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32Системный анализ System analysis
УДК 517.443 DOI 10.29141/2949-477X-2023-2-3-1 EDN MLQGCK
V. G. Labunets, V. P. Chasovskikh, E. N. Starikov
Ural State University of Economics, Ekaterinburg, Russia
New many-parameter Fourier-Clifford transforms
Abstract. The article shows how ordinary complex-valued Fourier transforms are extended to Cliffordean-valued many-parameter Fourier transforms (MPFCTs). Each MPFCT depends on finite set of independent parameters (angles), which could be changed independently one from another. When parameters are changed, MPFCT is also changed taking form of a set of known and unknown orthogonal transforms. Development of MPFCTs includes operator exponential representations, based on all parameterized imaginary units' square roots of minus one in Clifford algebra.
Key words: Fourier-Clifford transforms; many-parameter transforms; fast algorithms. Acknowledgments: This work was supported by the RFBR grant 19-29-09022\1.9. Paper submitted: July 23, 2023.
For citation: Labunets V. G., Chasovskikh V. P., Starikov E. V. New many-parameter Fourier-Clifford transforms. Digital models and solutions. 2023. Vol. 2, no. 3. Pp. 5-22. DOI: 10.29141/2949-477X-2023-2-3-1. EDN: MLQGCK.
В. Г. Лабунец1, В. П. Часовских1, Е. Н. Стариков1
'Уральский государственный экономический университет, г. Екатеринбург, Российская Федерация
Новые многопараметрические преобразования
Фурье - Клиффорда
Аннотация. В статье показано, как обычные комплекснозначные преобразования Фурье расширяются до клиффордовозначных многопараметрических преобразований Фурье (МПФКТ). Каждый МПФКТ зависит от конечного множества независимых параметров (углов), которые могут изменяться независимо друг от друга. При изменении параметров МПФКТ также изменяется в виде набора известных и неизвестных ортогональных преобразований. Разработка МПФКТ включает операторные экспоненциальные представления, основанные на квадратных корнях всех параметризованных мнимых единиц минус один в алгебре Клиффорда.
Системный анализ System analysis
Ключевые слова: преобразования Фурье - Клиффорда; многопараметрические преобразования; быстрый алгоритм.
Финансирование: Работа выполнена при поддержке гранта РФФИ 19-29-09022\1.9. Дата поступления статьи: 23 июля 2023 г.
Для цитирования: Labunets V. G., Chasovskikh V. P., Starikov E. V. Nonlinearizad of fast Fourier transform // Цифровые модели и решения. 2023. Т. 2, № 3. С. 5-22. DOI: 10.29141/2949-477X-2023-2-3-1. EDN: MLQGCK.
Introduction
In this paper, we present a unified view of many-parameter Fourier-Clifford transforms. Development of many-parameter Fourier-Clifford transforms includes operator exponential representations, based on all parameterized imaginary units square roots of V(-1) : i = i(0) =
= i ($1, $2, ..., $q) @ V(=1) e Alggpqr (RjJi, J2, ..., vector spaces R with n hyper imaginary units Ji, J2, of parameters (angles of cosines) of a pseudo-Riemannian sphere PRSMqV (+1) or a pseudo-hy
Jn) in Clifford algebras Algp,q,r over real ..., Jn, where 0 = (фь Ф2, ..., фq) is a vector
M-1
perbolic sphere PRS^ (-1).
Basically, the imaginary unit i = e C in the ordinary Fourier transformation kernel
«ï^n ___
eN = cos
2k
N
kn
+ isin
2 71
N
kn
is replaced by a /(0) = V-T G Algpq
q,r •
8^ = COS
^rrkn I + i(0)sin N 1
r 2%. Л Cn ,
\ 1У у
(T)
When the vector of parameters 0 = ($1, $2, ..., $q) runs completely the pseudo-Riemannian sphere, the ensemble of the Fourier-Clifford transforms is creating.
Usually those basic hyperimaginary units i = Js, s = 1, 2, ..., n for which Jj = -1 are used as V-1.
