A Generalization of the Octonion Fourier Transform to 3-D Octonion-Valued Signals -- Properties and Possible Applications to 3-D LTI Partial Differential Systems
aa r X i v : . [ m a t h . C A ] D ec A Generalization of the Octonion Fourier Transformto 3-D Octonion-Valued Signals– Properties and Possible Applicationsto 3-D LTI Partial Differential Systems
Łukasz Błaszczyk [email protected] of Mathematics and Information ScienceWarsaw University of Technologyul. Koszykowa 75, 00-662 Warszawa, Poland
Keywords : octonion Fourier transform, Cayley-Dickson numbers, hypercomplex alge-bras, multidimensional linear time-invariant systems.
Abstract
The paper is devoted to the development of the octonion Fourier transform(OFT) theory initiated in 2011 in articles by Hahn and Snopek. It is also a con-tinuation and generalization of earlier work by Błaszczyk and Snopek, where theyproved few essential properties of the OFT of real-valued functions, e.g. symmetryproperties. The results of this article focus on proving that the OFT is well-definedfor octonion-valued functions and almost all well-known properties of classical (com-plex) Fourier transform (e.g. argument scaling, modulation and shift theorems) havetheir direct equivalents in octonion setup. Those theorems, illustrated with some ex-amples, lead to the generalization of another result presented in earlier work, i.e.Parseval and Plancherel Theorems, important from the signal and system process-ing point of view. Moreover, results presented in this paper associate the OFT with3-D LTI systems of linear PDEs with constant coefficients. Properties of the OFTin context of signal-domain operations such as derivation and convolution of R -valued functions will be stated. There are known results for the QFT, but they usethe notion of other hypercomplex algebra, i.e. double-complex numbers. Consider-ations presented here require defining other higher-order hypercomplex structure,i.e. quadruple-complex numbers. This hypercomplex generalization of the Fouriertransformation provides an excellent tool for the analysis of 3-D LTI systems. Fourier analysis is one of the fundamental tools in signal and image processing. Fourierseries and Fourier transform enable us to look at the concept of signal in a dual manner –by studying its properties in the time domain (or in the space domain in case of images),where it is represented by amplitudes of the samples (or pixels), or by investigating it inthe frequency domain, where the signal can be represented by the infinite sums of complexharmonic functions, each with different frequency and amplitude [1].The classical signal theory deals with real- or complex-valued time series (or images).However, in some practical applications, signals are represented by more abstract struc-tures, e.g. hypercomplex algebras [13, 16, 18, 32]. A classic example is the use of them inthe processing of color images (where there are at least three color components) [13, 16],but also in the analysis of multispectral data (e.g. in satellite images where not only visible1ight is recorded, but also other frequency ranges) [21, 22]. Quaternions and octonions de-serve special attention in this considerations. They are examples of Cayley-Dickson (C-D)algebras [9]. C-D algebras are defined by a recursive procedure, so-called Cayley-Dicksonconstruction. They are algebras of the order N ( N ∈ N ) over the field of real numbers R .Each C-D algebra is created from the previous one and contains all previous algebras asproper sub-algebras.Recently, hypercomplex algebras drew scientists’ attention due to their numerous ap-plications, among others in the study of neural networks [25, 26, 34], in the analysis ofcolor and multispectral images [13, 14, 15, 16, 21, 22, 28], in biomedical signal process-ing [7, 19], in fluid mechanics [8] or in general signal processing [18, 32, 33]. Quaternionsmay be used in few different ways – to describe a vector-valued signal (with three or fourcoordinates) of one variable, i.e. u ( t ) = u ( t ) + u ( t ) · i + u ( t ) · j + u ( t ) · k , u , u , u , u : R → R , or to analyse a scalar or vector-valued signal of two variables, i.e. u : R → R or u : R → O . The basic tool in the second approach is the quaternion Fourier transform(QFT) [6]: U QFT ( f , f ) = Z R Z R u ( t , t ) e − π i f t e − π j f t d t d t . (1.1)It allows us (in contrast to the classical two-dimmensional Fourier transform) to analysetwo dimensions of the sampling grid independently. Each time-like dimension can beassociated with a different dimension of the four-dimensional quaternion space, while thecomplex transform mixes those two dimensions. It also allows us to study some symmetriespresent in certain signals (images), what was impossible before [13]. Similarly, octonionsare used to describe scalar or vector-valued signals of one or three variables.In the last few years some generalizations of the Fourier transform (defined as in (1.1))to the octonion and higher-order algebras appeared in the literature [17, 29, 30, 31, 32].They are defined on the basis of the Cayley-Dickson algebras and called the Cayley-Dickson Fourier transforms. The main goal of this paper is further development of suchgeneralization based on the Cayley-Dickson algebra of order 8 (octonions). Analysis of thecurrent state of knowledge on applications of octonions in the signal processing shows someareas previously unexplored or requiring thorough theoretical and experimental studies,although some gaps have recently been filled [3, 2, 4, 23].Properties of the quaternion Fourier transform (defined by (1.1)) are well studied inthe literature and it is fairly easy to notice some analogies to the properties of classical(complex) Fourier transform of functions of two variables [11]. They enable us to use theFourier transform in the analysis of some two-dimensional linear time-invariant systemsdescribed by systems of partial differential equations with constant coefficients [12]. In ourprevious investigations [4] we were able to show that the OFT is well defined for scalar(real-valued) functions of three variables (i.e. we proved the inverse transform theorem).In our research we also derived some properties of the OFT, analogous to the propertiesof the classical (complex) and quaternion Fourier transform, e.g. symmetry properties(analogue to the Hermitian symmetry properties), shift theorem, Plancherel and Parsevaltheorems, and Wiener-Khintchine theorem. Proofs of the those theorems were based onthe previous research of Hahn and Snopek, who used the fact that real–valued functionscan be expressed as a sum of components of different parity [17]. Despite these works, thestate of modern knowledge about octonion Fourier transform is negligible and requiresa thorough extension. This seems important especially in the context of new applications2hat have appeared in recent years (and which we described earlier in this section) –there is a tendency to describe the results of practical experiments, but without adequatetheoretical justification.Some of the results presented in this paper have been signaled in earlier works [3, 2],here we present a broader view of these issues and give details of the proofs. We alsoprovide some new results, mainly regarding the use of classical transformation techniquesfor calculating the OFT, as well as regarding the differentiation of the octonion transform.The paper is organized as follows. In Section 2 we recall the octonion algebra, someof its basic properties and the definition of the octonion Fourier transform, as well as theproof of its well-posedness. We also introduce the notion of the quadruple-complex algebra.In Section 3 we focus on deriving some important properties of the OFT, e.g. argumentscaling, modulation and shift theorems, relationship between the OFT of a function andthe OFT of its partial derivative, differentiation of the OFT and the convolution theorem.Those considerations lead to some remarks on applying the OFT to the analysis of 3-Dlinear time-invariant systems in Section 4, which also show the advantages of using OFTover classical transformation. The paper is conculed in Section 5 with a short discussionof obtained results. In this section, we introduce the definitions and the theorems necessary to presentthe main results of this work regarding the further properties of the octonion Fouriertransform.
An octonion o ∈ O is defined, according to the Cayley-Dickson construction, as theordered pair of quaternions [9]: o = ( q , q ) , where q = r + r e + r e + r e , q = r + r e + r e + r e ∈ H (we denote the quaternion imaginary units as e , e and e instead of traditional i , j and k ). Rules of octonion multiplication are given by the general Cayley-Dickson formula ( q , q ) · ( p , p ) = ( q · p − p ∗ · q , p · q + q · p ∗ ) , q , q , p , p ∈ H , (2.1)where multiplication of quaternions is defined as in [27] (it can be defined also by theformula (2.1) if we treat a quaternion as an ordered pair of complex numbers) and ∗ isquaternion conjugate. Applying those rules of multiplication (which can be presented inthe form of Tab. 1) we get four new imaginary units and octonions can be writen as o = r + r e + r e + r e | {z } = q +( r + r e + r e + r e | {z } = q ) · e = r + r e + r e + r e + r e + r e + r e + r e . Number r ∈ R is called the real part of o (denoted as Re o ) and the pure imaginaryoctonion r e + r e + . . . + r e is called the imaginary part of o (and denoted as Im o ).Octonions form a non-associative and a non-commutative algebra, which means that ingeneral, for o , o , o ∈ O ( o · o ) · o = o · ( o · o ) , o · o = o · o . e e e e e e e e e e e e e e e e − e − e e − e − e e e e − e − e e e − e − e e e e − e − e − e e − e e e − e − e − e − e e e e e e − e e − e − − e e e e e e − e − e e − − e e e − e e e − e − e e − Table 1: Multiplication rules in octonion algebra.On the other hand, it is true that for any o , o ∈ O we have ( o · o ) ∗ = o ∗ · o ∗ , where ∗ is the octonion conjugate, i.e. o ∗ = r − r e − r e − r e − r e − r e − r e − r e . As in case of complex numbers or quaternions, octonion conjugation is linear and we have o ∗∗ = o , which means that it is an involution. For any o , o ∈ O we also have that o · ( o · o ) = ( o · o ) · o , ( o · o ) · o = o · ( o · o ) (2.2)and o · ( o · o ) = ( o · o ) · o , (2.3)which means that the algebra of octonions is alternative (equations (2.2)) and flexible (equation (2.3)).In complex numbers we have the trigonometric form of a number and in octonionalgebra we can define a similar formula for any nonzero octonion o ∈ O : o = | o | · (cos θ + µ · sin θ ) , (2.4)where | o | = √ o · o ∗ is octonion norm, µ = Im o | Im o | is pure imaginary octonion and θ ∈ R isthe solution of the system of equations cos θ = Re o | o | , sin θ = | Im o || o | . To formulate the exponential form of an octonion, we have to define octonion exponentialfunction first. Similarly as for the complex numbers and quaternions [27], we use theinfinite series. For any