}{A}_{f,g}^R(2\mathbf{x},2\mathbf{\omega })\hfill & ={\int }_{{\mathbb{R}}^3}f\left(\mathbf{y}+\mathbf{x}\right)\overline{g\left(\mathbf{y}-\mathbf{x}\right)}·e^{-2\pi {\mathbf{e}}_1·2{\omega }_1{y}_1}{e}^{-2\pi {\mathbf{e}}_2·2{\omega }_2{y}_2}{e}^{-2\pi {\mathbf{e}}_4·2{\omega }_3{y}_3}\mathrm{d}\mathbf{y}\hfill \\ \hfill & =\frac{1}{8}{\int }_{{\mathbb{R}}^3}f\left(\mathbf{x}+\frac{\mathbf{t}}{2}\right)\overline{g\left(\frac{\mathbf{t}}{2}-\mathbf{x}\right)}·e^{-2\pi {\mathbf{e}}_1{\omega }_1{t}_1}{e}^{-2\pi {\mathbf{e}}_2{\omega }_2{t}_2}{e}^{-2\pi {\mathbf{e}}_4{\omega }_3{t}_3}\mathrm{d}\mathbf{t}\hfill \end{array}$$

which is almost the same formula, as in the definition of OWD. We conclude that

• If g is even, then $A_{f,g}^R(\mathbf{x},\mathbf{\omega })=\frac{1}{8}{W}_{f,g}^R\left(\frac{\mathbf{x}}{2},\frac{\mathbf{\omega }}{2}\right)$;
• If g is odd, then $A_{f,g}^R(\mathbf{x},\mathbf{\omega })=-\frac{1}{8}{W}_{f,g}^R\left(\frac{\mathbf{x}}{2},\frac{\mathbf{\omega }}{2}\right)$,

which is the same relation as in the classical case. Unfortunately, the interrelationships using Fourier transforms will not be true in the case of the octonion setup (due to non-associativity of octonion multiplication). We can only conclude that if we denote (as before)

$$h(\mathbf{y};\mathbf{x})=f\left(\mathbf{x}+\frac{\mathbf{y}}{2}\right)\overline{g\left(\mathbf{x}-\frac{\mathbf{y}}{2}\right)},$$

then $W_{f,g}^R(\mathbf{x},·)$ is the OFT of $h(·;\mathbf{x})$ and $A_{f,g}^R(\mathbf{x},·)$ is the OFT of $h(\mathbf{x};·)$.

6. OWD and Analytic Signals

One of the essential features of the Wigner distribution is its applicability to the time–frequency analysis of signals. In classical terms, one can approach this issue dually—considering the correlation product of functions in the time domain, but also considering the correlation product in the frequency domain (i.e., using Fourier transform). So far, we have only considered the first one. In the last part of this article, we try to provide a few comments on the second approach and indicate what effects this may have on other issues related to the topic under consideration.

6.1. 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.,

$$W_{f,g}(\mathbf{x},\mathbf{\omega })={\int }_{{\mathbb{R}}^3}\underset{=h(\mathbf{y};\mathbf{x})}{\underbrace{f\left(\mathbf{x}+\frac{\mathbf{y}}{2}\right)\overline{g\left(\mathbf{x}-\frac{\mathbf{y}}{2}\right)}}}{e}^{-2\pi \mathbf{i}\mathbf{\omega }·\mathbf{y}}\mathrm{d}\mathbf{y}.$$

If we consider the correlation product of Fourier transforms of two functions, i.e.,

$$H(\mathbf{\xi };\mathbf{\omega })=F\left(\mathbf{\omega }+\frac{\mathbf{\xi }}{2}\right)\overline{G\left(\mathbf{\omega }-\frac{\mathbf{\xi }}{2}\right)},$$

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 $H(·;\mathbf{\omega })$), we obtain

$$W_{F,G}(\mathbf{\omega },\mathbf{x})={\int }_{{\mathbb{R}}^3}F\left(\mathbf{\omega }+\frac{\mathbf{\xi }}{2}\right)\overline{G\left(\mathbf{\omega }-\frac{\mathbf{\xi }}{2}\right)}{e}^{-2\pi \mathbf{i}\mathbf{x}·\mathbf{\xi }}\mathrm{d}\mathbf{\xi }.$$

(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 $f,g:{\mathbb{R}}^3\to \mathbb{O}$, i.e.,

$$H(\mathbf{\xi };\mathbf{\omega })=F_{\mathrm{OFT}}\left(\mathbf{\omega }+\frac{\mathbf{\xi }}{2}\right)\overline{G_{\mathrm{OFT}}\left(\mathbf{\omega }-\frac{\mathbf{\xi }}{2}\right)}.$$

The cross right-sided OWD of those OFTs is

$$W_{F_{\mathrm{OFT}},G_{\mathrm{OFT}}}^R(\mathbf{\omega },\mathbf{x})={\int }_{{\mathbb{R}}^3}{F}_{\mathrm{OFT}}\left(\mathbf{\omega }+\frac{\mathbf{\xi }}{2}\right)\overline{G_{\mathrm{OFT}}\left(\mathbf{\omega }-\frac{\mathbf{\xi }}{2}\right)}·e^{-2\pi {\mathbf{e}}_1{x}_1{\xi }_1}{e}^{-2\pi {\mathbf{e}}_2{x}_2{\xi }_2}{e}^{-2\pi {\mathbf{e}}_4{x}_3{\xi }_3}\mathrm{d}\mathbf{\xi },$$

where the multiplication is performed from left to right. Unfortunately, the expected property of symmetry, i.e.,

$$W_{f,g}^R(\mathbf{x},\mathbf{\omega })=W_{F_{\mathrm{OFT}},G_{\mathrm{OFT}}}^R(\mathbf{\omega },-\mathbf{x})$$

(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 $H(·;\mathbf{\omega })$ computed at $\mathbf{x}$. This suggests the use of the inverse octonion Fourier transform as well, i.e., considering the expression

$${\int }_{{\mathbb{R}}^3}{F}_{\mathrm{OFT}}\left(\mathbf{\omega }+\frac{\mathbf{\xi }}{2}\right)\overline{G_{\mathrm{OFT}}\left(\mathbf{\omega }-\frac{\mathbf{\xi }}{2}\right)}·e^{2\pi {\mathbf{e}}_4{x}_3{\xi }_3}{e}^{2\pi {\mathbf{e}}_2{x}_2{\xi }_2}{e}^{2\pi {\mathbf{e}}_1{x}_1{\xi }_1}\mathrm{d}\mathbf{\xi }.$$

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.

6.2. Octonion Wigner–Ville Distribution

One of the important trends in research using Wigner distributions is considering the distribution of analytic signal associated with a real-valued function, i.e., signal with the single-orthant spectrum obtained by canceling the part at negative frequencies and doubling the part at positive frequencies [6]. The distributions of such functions are commonly called Wigner–Ville distributions [55].

The symmetry property mentioned in the previous section plays a vital role in these considerations. The research conducted both on the subject of Wigner distributions and analytic signals [26] raised hopes that similar properties would also be possible in the case of hypercomplex setup. Furthermore, while it is, of course, possible to define the Wigner–Ville octonion distribution of a real-valued signal as the Wigner distribution of the related octonion analytic signal, unfortunately, the lack of relation (5) means that these considerations do not lead to too optimistic conclusions. This remains within the scope of future work when (perhaps) other properties of the OFT and OWD become known.

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.