This produces a host of discrete Fourier transforms (see the historical sketch overview in [1]). However, in the algebra of quaternions H[R] = Algp,0,0 (RJ1, J2), in addition to basic hyper-imaginary units J1, J2, J3 = J1 ° J2, all pure vector quaternions J + J2 + J with unit norm (x2 + y2 + z2 = 1) are imaginary units, since i2 = (J + yJ2 + XJ3)2 = -1. They all lie on the classical unit sphere in 3D space and are parameterized by the Euler angles i = i (a, 9, 0). The new class Fourier-Hamilton transforms was proposed in [2-4], where each discrete harmonic has a classical imaginary unit i = e C is replaced by a i(a, 9, 0):
■"N
"N
COS
2ji
— fo2| + i'(a*,q>*,et)sin
Ai "N
2n j
JÏ ,
Системный анализ System analysis
where ik = i (ak, 9k,6k), k = 0, 1, ..., # - 1. As result we obtain wide family of quaternion man-yparameter Fourier-Hamilton transforms.
The paper is organized as follows. In Section 2 we present a brief introduction to the Clifford algebra and we go on to demonstrate how many-parameter imaginary units can be construct for design of many-parameter Fourier-Clifford transforms. Then, in Section 3, we present a new many-parameter ordinary-like and fraction-like Fourier-Clifford transforms.
Clifford Algebras
This section briefly describes the Clifford algebra. Clifford algebra Alggf^r(R) is an extension of real, complex numbers, and quaternions to higher dimensions, and has basis elements. The subscriptsp, q, r (such asp + q + r = n) determine signatures of the Clifford algebra. There are several ways to understand Alg2pqqr (R) from the very abstract to the very concrete. In [5], one can find five different (equivalent) definitions of Clifford algebra. Now let us discuss one definition of the Clifford algebra Alggpqq,r(R) with fixed basis that is useful for different applications.
Basic Concepts and Notation
Let {Ji}"i=\ be a set of n hyperimaginary units. We assume 1) J2 = n- = -1 for 1 < i < q; 2) Ji = n+ = +1 for q + 1 < i < q + p; 3) Ji = n0 = 0 for q + p + 1 < i < q + p + r = n and 4) J Jk = = -Jk Ji (k ^ i). Three integers (q, p, r) are the algebraic signature of {Ji\i = 1, 2, ..., n}. Let by definition Ji : = -Ji, then
1) IIJill2 = -J = -Jf = -n- = +1, 1 < i < q,
2) IIJil2 J Ji = -Jf = -n+ = -1, q + 1 < i < q + p,
3) IIJil2 = Ji Ji = -Jf = n0 = 0, q + p + 1 < i < q + p + r = n.
In mathematics and theoretical physics, a pseudo-Euclidean space Enp,q,r (R) is a n-D space spanned on the orthonormal basis of n hyperimaginary orts
(where q + p + r = n) together with a quadratic form
i=p+1 i=p+q+1 /=1 /=/7+1
which is called the magnitude of the vector x. Here smet = (p, q, r) is called the metric signature of Enp,q,r (R). It shows how many +1, -1 and 0, respectively, are containing in the sum
" i|2
¿=1
Obviously, (q, p, r)met = (p, q, r)aig. The pseudo-Euclidean space Enp,q,r (R) spanned on the orthonormal basis {J,- ei}"=1 we call «small» (or «parents») n-D space and units {Ji}1=1 we call the parents units.
Системный анализ System analysis
Let b = (bi, b2, ..., bn) e B2 be an arbitrary n-bit vector, where bi £ B2 = {0,1} and is the n-D Boolean. Let Wt(b) = bi + b2 + ... + bn and Wt2(b) = Wt(b) mod 2 be Z - and GF(2) -valued weights of b, respectively.