o ∈ O , e o = exp( o ) := ∞ X k =0 o k k ! . It can be shown that if we denote o = Im o , then e o = e Re o (cid:18) cos | o | + o | o | sin | o | (cid:19) . o , o ∈ O we have e o + o = e o · e o if and only if o · o = o · o , which follows from the fact that the octonion multiplication is non-commutative.From the above considerations it immediately follows that the exponential form of anoctonion o ∈ O , o = 0 , can be defined as o = | o | · e θ µ , where θ and µ are defined as in (2.4). We can also generalize well-known formulas fortrigonometric functions, i.e. for any α ∈ R we have that cos α = 12 (cid:0) e µ α + e − µ α (cid:1) , sin α = 12 µ (cid:0) e µ α − e − µ α (cid:1) , (2.5)where µ is any octonion such that | µ | = 1 and Re µ = 0 (i.e. µ is pure unitary octonion). Itshould be noted that every non-zero octonion is invertible and for pure unitary octonions µ we have µ − = − µ . Many formulas presented in Section 3, concerning the Fourier transforms, are quitecomplicated due to the fact that octionion multiplication is non-associative and non-commutative. Inspired by [12], we introduce the algebra of quadruple-complex numbers ,which will allow us to reformulate all presented properties and show them in a simplerform, very similar to those well-known for classic Fourier transform.Like octonions, we will define the algebra of order over the field of real numbers andeach element of this algebra will be identified with the -tuple of real numbers, i.e. p = p + p e + p e + p e + p e + p e + p e + p e ∈ F , p , . . . , p ∈ R . Addition in F is defined in a classical way – element-wise. Before we define the multi-plication, recall that in Cayley-Dickson construction, every octonion can be writen as anordered pair of quaternions. We are going now one step further and rewrite an octonionas a quadruple of complex numbers: p = ( p + p e ) + ( p + p e ) e + ( p + p e ) e + ( p + p e ) e e = s + s e + s e + s e e , (2.6)where s , . . . , s ∈ C and multiplication is done from left to right.We will identify each element of F with a quadruple of complex numbers ( s , s , s , s ) .Every element of F will correspond to exactly one octonion defined by (2.6). Multiplication ⊙ is given by the formula ( s , s , s , s ) ⊙ ( t , t , t , t ) = ( s t − s t − s t + s t , s t + s t − s t − s t ,s t + s t − s t − s t , s t + s t + s t + s t ) for every ( s , s , s , s ) , ( t , t , t , t ) ∈ F . After straightforward computation we get themultiplication rules table, like we have in case of octonions (Tab. 2). We can see thatimaginary units in F don’t follow the same rules that applied to octonions, i.e. e ⊙ e = e ⊙ e = − e ⊙ e = e ⊙ e = − e ⊙ e = − e ⊙ e = e ⊙ e = − . e e e e e e e e e e e e e e e e − e − e e − e e − e e e e − − e e e − e − e e e − e − e e − e − e e e e e e e − − e − e − e e e − e e − e − e − e e e e e − e − e − e − e e e e − e − e e − e e e − Table 2: Multiplication rules in F .There is a similarity to double-complex numbers [20], which have been used in the analysisof 2-D systems [12] and (though not so named) in hypercomplex representation of 2Dnuclear magnetic resonance spectra [7].The multiplication in F is commutative and associative. One can also show that thereare no zero divisors in F (i.e. if s ⊙ t = 0 , then s = 0 or t = 0 ), however not every non-zeroelement of F has its ⊙ -inverse. If inverse of ( s , s , s , s ) exists then it is the only one andis equal to ( s , s , s , s ) − = 1 δ (cid:18) s ( s + s + s − s ) + 2 s s s , − s ( s + s − s + s ) − s s s , − s ( s − s + s + s ) − s s s , s ( − s + s + s + s ) + 2 s s s (cid:19) , (2.7)where δ = (cid:0) ( s − s ) + ( s + s ) (cid:1)(cid:0) ( s + s ) + ( s − s ) (cid:1) . Elements of F for which δ = 0 (e.g. (1 , , , ±
1) = 1 ± e ∈ F ) have no ⊙ -inverse. One caneasily notice that the equation (2.7) is similar to formula (3.4) in [12] for double-complexnumbers, but every numer in (3.4) was a real number. In (2.7) we have (in the generalcase) complex numbers. Definition of the octonion Fourier transform (OFT) of the real-valued function ofthree variables was introduced in [29] and used in later publications concerning theoryof hypercomplex analytic functions [17, 29, 30, 31, 32]. In [4] we proved that the OFTof real-valued function is well-defined and has some interesting properties (such as theanalogue of the Hermitian symetry). In [2] we stated that the inverse transform formulais correct for the octonion-valued functions and we presented the sketch of the proof. Inthe further part of this section we will present previously omitted details.Consider the octonion-valued function of three variables u : R → O , i.e. u ( x ) = u ( x ) + u ( x ) e + . . . + u ( x ) e , u i : R → R , i = 0 , . . . , , x = ( x , x , x ) . u is givenby the formula U OFT ( f ) = Z R u ( x ) e − e πf x e − e πf x e − e πf x d x . (2.8)Recall that the octonion algebra is non-associative, so it is necessary to note that themultiplication in the above integrals is done from left to right. As we already explainedin [4], choice and order of imaginary units in the exponents is not accidental. In order forthe integral (2.8) to exist, it is be necessary for the function to be at least integrable. Ingeneral, conditions of existence of the OFT are the same as for the classical (complex)Fourier transform. It is worth noting here the advantage of using octonion transformationover the use of classical transformation. Unlike the classical Fourier transformation, theOFT kernel is no longer a one-dimensional function, but it changes independently in threeorthogonal directions. This is an analogous observation to that which was made in thecase of the quaternion Fourier transform [5, 6].In this section, we will focus on the invertibility of the OFT. For the special case of thereal-valued functions we proved the following theorem in [4]. Here we prove the generalversion of the theorem, where the tested function has octonion values. Theorem 1.
Let u : R → O be continuous and let both u and its OFT be integrable (inLebesgue sense). Then for all x ∈ R we have u ( x ) = Z R U OFT ( f ) e e πf x e e πf x e e πf x d f (where multiplication is performed from left to right). The assumptions given above are quite strong and in many cases can be mitigated.A number of other conditions are known in the literature for the classic Fourier transformto be invertible and the equivalent of the above formula occurs [1, 10]. Then we usuallydeal with equality almost everywhere, or the integral is understood in the sense of theprincipal value. In the case of an octonion transformation, these conditions are identicaland detailed considerations are left to the reader. The abovementioned result followsfrom Fourier Integral Theorem [10], which we state under the same assumptions as inTheorem 1.
Theorem 2.
Let u : R n → R be continuous and let both u and its OFT be integrable (inLebesgue sense). Then u ( x ) = Z R n Z R n u ( y ) e π i f · ( x − y ) d y d f , where i = e is complex imaginary unit, x = ( x , . . . , x n ) , y = ( y , . . . , y n ) , f = ( f , . . . , f n ) and · is classic scalar product.of Theorem 1. We need to prove the following equation u ( x ) = Z R Z R u ( y ) · e − e πf y · e − e πf y · e − e πf y · e e πf x · e e πf x · e e πf x d y d f , u as a sum u = u + u e + . . . + u e and use the distributive law on the algebra ofoctonions. It follows that the claim of the theorem is equivalent to the system of equations u ( x ) = Z R Z R u ( y ) · e − e πf y · e − e πf y · e − e πf y · e e πf x · e e πf x · e e πf x d y d f , (2.9) u i ( x ) e i = Z R Z R u i ( y ) e i · e − e πf y · e − e πf y · e − e πf y · e e πf x · e e πf x · e e πf x d y d f , i = 1 , . . . , . (2.10)Proof of (2.9) can be found in [4] and we only need to prove (2.10). We follow the samesteps as in the original proof and use the fact (derived by straightforward calculations)that for any imaginary unit e i , i = 1 , . . . , , we have (cid:16)(cid:0) ( e i · e − e πf y ) · e − e πf y (cid:1) · e − e πf y (cid:17) · e e πf x = (cid:0) ( e i · e − e πf y ) · e − e πf y (cid:1) · (cid:0) e − e πf y · e e πf x (cid:1) , (2.11) (cid:0) ( e i · e − e πf y ) · e − e πf y (cid:1) · e e πf x = ( e i · e − e πf y ) · ( e − e πf y · e e πf x ) (2.12) ( e i · e − e πf y ) · e e πf x = e i · ( e − e πf y · e e πf x ) . (2.13)Then, using (2.11)–(2.13), Fubini’s Theorem and Theorem 2 we have for i = 1 , . . . , Z R Z R (cid:18)(cid:18)(cid:16)(cid:0) ( u i ( y ) e i · e − e πf y ) · e − e πf y (cid:1) · e − e πf y (cid:17) · e e πf x (cid:19) · e e πf x (cid:19) · e e πf x d y d f (2.11) = Z R Z R (cid:18)(cid:18)(cid:0) ( e i · e − e πf y ) · e − e πf y (cid:1) · (cid:18)Z R Z R u i ( y ) · e − e πf y · e e πf x d y d f (cid:19) (cid:19) · e e πf x (cid:19) · e e πf x d y d y d f d f Th. 2 = Z R Z R (cid:18)(cid:16)(cid:0) ( e i · e − e πf y ) · e − e πf y (cid:1) · u i ( y , y , x ) (cid:17) · e e πf x (cid:19) · e e πf x d y d y d f d f = Z R Z R u i ( y , y , x ) · (cid:16)(cid:0) ( e i · e − e πf y ) · e − e πf y (cid:1) · e e πf x (cid:17) · e e πf x d y d y d f d f (2.12) = Z R Z R (cid:18) ( e i · e − e πf y ) · (cid:18)Z R Z R u i ( y , y , x ) · e − e πf y · e e πf x d y d f (cid:19) (cid:19) · e e πf x d y d f Th. 2 = Z R Z R (cid:0) ( e i · e − e πf y ) · u i ( y , x , x ) (cid:1) · e e πf x d y d f = Z R Z R u i ( y , x , x ) · ( e i · e − e πf y ) · e e πf x d y d f (2.13) = e i · Z R Z R u i ( y , x , x ) · e − e πf y · e e πf x d y d f h. 2 = e i · u i ( x , x , x ) . It concludes the proof.It is worth noting that the above theorem was independently proved also in a recentarticle [23], in which the author used other methods.Before we proceed to discuss the properties of octonion Fourier transforms, we shouldstart with the basic result formulated below. From now on, we will assume that all thefunctions under consideration have well-defined octonion Fourier transforms. We will usethe convention that the OFT of function u is denoted by U OFT or F OFT { u } . Analogously,the classic (complex) Fourier transform of u will be denoted by F CFT { u } . Theorem 3.
Octonion Fourier transform is R -linear operation, i.e. F OFT { a · u + b · v } = a · F OFT { u } + b · F OFT { v } , a, b ∈ R . (2.14)It should be noted here that, unlike the classical (complex) Fourier transform (andalso quaternion Fourier transform), OFT is not linear in general (to be more precise – itis not O -linear), i.e. property (2.14) is not true for any a, b ∈ O . This is due to the factthat the octonion multiplication is not associative.For many real- or complex-valued functions the form of the classic Fourier transform iswell known. To calculate the octonion Fourier transform of such function, we can use therelationship between these transformations instead of using formula (2.8). In particular,the following theorem holds, which is the generalization of the result of [31], where it wasproved for real-valued functions. This result was originally stated in [2], here we completethe details of the proof. Theorem 4.