Let us introduce the following 2n children units Jb : = Jib1 J|2... Jjibn. If Wt(b) = k then say that a children unit Jb : = Jf1 ... Jbn has weight k, i. e. Wt(Jb) = Wt(b) = k. For each algebraic parents signature sag = (q, p, r) we construct «big» 2n-D hypercomplex children space ep q r (R), spanned on 2n children units Jb, that form a basis of 2n-D space:
2"
(P,Q,R)
(R):=\x
^ =Z Vb =Ê Z Vbi = Sffi(Ei,(R)} (2)
beBj k=0 w(b)=k J *=1
with a children metric signature Smet = (P, Q, R) induced by parents algebraic signature (q, p, r), where
EpL(R)) := Spanwt(b)=t{Jb} = \(x)k (x)t = £ xbJb
l te*! J
w(b)=i
and X® is the symbol of direct vector sum. Obviously, E(p q r) (R) contains Enp q r (R) as subspace: Ep,q,r (R) c Ep, qr (R) and Ep,q,r (R) = (Ep" qr (R)').'
2n
Now we are going to transform 2n-D space Ep, q^r (R) into 2n-D hypercomplex algebra Algpnq,r(R) = (E(p q r) (R), o), defining a multiplication operation for elements
x = Z *»'Jb
beB?
2n
in space E(p q r) (R). It is easy to check that for children units we have the following multiplication rule:
Jb 0 Jc = (_1)(c|K|ft) nbnCn nbn-\cn-\ ... n22c2 nb1C1 Jb®c = [(-1)(b|-R|c)J n6c] Jb®
(3)
where
R:=
1 . 1 1
1111
Obviously, Jc o Jb = (-1)(c|R|b) ncb Jc®b). Hence,
Jb o Jc = [(-1)(b|«t+«|c)j Jc o Jb = [(—1)(b\E+Ic)(_1)(b|c)] Jc o Jb = [(-1)Wt2(c)Wt2(b)®(c|b)] Jc o Jb, (4)
where E + I = Rt + R + I are full unit matrix. From (4) we obtain (Jb)2 = (-1)(b|R|b) nb. Using (4),
If *=E*bJb, 7=2><jc
we define multiplication rule for arbitrary two vectors of
bEB; ceB"
are two elements then their product is
c
Системный анализ System analysis
z=xy-
VbeB2 У
vceB" y
Zvb 2>CJC - ZXv/J
ybeB; ceB"
where
Z Z Н^^Че.^ = Z
у de В2 cgB"
deBo
(5)
is the called Clifford convolution of x and y. Therefore, the space E(p q r) (R) forms 2n-D algebra Alg2p^r = (E(p q r) (R), o) which is called the Clifford algebras with algebraic signatures Salg = (p, q, r).
So, each Clifford algebra Algp,q,r = (E(p q r) (R), o) is a non-commutative associative algebra over R, which according to (2) naturally forms a graded linear space of dimension
2": = (Algi^iAlg^)®...®(Algg^).
Definition 1. Numbers of the form x = Z^v'1' are cliffordeons, where
1) J0 = 1 is the real unit; bsB*
2) Jb, b ф 0 are 2n - 1 hyperimaginary units. The addition and subtraction of two cliffordeons
* =Z*bjb and
ЬеВ?
beBÏ
are given by
x± y=z vb ± Z Уъ?= Z к Jb
beB?
beB?
Ьев;
So, a Clifford number x is a hypercomplex number over R, with 2n - 1 imaginary units, of the form
= Zvb=Z Z vb=ZK=*o+<*L„,
k=0 Wt(b)=£
ЬеВЗ
/t=0
where xi^n : = ((x)i + (x)2 + ... + (x)„) with (x)^ = Z the к - grade projector. Number component xo, direction component w(b)=i:
{x\= ZV" and(x)^ = %хь?={х\+(х\+...+{х)я. w(bH wt(b)>i
We recalled the real (or scalar), n-D vector and hyperimaginary (2n - 1)-D multivector) parts of cliffordeon, respectively. Now these components are denoted as Re(x) = Sc(x) = xo, Vec(x)= (x)i and Im(x) = Mvec(x) = (x)i + (x)2 + ... + (x)n. Sometimes the scalar part of a Clifford number Algpqqr is called the trace of and denoted as Tr(x) : = Tr({x))o = Tr(xo Io) = xo. Hence, every x is the sum of a real part (scalar number) and a hyperimaginary part (pure multivector) of a cliffordeon: x = xo + {x)i^n = Sc(x) + Mvec(x) = Re(x) + Im(x).
Системный анализ System analysis
Imaginary Units in Alg2"
p,q,r
In this subsection we are interesting in x2: x2=
( л 2>bJb r \ = XZVeJ'J' = Z^b(Jb)2 +
VbeB2 ) i^CEBj ) vbeB; се в; , се В;
+
x Xxbxcjbj<+x XW.H = X H)<b%^+
^b < С b > С ) beB!|
EEwbJe+11 wcJb] = I i-^W+Ц vc(JbJc+Jc4 (6)
V Ь < с
b < С
ьев;
< ь с
From (6) we see that if for some values b, c e B2 hyperimaginary units Jb, Jc anticommute Jc o Jb = - Jb o Jc, then the corresponding terms in the last double sum are zero. This is possible if (-i)(fe|«i+«|c>= -i, since JcJb = (-1)(b|fit+fi|c> JbJc.
To zero out the entire double sum, all hyperimaginary units included in (6) must anticommute.
Example 1. By definition the set of all parent hyperimaginary units is one B2
Bn2 = {b1,...,bM | wt(b1) = 1} since Ji Jk = -Jk Ji, (Vk, i = 1, 2, ..., M) & (k + i).
Let IB2 c B2 be a set of total anticommutating units B2 : = {b1,...,bM | Jb o Jbk = - Jbk o Jbi}, Vk, i = 1, 2, ...,M. Let the manifold
=2>„Jb
ьев;
be spanned on {Jb | b eB2}. Obviously, dim{Alg2"^} = M = | B2|. We see that if x e Alg2" is spanned on
{jb|beB ¡}, Le., x = £xbJb,
q,r
Ьев;
then x2 G R is a real number:
wt(bXwt(b>-i) ? = £ = 2
beBÎ beBÎ
nbx2bER.
Let (x G Alg2par) & ( xx2 ф 0). Then for
x = x = У Хъ Jb
6 >Ш Ä VîfF
we have and
xex = xe/(xe)z= xe/s =
\-xe,(xe)2=s = -l, I xe,(xe)2 = s = +l,
Definition 2. Cliffordeons for ( xe)2 = -1, and ( xe)2 = 1 are called elliptic, parabolic and hyperbolic imaginary units and denoted as i, e, d, respectively.
Системный анализ System analysis
Conjugation and Norm
For our purposes involution that exist on each Clifford algebra is interesting. Definition 3. The following operation in the Clifford algebra Algp^qr
y ■.= (-1)* J?, (41)+ := (-1)*2 ^,.... {Jb: y ■.= (-1)'" Jb:, (j y = (■■= (W = (-i)A" Jb; - ^(-1)" J? =
=[(-i)wt(b)]j„A" ...j^J^ = [(-i)(b|b>(-i)<b|fi|b>] J*2 • • • jhn- = [(-i)<b|A|b)] jb =
n n ZZVA П П
Jb = НГ' Jb =
' wt(bXwt(b)+i)' (-1) »
" к(к->1)
Jb = (-1) »
Jb,
where A = R + I, k = Wt(b), is called the Hermite-Clifford conjugation, which maps every Clifford number x = ^b Jb to the number
Wt(b)(Wt(b)+l)
= Vo+Zb^H)(b|A|b>Jb = Vo + Ib^(-l) 2 Jb-
(7)
The operation of Hermite-Clifford conjugation corresponds to the operation of complex conjugation of complex numbers in the case C and quaternion conjugation in the case H. The notion of conjugation allows to defines two maps
1) Alg^r - Alglv, Nci (x) : = xxt, N¿1 (x) : = xtx,
2) Alg-f^r - R, Nr(x) : = Tr(<xxt)o) = Tr(Xx)o) = Tr« x Nh (x))o)= Tr(<x N¿1 (x))o)
called right or left Algp,q,r-valued and ^-valued norms. From (J) = (-1)<blAb> Jb we obtain
J2 = (J)t Jb = (-1)<b|A|b> Jb Jb = (-1)<b|b> nb = (— 1)Wt2(b) nb
(8)
For x о x we have
Nrc(x):=x oxf ZV Z^c(Jc)t= XXVeAJ')'
vbeBj J ^ ceBj J ^ beB^ ceBj
= z ^jW+iz Z WW+Z Z-*л W
ceB^ Vb<C Ь>С у
= z (-i)wt2(b) л\2+iz ZvcJb(JC)t+Z Z *«*„ Jc(Jb)f
ЬеВ?