Let u : R → C , U = F CFT { u } and U OFT = F OFT { u } . Then U OFT ( f , f , f ) = 14 (cid:0) U ( f , f , f ) · (1 − e ) + U ( f , − f , f ) · (1 + e ) (cid:1) · (1 − e )+ 14 (cid:0) U ( f , f , − f ) · (1 − e ) + U ( f , − f , − f ) · (1 + e ) (cid:1) · (1 + e ) (2.15) where octonion multiplication is done from left to right. Remark 1.
Equation (39) proved in [31] may look slightly different from (2.15) , but afterstraightforward computation and application of the Hermitian symmetry of the Fouriertransform of the real-valued functions we get the abovementioned formula.of Theorem 4.
We carefully follow and modify steps presented in [31]. From the definitionof the classical Fourier transform we get U ( f , f , f ) = Z R u ( x ) e − e α e − e α e − e α d x , where α j = 2 πf j x j , j = 1 , , . From the equivalent definition of sine and cosine functionswe get (cid:0) U ( f , f , f ) + U ( f , − f , f ) (cid:1) = Z R u ( x ) e − e α (cos α ) e − e α d x , (2.16) (cid:0) U ( f , f , f ) − U ( f , − f , f ) (cid:1) = Z R u ( x ) e − e α ( − e sin α ) e − e α d x . (2.17)9y changing the sign of f in (2.17) and multiplying (from the left) by e we get (cid:0) U ( f , f , − f ) − U ( f , − f , − f ) (cid:1) e = Z R u ( x ) e − e α ( e sin α ) e − e α d x , (2.18)which follows from the fact that (cid:16)(cid:0) ( u · e − e α ) · e (cid:1) · e e α (cid:17) · e = (cid:0) ( u · e − e α ) · ( e · e ) (cid:1) · e − e α (from the fact that octonion multiplication is alternative). Subtracting (2.18) from (2.16)we then obtain (cid:0) U ( f , f , f ) + U ( f , − f , f ) (cid:1) + 12 (cid:0) U ( f , − f , − f ) − U ( f , f , − f ) (cid:1) e = Z R u ( x ) e − e α e − e α e − e α d x . We introduce the following notation: V ( f , f , f )= 12 (cid:0) U ( f , f , f ) + U ( f , − f , f ) (cid:1) + 12 (cid:0) U ( f , − f , − f ) − U ( f , f , − f ) (cid:1) e . (2.19)By following similar steps as before we get (cid:0) V ( f , f , f ) + V ( f , f , − f ) (cid:1) = Z R u ( x ) e − e α e − e α (cos α ) d x , (2.20) (cid:0) V ( f , f , f ) − V ( f , f , − f ) (cid:1) = Z R u ( x ) e − e α e − e α ( − e sin α ) d x . (2.21)As earlier we change the sign of f in (2.21) and multiply (from the left) by e and obtain (cid:0) V ( f , − f , f ) − V ( f , − f , − f ) (cid:1) e = Z R u ( x ) e − e α e − e α ( e sin α ) d x (2.22)(again from the fact that octonion multiplication is alternative). By subtracting equa-tion (2.22) from (2.20) we get (cid:0) V ( f , f , f ) + V ( f , f , − f ) (cid:1) + 12 (cid:0) V ( f , − f , − f ) − V ( f , − f , f ) (cid:1) e = Z R u ( x ) e − e α e − e α e − e α d x . (2.23)We conclude the proof by substituting equation (2.19) in (2.23) and regrouping all terms.In the general case of an octonion-valued function, well-known formulas for the classicFourier transform can also be used. If we factor out the complex components of theoctonion-valued function, we get u = u + u e + u e + u e + u e + u e + u e + u e = ( u + u e ) + ( u + u e ) e + ( u + u e ) e + ( u + u e ) e e =: v + v e + v e + v e e and v , . . . , v are complex-valued functions. Using Theorem 4 we can easily calculatethe OFTs of those functions, denote them by V , . . . , V , respectively. By straightforwardcalculations we obtain the following properties of octonions: (cid:0) (( o · e ) · e − e α ) · e − e α (cid:1) · e − e α = (cid:0) (( o · e e α ) · e − e α ) · e e α (cid:1) · e , (2.24)10 (( o · e ) · e − e α ) · e − e α (cid:1) · e − e α = (cid:0) (( o · e e α ) · e e α ) · e − e α (cid:1) · e , (2.25) (cid:0) (( o · e · e ) · e − e α ) · e − e α (cid:1) · e − e α = (cid:0) (( o · e − e α ) · e e α ) · e e α (cid:1) · e · e , (2.26)for any o ∈ O and α , α , α ∈ R . From those calculations, the corollary below immediatelyfollows. Corollary 1.
Let v , . . . , v : R → C , V i = F OFT { v i } and u = v + v e + v e + v e e , U OFT = F OFT { u } . Then U OFT ( f , f , f ) = V ( f , f , f ) + V ( − f , f , − f ) · e + V ( − f , − f , f ) · e + V ( f , − f , − f ) · e · e . (2.27)Inverse octonion Fourier transformation can also be done using classic tools. However,the situation is more complicated from the beginning. In general, the OFT of any functionis a function with octonion values. However, we will start, as in the case of forwardtransform, from the case when a function has OFT with complex values, but in thespecific subfield of the octonion algebra, i.e. C e = { x + x e ∈ O : x , x ∈ R } . It is enough to note that in every subfield of this type we can define the classic Fouriertransform. In the above case, we have formulas for forward and inverse transforms: U ( f , f , f ) = Z R u ( x , x , x ) e − e α e − e α e − e α d x ,u ( x , x , x ) = Z R U ( f , f , f ) e e α e e α e e α d f (2.28)where u : R → C e , α j = 2 πf j x j , j = 1 , , . We have then, of course U : R → C e . Theorem 5.
Let u : R → O be such that U OFT = F OFT { u } : R → C e . Moreover, let ˆ u = F − { U OFT } (in the C e complex subfield of O , i.e. (2.28) ). Then u ( x , x , x ) = 14 (cid:0) ˆ u ( x , x , x ) · (1 + e ) + ˆ u ( x , − x , x ) · (1 − e ) (cid:1) · (1 + e )+ 14 (cid:0) ˆ u ( − x , x , x ) · (1 + e ) + ˆ u ( − x , − x , x ) · (1 − e ) (cid:1) · (1 − e ) (2.29) where octonion multiplication is done from left to right.Proof. From the modified definition of the classical Fourier transform (2.28) we get ˆ u ( x , x , x ) = Z R U OFT ( f ) e e α e e α e e α d f , where α j = 2 πf j x j , j = 1 , , . From the equivalent definition of sine and cosine functionswe get (cid:0) ˆ u ( x , x , x ) + ˆ u ( x , − x , x ) (cid:1) = Z R U OFT ( f ) e e α (cos α ) e e α d f , (2.30) (cid:0) ˆ u ( x , x , x ) − ˆ u ( x , − x , x ) (cid:1) = Z R U OFT ( f ) e e α ( e sin α ) e e α d f . (2.31)11y changing the sign of x in (2.31) and multiplying (from the left) by e we get (cid:0) ˆ u ( − x , x , x ) − ˆ u ( − x , − x , x ) (cid:1) e = Z R U OFT ( f ) e e α ( e sin α ) e e α d f , (2.32)which follows from the fact that (cid:16)(cid:0) ( U OFT · e e α ) · e (cid:1) · e − e α (cid:17) · e = (cid:0) ( U OFT · e e α ) · ( e ) (cid:1) · e e α (from the fact that octonion multiplication is alternative). Subtracting (2.32) from (2.30)we then obtain (cid:0) ˆ u ( x , x , x ) + ˆ u ( x , − x , x ) (cid:1) + 12 (cid:0) ˆ u ( − x , x , x ) − ˆ u ( − x , − x , x ) (cid:1) e = Z R U OFT ( f ) e e α e e α e e α d f . We introduce the following notation: w ( x , x , x )= 12 (cid:0) ˆ u ( x , x , x ) + ˆ u ( x , − x , x ) (cid:1) + 12 (cid:0) ˆ u ( − x , x , x ) − ˆ u ( − x , − x , x ) (cid:1) e . (2.33)By following similar steps as before we get (cid:0) w ( x , x , x ) + w ( − x , x , x ) (cid:1) = Z R U OFT ( f ) e e α e e α (cos α ) d f , (2.34) (cid:0) w ( x , x , x ) − w ( − x , x , x ) (cid:1) = Z R U OFT ( f ) e e α e e α ( e sin α ) d f . (2.35)As earlier we change the sign of x in (2.35) and multiply (from the left) by e and obtain (cid:0) w ( x , − x , x ) − w ( − x , − x , x ) (cid:1) e = Z R U OFT ( f ) e e α e e α ( e sin α ) d f (2.36)(again from the fact that octonion multiplication is alternative). By subtracting equa-tion (2.36) from (2.34) we get (cid:0) w ( x , x , x ) + w ( − x , x , x ) (cid:1) + 12 (cid:0) w ( x , − x , x )) − w ( − x , − x , x ) (cid:1) e = Z R U OFT ( f ) e e α e e α e e α d f . (2.37)We conclude the proof by substituting equation (2.33) in (2.37) and regrouping all terms.We can now return to the general case. Similarly as before, we factor out the complex( C e ) components of the octonion-valued function: U OFT = U + U e + U e + U e + U e + U e + U e + U e = ( U + U e ) + ( U − U e ) e + ( U − U e ) e + ( U + U e ) e e := V + V e + V e + V e e and V , . . . , V are functions with values in C e . Using Theorem 5 we can calculate theinverse OFTs of those functions, denote them by v , . . . , v , respectively. We also use thefact that (cid:0) (( z · e ) · e e α ) · e e α (cid:1) · e e α = (cid:0) (( z · e − e α ) · e − e α ) · e e α (cid:1) · e , (( z · e ) · e e α ) · e e α (cid:1) · e e α = (cid:0) (( z · e − e α ) · e e α ) · e − e α (cid:1) · e , (cid:0) (( z · e · e ) · e e α ) · e e α (cid:1) · e e α = (cid:0) (( z · e e α ) · e − e α ) · e − e α (cid:1) · e · e , for every z ∈ C e . From those calculation we get the corollary below. Corollary 2.