V b < С
b < С
For this reason,
Nr(x) := Tr(x oxt)=£ (-l)wt>(b)i,Vb.
Системный анализ System analysis
Definition 4. We shall say that an element hx e Algp,q,r is antihermitean if V = - hx. Hence, for antihermitean element
M =-<**), .^O.u.^t-lflj^-^J'
b*0 b*0
and all xb = 0, for which (-1)(b|A|b> = 1, i. e., for (b|A|b> = 0. Hence,
= X xbJb e ~hAlg^).
1—i
beBj (b*0) &((Ь|Д|Ь)=1)
Definition 5. An element hx Algpqs is called hermitean if hxi= hx. Therefore, for hermitean cliffordeon its scalar part xo is nonzero. From,
we see, that xb = 0, for which (-1)(b|A|b> = -1, i. e., for (b|A|b> = 1. Hence, hx = x0 + (hx)^=x0 + £ xbrhehAlg™'r(R).
bEB^ (b*0)
&«b|A|b)=0)
For 'hxe'hAlglp/.= "Algl^Alg2^ and "xe kAlg^:= "Alg^OAIg^ we have
NcC~x) = NrCx) = -C~x)2= I (-l)wt2(b)riVbeR,
bEB; (b*0) &C<b|A|b>=l)
Nc(hx) = NR(hx) = +(h~xf= X (-l)Wt2(b)tlb^beR-
beBj (b*0) &({Ь|Д|Ь)=0)
So, for hermitean and antihermitean elements
Cxf = -NRêx) = -~hx 0^=- £ (-l)w,î(b)tib^R,
ЬеВ" (b#0) &«Ь|А|Ь>=1)
("xf = + NR(hx) = X (-l)wt'(b)iib^R.
beBj (b*0) &((Ь|Д|Ь)=0)
Geometry of Imaginary Units Sets
Let the manifold
=Evb
ьев;
be spanned on {Jb | b e B2}. Obviously, dim{Alg"^} = M = | B2|. Manifold
Ä1sl,P,ry=\*
=Zvb
x
ьев;
Системный анализ System analysis
has a linear cone of null cliffordions (see Fig. 1) given by
Cone {J IgPqrr} : = (X \Nr (x) = x o ~ = 0}. The null cone separates two open sets, for which Nr(x) > 0 and Nr(x) <0.
Fig. 1. Cone{Algp,q,o}
We define the sets of "positive" and "negative" cliffordions,
Pos{Algfqr] : = {x eAlglqr | Nr(X) > 0}, Neg{Alg$qrr} : = {x eAlg2^ | Nr(X) <0}.
Obviously, Pos{Algp,q,r} and Neg{Algp,q,r} are pseudo-Euclidean spaces. We can normalize numbers x eAlgp,q,r.
IIf x o X ^ 0, then cliffordeon xfl = x / || x ||r will has the following norm
Nr(xs) = ~xa»\ = = e{-l,0,l},
II ^ IIr I X o XI
where II x||R:=| VII x ||R. Hence, N^xJ = "xn ° = -(*xe)2e {-1,0,1}, NR(hx) = hxa. hxa = = +(Axe)2e {-1,0,1}. Whereas Euclidean space has a unit sphere, pseudo-Euclidean spaces have the hypersurfaces
?>1)-={x^ Pos^/g^} I Nr(x) = +l},
prsm-I
PHC
' (-1) : = j хд e Neg| Alg^qpr^ \ NK{x) = -1
of squared radiuses +1 and -1, respectively, whereM = | B2| = dim{Algp,q,r}. Such a hypersur-faces, called a quasi-spheres or pseudo-Riemannian sphere PRSMq1,,-(+1) c Pos{Algp,q,r} (see Fig. 2) and pseudo-hyperbolic sphere PRSM,-q,r(-1) c Neg{Algp,q,r} (see Fig. 3). We need in ( x)2 = i = -1. It is possible in two cases
{(hi)2 = -l) & = ~hi o*7 = +l)« (hi e FRSj£(+l)), [Cif =-l) &(nr(\) = hi °hi=-l)<^ (hi e PHS^(-l)).