Let V , . . . , V : R → C e , v i = F − { V i } , i = 0 , . . . , , and let U OFT = V + V e + V e + V e e , u = F − { U OFT } . Then u ( x , x , x ) = v ( x , x , x ) + v ( x , − x , − x ) · e + v ( − x , x , − x ) · e + v ( − x , − x , x ) · e · e . From Theorem 4 and Corollary 1 one can draw several direct conclusions relatedto the behavior of the octonion transformation in infinity and the composition of thetransformations.
Theorem 6.
Let u : R → O be integrable (in Lebesgue sense) and U OFT = F OFT { u } .Then lim | f |→∞ U OFT ( f ) = 0 . Proof.
It is a direct corollary from the classical Riemann-Lebesgue theorem [10], Theo-rem 4 and Corollary 1.Before we formulate and prove the next result, let us recall the classical theorem knownfrom Fourier analysis [10].
Theorem 7.
Let u : R n → C be smooth and rapidly decreasing (i.e. element of Schwartzclass). Then F CFT {F CFT { u }} ( x ) = u ( − x ) , and so the classical Fourier transform has period (i.e. if we apply it four times, we getthe identity operator). In the case of OFT, the analogous result is very similar, but slightly more complicated.
Theorem 8.
Let u : R → C be smooth and rapidly decreasing (i.e. element of Schwartzclass). Then F OFT {F OFT { u }} ( x , x , x ) = 12 (cid:0) u ( x , x , x ) + u ( − x , x , − x )+ u ( − x , − x , x ) − u ( x , − x , − x ) (cid:1) , and so the OFT has period .Proof. We begin with the case of u : R → C . Let U = F CFT { u } . By carrying out directcalculations, we can write the claim of Theorem 4 in the form F OFT { u } ( f ) = U ee ( f ) − U oe ( f ) · e · e − U eo ( f ) · e · e − U oo ( f ) · e · e , where U yz , y, z ∈ { e, o } , are four components of U of different parity with respect to f and f , i.e. U yz ( f , f , f ) = ( U ( f , f , f ) + ε y U ( f , − f , f )+ ε z U ( f , f , − f ) + ε y ε z U ( f , − f , − f )) / , ε y = 1 if y = e and ε y = − if y = o , etc.Note that this is the form as in the assumptions of Corollary 1, so when calculatingthe OFT of function U OFT we get F OFT {F OFT { u }} ( x , x , x ) = F OFT { U ee } ( x , x , x )+ F OFT {− U oe · e } ( − x , x , − x ) · e + F OFT {− U eo · e } ( − x , − x , x ) · e + F OFT {− U oo } ( x , − x , − x ) · e · e . All functions which OFTs we want to calculate are C -valued functions, so we can againuse Theorem 4. After tedious calculations we get that F OFT { U ee } ( x , x , x ) = u ee ( − x , − x , − x ) , F OFT {− U oe · e } ( − x , x , − x ) · e = − u oe ( x , − x , − x ) , F OFT {− U eo · e } ( − x , − x , x ) · e = − u eo ( x , − x , − x ) , F OFT {− U oo } ( x , − x , − x ) · e · e = − u oo ( − x , − x , − x ) , where u yz , y, z ∈ { e, o } , are four components of u of different parity with respect to x and x , just like earlier. By expanding the above functions and rearranging the componentswe will receive a claim in the case of functions with complex values. It is easy to calculatethat by applying this "double OFT" transformation twice, we get identity.In the general case of u : R → O we proceed like in the proof of Corollary 1. Let u = v + v e + v e + v e e , where v , . . . , v : R → C . Then, from (2.27) and (2.24)–(2.26) we get F OFT {F OFT { u }} = F OFT {F OFT { v }} + F OFT {F OFT { v }} e + F OFT {F OFT { v }} e + F OFT {F OFT { v }} e e . We immediately receive a claim of the theorem from the previous part of the proof.In [17] it was proved that the octonion Fourier transform of the real-valued functioncan be represented as the octonion sum of components of different parity, i.e. U OFT = U eee − U oee e − U eoe e + U ooe e − U eeo e + U oeo e + U eoo e − U ooo e , (2.38)where U xyz , x, y, z ∈ { e, o } are defined as U eee ( f ) = Z R u eee ( x ) cos(2 πf x ) cos(2 πf x ) cos(2 πf x ) d x , (2.39) U oee ( f ) = Z R u oee ( x ) sin(2 πf x ) cos(2 πf x ) cos(2 πf x ) d x , (2.40) U eoe ( f ) = Z R u eoe ( x ) cos(2 πf x ) sin(2 πf x ) cos(2 πf x ) d x , (2.41) U ooe ( f ) = Z R u ooe ( x ) sin(2 πf x ) sin(2 πf x ) cos(2 πf x ) d x , (2.42) U eeo ( f ) = Z R u eeo ( x ) cos(2 πf x ) cos(2 πf x ) sin(2 πf x ) d x , (2.43) U oeo ( f ) = Z R u oeo ( x ) sin(2 πf x ) cos(2 πf x ) sin(2 πf x ) d x , (2.44) U eoo ( f ) = Z R u eoo ( x ) cos(2 πf x ) sin(2 πf x ) sin(2 πf x ) d x , (2.45)14 ooo ( f ) = Z R u ooo ( x ) sin(2 πf x ) sin(2 πf x ) sin(2 πf x ) d x , (2.46)where f = ( f , f , f ) , x = ( x , x , x ) , and functions u xyz ( x ) , x, y, z ∈ { e, o } , are eightcomponents of u of different parity with respect to x , x and x , i.e. u xyz ( x , x , x ) = ( u ( x , x , x ) + ε x u ( − x , x , x )+ ε y u ( x , − x , x ) + ε x ε y u ( − x , − x , x )+ ε z u ( x , x , − x ) + ε x ε z u ( − x , x , − x )+ ε y ε z u ( x , − x , − x ) + ε x ε y ε z u ( − x , − x , − x )) / , where ε x = 1 if x = e and ε x = − if x = o , etc.In this notation we use indices e and o to denote that the function is even ( e ) or odd( o ) with respect to the proper variable, e.g. function u eeo ( x ) is even with respect to x and x and odd with respect to x . Analogous considerations can be performed for U xyz ,e.g. U eoo is even with respect to f and odd with respect to f and f .It should be noted that in case of the real-valued functions u , all terms U xyz in (2.38) arereal-valued functions. Similar formulas can be obtained for the octonion-valued functionsbut we omit the details here. Properties of the complex Fourier transform and its quaternion counterpart are wellknown in literature [1, 5, 10, 13]. In [4] we already proved some of their octonion analogues(i.e. shift theorem stated in Theorem 13 and Hermitian symmetry analogue) and in thissection we will derive equivalents of other classical properties such as argument scaling andmodulation theorem. Sketches of some proofs can be found in [2], here we will present allpreviously omitted details, but also present previously unpublished results. The summaryof the content of this section is shown in Table 3.If we do not state otherwise, then in each of the following statements we assume that u : R → O and U = F OFT { u } . Theorem 9.
Let a, b, c ∈ R \{ } and v : R → O be defined by v ( x , x , x ) = u ( x a , x b , x c ) , V = F OFT { v } . Then V ( f , f , f ) = | abc | U ( af , bf , cf ) . Proof.
Proof is very similar to the classical case and utilizes integration by substitution.From the definition of the OFT we have V ( f ) = Z R u (cid:16) x a , x b , x c (cid:17) e − e πf x e − e πf x e − e πf x d x = ( ⋆ ) . We introduce the substitution ( y , y , y ) = (cid:0) x a , x b , x c (cid:1) . Let us note that determinant ofthe Jacobian matrix of this substitution is equal to (cid:12)(cid:12)(cid:12) ∂ ( x ,x ,x ) ∂ ( y ,y ,y ) (cid:12)(cid:12)(cid:12) = | abc | . Then ( ⋆ ) = Z R u ( y , y , y ) e − e πaf y e − e πbf y e − e πcf y | abc | d y = | abc | U ( af , bf , cf ) , which concludes the proof. 15 unction octonion Fourier transform1. u ( x a , x b , x c ) | abc | U ( af , bf , cf ) u ( x ) · cos(2 πf x ) (cid:0) U ( f + f , f , f ) + U ( f − f , f , f ) (cid:1) · u ( x ) · cos(2 πf x ) (cid:0) U ( f , f + f , f ) + U ( f , f − f , f ) (cid:1) · u ( x ) · cos(2 πf x ) (cid:0) U ( f , f , f + f ) + U ( f , f , f − f ) (cid:1) · u ( x ) · sin(2 πf x ) (cid:0) U ( f + f , − f , − f ) − U ( f − f , − f , − f ) (cid:1) · e u ( x ) · sin(2 πf x ) (cid:0) U ( f , f + f , − f ) − U ( f , f − f , − f ) (cid:1) · e u ( x ) · sin(2 πf x ) (cid:0) U ( f , f , f + f ) − U ( f , f , f − f ) (cid:1) · e u ( x ) · exp( − e πf x ) U ( f + f , f , f ) u ( x ) · exp( − e πf x ) (cid:0) U ( f , f + f , f ) + U ( f , f − f , f )+ U ( − f , f + f , f ) − U ( − f , f − f , f ) (cid:1) · u ( x ) · exp( − e πf x ) (cid:0) U ( f , f , f + f ) + U ( f , f , f − f )+ U ( − f , − f , f + f ) − U ( − f , − f , f − f ) (cid:1) · u ( x − α, x , x ) cos(2 πf α ) U ( f , f , f ) − sin(2 πf α ) U ( f , − f , − f ) · e u ( x , x − β, x ) cos(2 πf β ) U ( f , f , f ) − sin(2 πf β ) U ( f , f , − f ) · e u ( x , x , x − γ ) cos(2 πf γ ) U ( f , f , f ) − sin(2 πf γ ) U ( f , f , f ) · e u x ( x ) U ( f , − f , − f ) · (2 πf e ) u x ( x ) U ( f , f , − f ) · (2 πf e ) u x ( x ) U ( f , f , f ) · (2 πf e ) πx · u ( x ) U f ( f , − f , − f ) · e πx · u ( x ) U f ( f , f , − f ) · e πx · u ( x ) U f ( f , f , f ) · e ( u ∗ v )( x ) V ( f , f , f ) · ( U eee ( f ) − U eeo ( f ) e )+ V ( f , − f , − f ) · ( − U oee ( f ) e + U ooe ( f ) e )+ V ( f , f , − f ) · ( − U eoe ( f ) e + U oeo ( f ) e )+ V ( − f , f , − f ) · ( U eoo ( f ) e − U ooo ( f ) e ) Table 3: Summary of octonion Fourier transform properties.Theorem 9 can be generalized to all linear maps of x . In the case of quaternion Fouriertransform one can find similar result in [5] for functions u : R → R and v ( x ) = u ( Ax ) ,where A real-valued × matrix such that det( A ) = 0 . Then V ( f , f ) = 12 det A (cid:0) U (˜ a f + ˜ a f , ˜ a f + ˜ a f ) + U (˜ a f − ˜ a f , − ˜ a f + ˜ a f ) − e U ( − ˜ a f + ˜ a f , − ˜ a f + ˜ a f ) + e U ( − ˜ a f − ˜ a f , ˜ a f + ˜ a f ) (cid:1) , (cid:0) ˜ a ˜ a ˜ a ˜ a (cid:1) = A A .In the octonion setup, considering v ( x ) = u ( Ax ) , where A is some arbitrary non-singular × matrix, we would get a result containing 64 different terms. Due to thecomplication of calculations and slight significance for further research we skip this for-mula.The next three theorems are known in signal and system theory as the modulationtheorem. One can notice that the claim of cosine modulation theorem (with cosine functionas a carrier) is exactly the same as in the case of complex Fourier transform. This can notbe said about the sine modulation theorem. Theorem 10.
Let f ∈ R and denote u cos ,i ( x ) = u ( x ) · cos(2 πf x i ) , U cos ,i = F OFT { u cos ,i } , i = 1 , , . Then U cos , ( f , f , f ) = (cid:0) U ( f + f , f , f ) + U ( f − f , f , f ) (cid:1) · ,U cos , ( f , f , f ) = (cid:0) U ( f , f + f , f ) + U ( f , f − f , f ) (cid:1) · ,U cos , ( f , f , f ) = (cid:0) U ( f , f , f + f ) + U ( f , f , f − f ) (cid:1) · . Proof.
We will use the equivalent definition of the cosine function, i.e. equation (2.5).Then cos α = 12 (cid:0) e e α + e − e α (cid:1) = 12 (cid:0) e e α + e − e α (cid:1) = 12 (cid:0) e e α + e − e α (cid:1) . (3.1)Then for i = 1 we have U cos , ( f , f , f ) = Z R (cid:0) u ( x ) · cos(2 πf x ) (cid:1) e − e πf x e − e πf x e − e πf x d x = Z R u ( x ) (cid:0) e − e πf x cos(2 πf x ) (cid:1) e − e πf x e − e πf x d x = 12 Z R u ( x ) (cid:0) e − e πf x ( e e πf x + e − e πf x ) (cid:1) e − e πf x e − e πf x d x = 12 Z R u ( x ) (cid:0) e − e π ( f − f ) x + e − e π ( f + f ) x (cid:1) e − e πf x e − e πf x d x = 12 (cid:0) U ( f − f , f , f ) + U ( f + f , f , f ) (cid:1) , which concludes the proof in this case. For i = 2 , proceed analogously. Theorem 11.
Let f ∈ R and denote u sin ,i ( x ) = u ( x ) · sin(2 πf x i ) , U sin ,i = F OFT (cid:8) u sin ,i (cid:9) , i = 1 , , . Then U sin , ( f , f , f ) = (cid:0) U ( f + f , − f , − f ) − U ( f − f , − f , − f ) (cid:1) · e ,U sin , ( f , f , f ) = (cid:0) U ( f , f + f , − f ) − U ( f , f − f , − f ) (cid:1) · e ,U sin , ( f , f , f ) = (cid:0) U ( f , f , f + f ) − U ( f , f , f − f ) (cid:1) · e . roof. We proceed similarly as in proof of Theorem 10. We will also use the equivalentdefinition of the sine function formulated in equation (2.5), i.e. sin α = 12 e (cid:0) e e α − e − e α (cid:1) = 12 e (cid:0) e e α − e − e α (cid:1) = 12 e (cid:0) e e α − e − e α (cid:1) . (3.2)The following properties of octonion numbers, which can be derived using direct calcula-tions, will also be necessary. For any o ∈ O and α , α , α ∈ R we have (cid:16)(cid:0) o · ( e − e α · e ) (cid:1) · e − e α (cid:17) · e − e α = (cid:16)(cid:0) ( o · e − e α ) · e e α (cid:1) · e e α (cid:17) · e , (3.3) (cid:0) ( o · e − e α ) · ( e − e α · e ) (cid:1) · e − e α = (cid:16)(cid:0) ( o · e − e α ) · e − e α (cid:1) · e e α (cid:17) · e , (3.4) (cid:0) ( o · e − e α ) · e − e α (cid:1) · ( e − e α · e ) = (cid:16)(cid:0) ( o · e − e α ) · e − e α (cid:1) · e − e α (cid:17) · e . (3.5)Then, for i = 1 we have U sin , ( f , f , f ) = Z R (cid:0) u ( x ) · sin(2 πf x ) (cid:1) e − e πf x e − e πf x e − e πf x d x = Z R u ( x ) (cid:0) e − e πf x sin(2 πf x ) (cid:1) e − e πf x e − e πf x d x = − Z R u ( x ) (cid:0) e − e πf x · e sin(2 πf x ) · e (cid:1) e − e πf x e − e πf x d x (3.3) = − Z R u ( x ) (cid:0) e − e πf x ( e e πf x − e − e πf x ) (cid:1) e e πf x e e πf x d x · e = − Z R u ( x ) (cid:0) e − e π ( f − f ) x − e − e π ( f + f ) x (cid:1) e − e πf x e − e πf x d x = − (cid:0) U ( f − f , f , f ) − U ( f + f , f , f ) (cid:1) · e , which concludes the proof. For i = 2 , the property is proved analogously, using equa-tions (3.4) and (3.5).It may seem that the next theorem is a simple consequence of Theorem 10 and 11.However, that is not the case since OFT is not a O -linear operation. We need to prove itindependently. Theorem 12.
Let f ∈ R and u exp ,i ( x ) = u ( x ) · exp( − e i − πf x i ) , U exp ,i = F OFT { u exp ,i } , i = 1 , , . Then U exp , ( f , f , f ) = U ( f + f , f , f ) ,U exp , ( f , f , f ) = (cid:0) U ( f , f + f , f ) + U ( f , f − f , f )+ U ( − f , f + f , f ) − U ( − f , f − f , f ) (cid:1) · ,U exp , ( f , f , f ) = (cid:0) U ( f , f , f + f ) + U ( f , f , f − f )+ U ( − f , − f , f + f ) − U ( − f , − f , f − f ) (cid:1) · . Proof.
Properties in the claim of this theorem are proved similarly as those of Theorem 10and 11 – using equations (3.1) and (3.2). Furthermore we will use the following fact – foreach o ∈ O and α , α , α ∈ R we have: (cid:16)(cid:0) ( o · e ) · e − e α (cid:1) · e − e α (cid:17) · e − e α = (cid:16)(cid:0) o · ( e − e α · e ) (cid:1) · e − e α (cid:17) · e − e α , (3.6) (cid:16)(cid:0) ( o · e ) · e − e α (cid:1) · e − e α (cid:17) · e − e α = (cid:0) ( o · e e α ) · ( e − e α · e ) (cid:1) · e − e α , (3.7) (cid:16)(cid:0) ( o · e ) · e − e α (cid:1) · e − e α (cid:17) · e − e α = (cid:0) ( o · e e α ) · e e α (cid:1) · ( e − e α · e ) . (3.8)18hen for i = 3 we have U exp , ( f , f , f ) = Z R (cid:0) u ( x ) · e − e πf x (cid:1) e − e πf x e − e πf x e − e πf x d x = Z R (cid:0) u ( x ) · (cos(2 πf x ) − e sin(2 πf x )) (cid:1) e − e πf x e − e πf x e − e πf x d x = Z R (cid:0) u ( x ) · cos(2 πf x ) (cid:1) e − e πf x e − e πf x e − e πf x d x − Z R (cid:0) u ( x ) · e sin(2 πf x ) (cid:1) e − e πf x e − e πf x e − e πf x d x (3.8) = Z R u ( x ) e − e πf x e − e πf x (cid:0) e − e πf x · cos(2 πf x ) (cid:1) d x − Z R u ( x ) e e πf x e e πf x (cid:0) e − e πf x · e sin(2 πf x ) (cid:1) d x = 12 (cid:0) U ( f , f , f − f ) + U ( f , f , f + f ) (cid:1) − (cid:0) U ( − f , − f , f − f ) − U ( − f , − f , f + f ) (cid:1) , which concludes the proof in this case. Proofs for i = 1 , are similar and use equa-tions (3.6) and (3.7).To complete our considerations about properties of the octonion Fourier transformsof octonion-valued functions, we should also state and prove the shift theorem. In case ofreal-valued functions we already stated this theorem in our earlier work, i.e. article [4]. Theorem 13.
Let α, β, γ ∈ R and denote u α ( x ) = u ( x − α, x , x ) , u β ( x ) = u ( x , x − β, x ) and u γ ( x ) = u ( x , x , x − γ ) . Let U ℓ = F OFT (cid:8) u ℓ (cid:9) , ℓ = α, β, γ . Then U α ( f , f , f ) = cos(2 πf α ) U ( f , f , f ) − sin(2 πf α ) U ( f , − f , − f ) · e , (3.9) U β ( f , f , f ) = cos(2 πf β ) U ( f , f , f ) − sin(2 πf β ) U ( f , f , − f ) · e , (3.10) U γ ( f , f , f ) = cos(2 πf γ ) U ( f , f , f ) − sin(2 πf γ ) U ( f , f , f ) · e . (3.11) Proof.
We will use again the tools used in proofs of previous claims. Consider the OFTof function u α . Using integration by substitution we get U α ( f , f , f ) = Z R u ( x − α, x , x ) e − e πf x e − e πf x e − e πf x d x = Z R u ( x , x , x ) e − e πf ( x + α ) e − e πf x e − e πf x d x = Z R u ( x )( e − e πf x · e − e πf α ) e − e πf x e − e πf x d x = Z R u ( x ) (cid:0) e − e πf x · (cos(2 πf α ) − e sin(2 πf α )) (cid:1) e − e πf x e − e πf x d x (3.3) = cos(2 πf α ) Z R u ( x ) e − e πf x e − e πf x e − e πf x d x − sin(2 πf α ) Z R u ( x ) e − e πf x e e πf x e e πf x d x · e = cos(2 πf α ) U ( f , f , f ) − sin(2 πf α ) U ( f , − f , − f ) · e , which concludes the proof for u α . The derivation of (3.10) and (3.11) is very similar andutilises properties (3.4) and (3.5). 19ext properties that we will prove are a key element in the analysis of multidimensionallinear time-invariant systems described by a system of partial differential equations. Inour considerations, however, we will limit ourselves to real-valued functions and from nowon we assume that u, v : R → R and U and V are the OFTs of u and v , respectively. Wewill also assume that the relevant derivatives of u exist, as well as their OFTs. Theorem 14.
Let U ∂x , U ∂x and U ∂x denote the OFTs of u x , u x and u x , respectively.Then U ∂x ( f , f , f ) = U ( f , − f , − f ) · (2 πf e ) , (3.12) U ∂x ( f , f , f ) = U ( f , f , − f ) · (2 πf e ) , (3.13) U ∂x ( f , f , f ) = U ( f , f , f ) · (2 πf e ) . (3.14) Proof.
We will prove only the first formula. Consider derivative u x and its octonionFourier transform U ∂x . u x is a real-valued function, hence we can write U ∂x as a sumof eight components of different parity U ∂x = U ∂x eee − U ∂x oee e − U ∂x eoe e + U ∂x ooe e − U ∂x eeo e + U ∂x oeo e + U ∂x eoo e − U ∂x ooo e , where U ∂x eee ( f ) = Z R u x ( x ) cos(2 πf x ) cos(2 πf x ) cos(2 πf x ) d x ,U ∂x oee ( f ) = Z R u x ( x ) sin(2 πf x ) cos(2 πf x ) cos(2 πf x ) d x ,U ∂x eoe ( f ) = Z R u x ( x ) cos(2 πf x ) sin(2 πf x ) cos(2 πf x ) d x ,U ∂x ooe ( f ) = Z R u x ( x ) sin(2 πf x ) sin(2 πf x ) cos(2 πf x ) d x ,U ∂x eeo ( f ) = Z R u x ( x ) cos(2 πf x ) cos(2 πf x ) sin(2 πf x ) d x ,U ∂x oeo ( f ) = Z R u x ( x ) sin(2 πf x ) cos(2 πf x ) sin(2 πf x ) d x ,U ∂x eoo ( f ) = Z R u x ( x ) cos(2 πf x ) sin(2 πf x ) sin(2 πf x ) d x ,U ∂x ooo ( f ) = Z R u x ( x ) sin(2 πf x ) sin(2 πf x ) sin(2 πf x ) d x , where f = ( f , f , f ) , x = ( x , x , x ) .We will use integration by parts and utilize the fact that for every integrable andsmooth function u and every x , x ∈ R we have lim x →±∞ u ( x ) = 0 . Then U ∂x eee ( f ) = Z R u ( x ) sin(2 πf x ) cos(2 πf x ) cos(2 πf x ) d x · πf = U oee ( f ) · πf ,U ∂x oee ( f ) = − Z R u ( x ) cos(2 πf x ) cos(2 πf x ) cos(2 πf x ) d x · πf = − U eee ( f ) · πf ,U ∂x eoe ( f ) = Z R u ( x ) sin(2 πf x ) sin(2 πf x ) cos(2 πf x ) d x · πf = U ooe ( f ) · πf ,U ∂x ooe ( f ) = − Z R u ( x ) cos(2 πf x ) sin(2 πf x ) cos(2 πf x ) d x · πf = − U eoe ( f ) · πf , ∂x eeo ( f ) = Z R u ( x ) sin(2 πf x ) cos(2 πf x ) sin(2 πf x ) d x · πf = U oeo ( f ) · πf ,U ∂x oeo ( f ) = − Z R u ( x ) cos(2 πf x ) cos(2 πf x ) sin(2 πf x ) d x · πf = − U eeo ( f ) · πf ,U ∂x eoo ( f ) = Z R u ( x ) sin(2 πf x ) sin(2 πf x ) sin(2 πf x ) d x · πf = U ooo ( f ) · πf ,U ∂x ooo ( f ) = − Z R u ( x ) cos(2 πf x ) sin(2 πf x ) sin(2 πf x ) d x · πf = − U eoo ( f ) · πf , where U = U eee − U oee e − U eoe e + U ooe e − U eeo e + U oeo e + U eoo e − U ooo e is a sum of eight components of different parity, as explained in Section 2. Hence U ∂x = (cid:0) U oee + U eee e − U ooe e − U eoe e − U oeo e − U eeo e + U ooo e + U eoo e (cid:1) · πf = (cid:0) U eee − U oee e + U eoe e − U ooe e + U eeo e − U oeo e + U eoo e − U ooo e (cid:1) · (2 πf e ) . Considering the parity of each component we get formula (3.12).Let us note that the statement of the above theorem is analogous to the claim of theclassic version of the Fourier transform of the derivative. The difference is first of all thekind of imaginary unit by which the Fourier transform is multiplied and the change of signat some variables. This is a characteristic feature of the octonion Fourier transformation.Similar argument leads to the following corollaries for the OFTs of partial derivatives ofhigher order.
Corollary 3.
Let U ∂x i ...x j denote the OFT of u x i ...x j . Then U ∂x x ( f , f , f ) = U ( f , − f , − f ) · (2 πf )(2 πf ) e ,U ∂x x ( f , f , f ) = U ( f , f , − f ) · (2 πf )(2 πf ) e ,U ∂x x ( f , f , f ) = U ( − f , f , − f ) · (2 πf )(2 πf ) e ,U ∂x x x ( f , f , f ) = U ( − f , f , − f ) · (2 πf )(2 πf )(2 πf ) e . An analogous conclusion can also be drawn for pure partial derivatives of the secondorder. It is worth noting that the claim of the theorem is no different from the corre-sponding theorem for the classic Fourier transform. We leave claims of Corollary 3 and 4without proof.
Corollary 4.
Let U ∂x i x i denote the OFT of u x i x i . Then U ∂x x ( f , f , f ) = − U ( f , f , f ) · (2 πf ) ,U ∂x x ( f , f , f ) = − U ( f , f , f ) · (2 πf ) ,U ∂x x ( f , f , f ) = − U ( f , f , f ) · (2 πf ) . Another significant result that can be demonstrated is the OFT differentiation the-orem. First of all, however, the concept of differentiation of octonion-valued functionshould be defined. We will say that the partial derivative v x i , i = 1 , , , of the function v : R → O , v = v + v e + . . . + v e , exists if and only if all the partial derivatives v j,x i , i = 1 , , , j = 0 , . . . , , exist and then: v x i ( x ) = v ,x i + v ,x i ( x ) e + . . . + v ,x i ( x ) e . We can now formulate the theorem on partial derivatives of the OFT U of u : R → R ,analogous to the theorem known from the classical Fourier analysis. As before, we willassume that all considered derivatives and transforms exist.21 heorem 15. Let V i and W i , where i = 1 , , , denote the OFTs of v i ( x ) = − πx i u ( x ) and w i ( x ) = v i ( x ) · e i − , i = 1 , , , respectively. Then U f ( f , f , f ) = V ( f , − f , − f ) · e = W ( f , f , f ) , (3.15) U f ( f , f , f ) = V ( f , f , − f ) · e = W ( − f , f , f ) , (3.16) U f ( f , f , f ) = V ( f , f , f ) · e = W ( − f , − f , f ) . (3.17) Proof.
We will apply methods analogous to those used in the proof of Theorem 14. Wewill prove only the first of the given formulas, the remaining ones are shown in the sameway. Consider function v ( x ) = − πx u ( x ) and notice that v eyz ( x ) = − πx u oyz ( x ) , y, z ∈ { e, o } ,v oyz ( x ) = − πx u eyz ( x ) , y, z ∈ { e, o } . As in the previous considerations, we can write U and V (the OFT of v ) as sums of eightcomponents of different parity U = U eee − U oee e − U eoe e + U ooe e − U eeo e + U oeo e + U eoo e − U ooo e ,V = V eee − V oee e − V eoe e + V ooe e − V eeo e + V oeo e + V eoo e − V ooo e , where U xyz , x, y, z ∈ { e, o } are defined as in (2.39)–(2.46) and V xyz analogously.Assuming that the functions u and v are integrable (in the Lebesgue sense), we candifferentiate under the integral sign and then we get U eee,f ( f ) = − Z R πx u eee ( x ) sin(2 πf x ) cos(2 πf x ) cos(2 πf x ) d x = V oee ( f ) ,U oee,f ( f ) = Z R πx u oee ( x ) cos(2 πf x ) cos(2 πf x ) cos(2 πf x ) d x = − V eee ( f ) , ... U eoo,f ( f ) = − Z R πx u eoo ( x ) sin(2 πf x ) sin(2 πf x ) sin(2 πf x ) d x = V ooo ( f ) ,U ooo,f ( f ) = Z R πx u ooo ( x ) sin(2 πf x ) sin(2 πf x ) sin(2 πf x ) d x = − V eoo ( f ) . Hence U f = V oee + V eee e − V ooe e − V eoe e − V oeo e − V eeo e + V ooo e + V eoo e = ( V eee − V oee e + V eoe e − V ooe e + V eeo e − V oeo e + V eoo e − V ooo e ) · e . Considering the parity of each component we get the first equality in formula (3.15). Usingthe fact that for any o ∈ O and α , α , α ∈ R we have (cid:0) (( o · e ) · e − e α ) · e − e α (cid:1) · e − e α = (cid:0) (( o · e − e α ) · e e α ) · e e α (cid:1) · e we get the second equality in (3.15), which concludes the proof.At the end of this section we go to one of the most important Fourier transformproperties and the most frequently used in signal analysis – the so-called Convolutiontheorem. This claim was already signaled in [3], here we present the details of the proof.22 heorem 16. Let F OFT { u ∗ v } denote the OFT of the convolution of u and v , i.e. ( u ∗ v )( x ) = Z R u ( y ) · v ( x − y ) d y . Then F OFT { u ∗ v } ( f ) = V ( f , f , f ) · ( U eee ( f ) − U eeo ( f ) e )+ V ( f , − f , − f ) · ( − U oee ( f ) e + U ooe ( f ) e )+ V ( f , f , − f ) · ( − U eoe ( f ) e + U oeo ( f ) e )+ V ( − f , f , − f ) · ( U eoo ( f ) e − U ooo ( f ) e ) , where U = U eee − U oee e − U eoe e + U ooe e − U eeo e + U oeo e + U eoo e − U ooo e is a sum of terms with different parity with relation to x , x , and x , as in (2.39) – (2.46) . This theorem is the generalization of results presented in [5] and [12]. Moreover, ifat least one of the functions u or v is even with respect to both x and x then theabovementioned formula reduces to the well-known form. However, one should bear inmind the fact that in general the above complicated form this claim is of little use. In thenext section we will provide the argument similar to one presented in [12] which will givemuch simpler formula. of Theorem 16. We will use the fact that for every α , α , α ∈ R we have (cid:0) ( e − e α ) · ( e − e α ) (cid:1) · ( e − e α ) = (cid:0) ( e − e α · e − e α ) · e − e α (cid:1) , (3.18) (cid:0) ( e − e α · e ) · ( e − e α ) (cid:1) · ( e − e α ) = (cid:0) ( e − e α · e e α ) · e e α (cid:1) · e , (3.19) (cid:0) ( e − e α ) · ( e − e α · e ) (cid:1) · ( e − e α ) = (cid:0) ( e − e α · e − e α ) · e e α (cid:1) · e , (3.20) (cid:0) ( e − e α · e ) · ( e − e α · e ) (cid:1) · ( e − e α ) = (cid:0) ( e − e α · e e α ) · e e α (cid:1) · e , (3.21) (cid:0) ( e − e α ) · ( e − e α ) (cid:1) · ( e − e α · e ) = (cid:0) ( e − e α · e − e α ) · e − e α (cid:1) · e , (3.22) (cid:0) ( e − e α · e ) · ( e − e α ) (cid:1) · ( e − e α · e ) = (cid:0) ( e − e α · e − e α ) · e e α (cid:1) · e , (3.23) (cid:0) ( e − e α ) · ( e − e α · e ) (cid:1) · ( e − e α · e ) = (cid:0) ( e e α · e − e α ) · e e α (cid:1) · e , (3.24) (cid:0) ( e − e α · e ) · ( e − e α · e ) (cid:1) · ( e − e α · e ) = (cid:0) ( e e α · e − e α ) · e e α (cid:1) · e . (3.25)From the definition of the OFT and the convolution, the Fubini theorem and usingintegration by substitution we have Z R (cid:18)Z R u ( y ) · v ( x − y ) d y (cid:19) · e − e πf x e − e πf x e − e πf x d x = Z R u ( y ) · (cid:18)Z R v ( x − y ) · e − e πf x e − e πf x e − e πf x d x (cid:19) d y = Z R u ( y ) · (cid:18)Z R v ( x ) · e − e πf ( x + y ) e − e πf ( x + y ) e − e πf ( x + y ) d x (cid:19) d y = ( ⋆ ) . Consider the inner integral. We can write the transformation kernel in the following way: e − e πf ( x + y ) e − e πf ( x + y ) e − e πf ( x + y ) = (cid:0) ( e − e α · cos( β )) · ( e − e α · cos( β )) (cid:1) · ( e − e α · cos( β )) (cid:0) ( e − e α · e sin( β )) · ( e − e α · cos( β )) (cid:1) · ( e − e α · cos( β )) − (cid:0) ( e − e α · cos( β )) · ( e − e α · e sin( β )) (cid:1) · ( e − e α · cos( β ))+ (cid:0) ( e − e α · e sin( β )) · ( e − e α · e sin( β )) (cid:1) · ( e − e α · cos( β )) − (cid:0) ( e − e α · cos( β )) · ( e − e α · cos( β )) (cid:1) · ( e − e α · e sin( β ))+ (cid:0) ( e − e α · e sin( β )) · ( e − e α · cos( β )) (cid:1) · ( e − e α · e sin( β ))+ (cid:0) ( e − e α · cos( β )) · ( e − e α · e sin( β )) (cid:1) · ( e − e α · e sin( β )) − (cid:0) ( e − e α · e sin( β )) · ( e − e α · e sin( β )) (cid:1) · ( e − e α · e sin( β )) , where α i = 2 πf i x i , β i = 2 πf i y i , i = 1 , , .Using equations (3.18)–(3.25) we get ( ⋆ ) = V ( f , f , f ) · U eee ( f ) − V ( f , − f , − f ) · U oee ( f ) e − V ( f , f , − f ) · U eoe ( f ) e + V ( f , − f , − f ) · U ooe ( f ) e − V ( f , f , f ) · U eeo ( f ) e + V ( f , f , − f ) · U oeo ( f ) e + V ( − f , f , − f ) · U eoo ( f ) e − V ( − f , f , − f ) · U ooo ( f ) e , which, after rearranging the terms, concludes the proof.Note that (due to the commutativity of convolution) the following formula is alsovalid: F OFT { u ∗ v } ( f ) = U ( f , f , f ) · ( V eee ( f ) − V eeo ( f ) e )+ U ( f , − f , − f ) · ( − V oee ( f ) e + V ooe ( f ) e )+ U ( f , f , − f ) · ( − V eoe ( f ) e + V oeo ( f ) e )+ U ( − f , f , − f ) · ( V eoo ( f ) e − V ooo ( f ) e ) , where V = V eee − V oee e − V eoe e + V ooe e − V eeo e + V oeo e + V eoo e − V ooo e is a sum of terms with different parity.At the end of this section, we will cite several other results that are important from thepoint of view of system analysis, i.e. octonion analogues of Parseval-Plancherel Theoremsfor real-valued functions, which we proved in [4]. Theorem 17.
Let u, v : R → R be square-integrable functions (in Lebesgue sense). Then ( u, v ) = ( U OFT , V
OFT ) , where ( f, g ) = Z R f ( x ) · g ∗ ( x ) d x denotes the classical scalar product of functions (real- or octonion-valued) of variables. In [2] we presented a detailed commentary on these assertions, including indicatingthe significance of the assumption in Theorem 17 that the considered functions are real-valued – for the octonion-valued functions the claim of Theorem 17 doesn’t hold. In caseof real-valued functions Theorem 18 (also known in classical theory as Rayleigh Theorem)is direct corollary of Theorem 17, but (as we proved in [2] and was shown independentlyin [23]) is valid also in the general case of octonion-valued functions.24 heorem 18. L -norm of any function u : R → O (square-integrable in Lebesgue sense)is equal to the L -norm of its octonion Fourier transform, i.e. k u k L ( R ) = k U OFT k L ( R ) , where k f k L ( R ) = (cid:0)R R | f ( x ) | d x (cid:1) / for any square-integrable function f : R → O . Of course, the above theorem shows that OFT preserves the energy of octonion-valuedfunctions. However, it is worth mentioning the recent result in [23], where the authorargues that OFT of octonion-valued function also satisfies the Hausdorff-Young inequality.
As we mentioned in the previous section, the formulas in the theorems on the OFTproperties are quite complicated and it seems that they can not be applied in practice. Inthis section we will show that using the notation of quadruple-complex numbers, we cansimplify these expressions.We will focus on using the OFT and notion of quadruple-complex numbers in theanalysis of 3-D linear time-invariant (LTI) systems of linear partial differential equations(PDEs) with constant coefficients. The classical Fourier transform is well recognized tool insolving linear PDEs with constant coefficients due to the fact, that it reduces differentialequations into algebraic equations [1]. It is true also in case of the quaternion Fouriertransform [12] and, as we will present in this section, the octonion Fourier transform. Wehave already signaled the possibility of this application in [3], here we will develop theseconsiderations and show additional examples.In Section 3 we derived formulas for the OFT of partial derivatives. We can nowreformulate (by straightforward computations) those formulas using the multiplication in F algebra. Corollary 5.
Let u : R → R and U = F OFT { u } . Then U ∂x ( f ) = U ( f ) ⊙ (2 πf ) e ,U ∂x ( f ) = U ( f ) ⊙ (2 πf ) e ,U ∂x x ( f ) = U ( f ) ⊙ (2 πf )(2 πf ) e ,U ∂x ( f ) = U ( f ) ⊙ (2 πf ) e ,U ∂x x ( f ) = U ( f ) ⊙ (2 πf )(2 πf ) e ,U ∂x x ( f ) = U ( f ) ⊙ (2 πf )(2 πf ) e ,U ∂x x x ( f ) = U ( f ) ⊙ (2 πf )(2 πf )(2 πf ) e . It is worth noting here the advantages that the above theorem on the partial derivativestransform brings. Let u : R → R be a function even with respect to each variable. Bothclassic and octonion Fourier transforms of u are real-valued functions. Using the classicalFourier transform for the function u x x we get − U ( f ) · (2 πf )(2 πf ) , and thus also the real-valued function. In a sense, we lose information that the function has been differentiatedat all. In turn, OFT of function u x x is equal to U ( f ) · (2 πf )(2 πf ) e , therefore it isa purely imaginary function (only the imaginary part standing next to the unit e will benon-zero). This clearly indicates differentiation with respect to variables x and x .25very linear partial differential equation with constant coefficients can be reduced toalgebraic equation (with respect to multiplication in F ). Note that in the case of secondorder equations in which there are no mixed derivatives, this is also true in the sense ofmultiplication of octonions, e.g. for a nonhomogeneous wave equation, i.e. u tt = u x x + u x x + f ( t, x , x ) we have (cid:0) (2 πf ) + (2 πf ) − (2 πτ ) (cid:1) · U ( τ, f , f ) = F ( τ, f , f ) , where U = F OFT { u } and F = F OFT { f } . But on the other hand, if we consider the heatequation, i.e. u t = u x x + u x x + f ( t, x , x ) , where we get (cid:0) (2 πf ) + (2 πf ) + (2 πτ ) e (cid:1) ⊙ U ( τ, f , f ) = F ( τ, f , f ) . An inverse (in sense of multiplication in F ) of (cid:0) (2 πf ) + (2 πf ) + (2 πτ ) e (cid:1) exists if andonly if ( τ, f , f ) = (0 , , and is equal to (cid:0) (2 πf ) + (2 πf ) + (2 πτ ) e (cid:1) − = (2 πf ) + (2 πf ) − (2 πτ ) e (cid:0) (2 πf ) + (2 πf ) (cid:1) + (2 πτ ) . Hence U ( τ, f , f ) = (2 πf ) + (2 πf ) − (2 πτ ) e (cid:0) (2 πf ) + (2 πf ) (cid:1) + (2 πτ ) ⊙ F ( τ, f , f ) . You can not get such a simple formula using multiplication in octonion algebra. Additionaltheoretical considerations regarding partial differential equations and the use of integraltransforms in Cayley-Dickson algebras can be found in [24].Moreover, the notion of quadruple-complex number multiplication can be used todescribe general linear time-invariant systems of three variables and to reduce parallel,cascade and feedback connections of linear systems into simple algebraic equations, as inclassical system theory.Consider a 3-D LTI system. We know from the classical signal and system theorythat it can be described by its impulse response h : R → R (sometimes called its Greenfunction) and then, given the input signal u : R → R , the output v : R → R of suchsystem is given by the formula: v ( x ) = Z R u ( y ) · h ( x − y ) d y = ( u ∗ h )( x ) . The output is the convolution of the input signal and the impulse response of the system,which can be schematicaly presented as in Fig. 1.
Corollary 6.
The dependence between the OFTs of an input u and an output v of the3-D LTI system with the impulse response h is given by V ( f , f , f ) = H OFT ( f , f , f ) · ( U eee ( f ) − U eeo ( f ) e )+ H OFT ( f , − f , − f ) · ( − U oee ( f ) e + U ooe ( f ) e )+ H OFT ( f , f , − f ) · ( − U eoe ( f ) e + U oeo ( f ) e )+ H OFT ( − f , f , − f ) · ( U eoo ( f ) e − U ooo ( f ) e ) , (4.1) where V = F OFT { v } , U = F OFT { u } and H OFT = F OFT { h } will be called the octonionfrequency response of the system. u ( x ) ✲ v ( x ) h ( x ) Figure 1: 3-D LTI system.
Corollary 7.
Formula (4.1) can be restated using the multiplication in F algebra. Wehave V ( f ) = H OFT ( f ) ⊙ U ( f ) which is the same as the very well known input-output relation in classic signal and systemtheory. Consider now the classical connections between the systems, i.e. cascade, parallel andfeedback connections. We start with the cascade connection (Fig. 3), for which we canwrite V ( f ) = H , OFT ( f ) ⊙ W ( f ) ,W ( f ) = H , OFT ( f ) ⊙ U ( f ) , where V = F OFT { v } , W = F OFT { w } , U = F OFT { u } , H , OFT = F OFT { h } and H , OFT = F OFT { h } . Since the multiplication in F is commutative and alternative,we obtain V ( f ) = H OFT ( f ) ⊙ U ( f ) , where H OFT ( f ) = H , OFT ( f ) ⊙ H , OFT ( f ) , as we have for classical (complex) Fourier transform. ✲ u ( x ) h ( x ) ✲ w ( x ) h ( x ) ✲ v ( x ) Figure 2: Cascade connection of 3-D LTI systems.In the case of parallel connection (Fig. 3) the computations are much simpler. We get V ( f ) = H OFT ( f ) ⊙ U ( f ) , where H OFT ( f ) = H , OFT ( f ) + H , OFT ( f ) , the same as in the classical theory. ✲ u ( x ) r ✲ ❄✲ ✻ ❤ ✲ v ( x ) h ( x ) h ( x ) ++ Figure 3: Parallel connection of 3-D LTI systems.27t gets more complicated if we consider the feedback connection, as in Fig.4. We canwrite the system of equations: V ( f ) = H , OFT ( f ) ⊙ W ( f ) ,W ( f ) = U ( f ) − H , OFT ( f ) ⊙ V ( f ) . Using the commutativity and associativity of ⊙ we get (1 + H , OFT ( f ) ⊙ H , OFT ( f )) ⊙ V ( f ) = H , OFT ( f ) ⊙ U ( f ) , which leads to V ( f ) = H OFT ( f ) ⊙ U ( f ) , where H OFT ( f ) = (1 + H , OFT ( f ) ⊙ H , OFT ( f )) − ⊙ H , OFT ( f ) , and the inverse is in sense of multiplication in F . The formula is very well-known, but weneed to remember that not every element of F has its inverse. Hence we can see that notevery 3-D LTI system can be described with the convolution formula and analyzed withthe OFT. ✲ u ( x ) ❤ ✲✻ + − w ( x ) ✲ r ✛ ✲ v ( x ) h ( x ) h ( x ) Figure 4: Feedback connection of 3-D LTI systems.
It has been shown that the theory of octonion Fourier transforms can be general-ized to the case of functions with values in higher-order algebras. Those transforms haveproperties that are similar to their classical (complex) counterparts. Octonion analoguesof scaling, modulation and shift theorems proved in Section 3 form the foundation ofoctonion-based signal and system theory. Properties of the octonion Fourier transform incontext of other signal-domain operations, i.e. derivation and convolution of real-valuedfunctions, show that it is a fine tool in the analysis of multidimensional LTI systems.It remains to study the discrete case, i.e. discrete-space octonion Fourier transform(DSOFT). Preliminary studies show that the notion of quadruple-complex numbers canbe applied to define the DSOFT and to analyze linear difference equations.
Acknowledgments
The research conducted by the author was supported by National Science Centre (Poland)grant No. 2016/23/N/ST7/00131. 28 eferences [1] R. L. Allen and D. Mills.
Signal Analysis: Time, Frequency, Scale, and Structure .Wiley-IEEE Press, 2003.[2] Ł. Błaszczyk. Octonion spectrum of 3d octonion-valued signals – properties andpossible applications. In
Proc. 2018 26th European Signal Processing Conference(EUSIPCO) , pages 509–513, 2018.[3] Ł. Błaszczyk. Hypercomplex fourier transforms in the analysis of multidimensionallinear time-invariant systems. In
Progress in Industrial Mathematics at ECMI 2018 .Springer Nature Switzerland AG, 2019. (in press).[4] Ł. Błaszczyk and K. M. Snopek. Octonion fourier transform of real-valued functionsof three variables – selected properties and examples.
Signal Process. , 136:29–37,2017.[5] T. Bülow.
Hypercomplex Spectral Signal Representations for the Processing and Anal-ysis of Images . PhD thesis, Institut für Informatik und Praktische Mathematik,Christian-Albrechts-Universität Kiel, 1999.[6] T. Bülow and G. Sommer. The hypercomplex signal – a novel extension of the analyticsignal to the multidimensional case.
IEEE Trans. Signal Process. , 49(11):2844–2852,2001.[7] M.-A. Delsuc. Spectral representation of 2d nmr spectra by hypercomplex numbers.
J. Magn. Reson. , 77:119–124, 1988.[8] S. Demir and M. Tanişli. Hyperbolic octonion formulation of the fluid maxwellequations.
J. Korean Phys. Soc. , 68(5):616–623, 2016.[9] L. E. Dickson. On quaternions and their generalization and the history of the eightsquare theorem.
Ann. Math. , 20(3):155–171, 1919.[10] J. Duoandikoetxea.
Fourier Analysis , volume 29 of
Graduate Studies in Mathematics .American Mathematical Society, 2001.[11] T. A. Ell.
Hypercomplex Spectral Transformations . PhD thesis, Univ. of Minnesota,Minneapolis, USA, 1992.[12] T. A. Ell. Quaternion-fourier transforms for analysis of 2-dimensional linear time-invariant partial-differential systems. In
Proc. 32nd IEEE Conf. on Decision andControll , volume 1–4, pages 1830–1841, 1993.[13] T. A. Ell, N. Le Bihan, and S. J. Sangwine.
Quaternion Fourier Transforms forSignal and Image Processing . Wiley-ISTE, 2014.[14] H.-Y. Gao and K.-M. Lam. From quaternion to octonion: Feature-based imagesaliency detection. In , pages 2808–2812, 2014.[15] N. Gomes, S. Hartmann, and U. Kähler. Compressed sensing for quaternionic signals.
Complex Anal. Oper. Theory , 11:2017, 417–455.[16] A. M. Grigoryan and S. S. Agaian.
Quaternion and Octonion Color Image Processingwith MATLAB . SPIE, 2018. 2917] S. L. Hahn and K. M. Snopek. The unified theory of n-dimensional complex andhypercomplex analytic signals.
Bull. Polish Ac. Sci., Tech. Sci. , 59(2):167–181, 2011.[18] S. L. Hahn and K. M. Snopek.
Complex and Hypercomplex Analytic Signals: Theoryand Applications . Artech House, 2016.[19] P. Klco, M. Smetana, M. Kollarik, and M. Tatar. Application of octonions in thecough sounds classification.
Advances in Applied Science Research , 8(2):30–37, 2017.[20] K. Kurman. Liczby podwÃşjne zespolone i możliwość ich zastosowania. Technicalreport, Politechnika Warszawska, Katedra Automatyki i Telemechaniki, 1958.[21] S. Lazendić, H. De Bie, and A. Pižurica. Octonion sparse representation for colorand multispectral image processing. In
Proc. 2018 26th European Signal ProcessingConference (EUSIPCO) , pages 608–612, 2018.[22] S. Lazendić, A. Pižurica, and H. De Bie. Hypercomplex algebras for dictionarylearning. In
Proc. The 7th Conference on Applied Geometric Algebras in ComputerScience and Engineering–AGACSE 2018 , pages 57–64, 2018.[23] P. Lian. The octonionic fourier transform: Uncertainty relations and convolution.
Signal Processing , 164:295–300, 2019.[24] S. Ludkovsky.
Analysis over Cayley-Dickson numbers and its applications . LAPLAMBERT Academic Publishing, 2010.[25] C. A. Popa. Octonion-valued neural networks. In
Artificial Neural Networks andMachine Learning âĂŞ ICANN 2016 , pages 435–443, 2016.[26] C. A. Popa. Global exponential stability of octonion-valued neural networks withleakage delay and mixed delays.
Neural Networks , 105:277–293, 2018.[27] L. Rodman.
Topics in Quaternion Algebra . Princeton University Press, 2014.[28] H. Sheng, X. Shen, Y. Lyu, and Z. Shi. Image splicing detection based on markovfeatures in discrete octonion cosine transform domain.
IET Image Processing ,12(10):1815–1823, 2018.[29] K. M. Snopek. New hypercomplex analytic signals and fourier transforms in cayley-dickson algebras.
Electronics and Telecommunications Quaterly , 55(3):403–415, 2009.[30] K. M. Snopek. The n-d analytic signals and fourier spectra in complex and hy-percomplex domains. In
Proc. 34th Int. Conf. on Telecommunications and SignalProcessing, Budapest , pages 423–427, 2011.[31] K. M. Snopek. The study of properties of n-d analytic signals in complex and hy-percomplex domains.
Radioengineering , 21(2):29–36, 2012.[32] K. M. Snopek. Quaternions and octonions in signal processing – fundamentals andsome new results.
Telecommunication Review + Telecommunication News, Tele-Radio-Electronic, Information Technology , 6:618–622, 2015.[33] R. Wang, G. Xiang, and F. Zhang. L1-norm minimization for octonion signals. , pages552–556, 2017. 3034] J. Wu, L. Xu, Y. Kong, L. Senhadji, and H. Shu. Deep octonion networks.