Системный анализ System analysis
Fig. 2. HS2,1(-1) sits lies Cone{E2,1(R)} and RS2,1(+1) lies outside one
Fig. 3. HS2,1(-1) lies inside {e1,2(R)} and RS1,2(+1) lies outside one
Example 2. By definition the set of all parent hyperimaginary units is one of B2: B2 = {b1,...,bM | WT2(bk) = 1, k= 1, 2, ..., M}, since Ji Jk = -Jk J, (Vk, i = 1, 2, ..., M) & (k ^ i). Let
^¿,0 10 = &/)&(b e'bOUe^CR). I bEB? J
Hence, for all x e Enpqr (R) we have
n v
I Jk
k=1 Vll * IIh
and || xfl ||2 =|| xfl ||2 = x2 e {-1,0,1}, i. e., xfl e Cone{Algpn^} or xfl e PRSp",q1(+1), or
хд e PRSp"q(-1). So, all pseudo-Riemannian spheres {PRSp"q(+1)}"=0 are diffeomorphic to
n-1/ _____________________ Гтто n-1, , пл
{Eq(R) x S"-1-q}q:10 : diff{PRSp">q1(+1)} = Eq(R) x Sn-1-q, q = 0, 1, ..., n-1
iq=0 : U111\ i^pqy
|n
and the pseudo-hyperbolic spaces {PRSp"q(-1)}q=1 are homeomorphic to
^ "-i
Spq
Sq-1 x En-q(R) : diff{PRSpn,q1(-1)} = Sq-1 x En-q(R), q = 1, 2, ..., n,
where Sm is the (Euclidian) unit sphere of dimension m. PRS^o^+l) « S"-1 is just the standard sphere in Euclidean space E",o(R) = E"(R). By convention E0 is a single point; by definition S0 consists of two points. So,
PRS^+l) « S"-1-9 x Eq(R), PRS;,q1(-1) « S9"1 x E"-9(R)
Системный анализ System analysis
and
PRSpT^-l) - PRS^pn-q (+1) - Sq-1 x En-q(R) for q = 1, 2, ..., n.
~ " x
In particular, if X = x e Eo,n,o(R) we have дсд * Jke S""1 (see Fig. 4).
k=1
Fig. 4. Each n-D vector Хд e Sn 1 с Eo,n,o(R) = En(R)
Discrete Fourier Clifford Transforms
Quaternions and Clifford hypercomplex number were first simultaneously and independently applied to quaternion-valued Fourier and Clifford-valued Fourier transforms by Labunets [6] and Sommen [7; 8], respectively, at the 1981. The Labunets quaternion transforms were over quaternion with real and Galois coefficients (i. e., over H[R] and H[GF(p)]). They generalize both classical and co-called number theoretical transforms (NNTs) and proposed for application to fast signal processing. Ernst et al. [9] and Delsuc [10] in the late 1981s, seemingly without knowledge of the earlier works of Labunets and Sommen proposed bicomplex Fourier transforms over 4D commutative hypercomplex algebra of bicomplex numbers (C 0C).
Note that the bicomplex algebra is quite different from the quaternion algebra; among general things, bicomplex multiplication is commutative, but quaternion one is noncommutative. For this reason, the Ernst and Delsuc transforms are direct sum of ordinary Fourier transforms (i. e., duplex Fourier transform). They are a little bit similar in kind to quaternion Fourier transforms. Ernst and Delsuc's transforms were two-dimensional and proposed for application to nuclear magnetic resonance imaging. Two new ideas emerged in 1998-1999 in a paper by Labunets et al. [11] and Sangwine & Ell [12]. These were, firstly, the choice of a general root i = i(y, 0) of -1 (a unit quaternion with zero scalar part) rather than a basis unit (i, j or k) of the quaternion algebra, and secondly, the choice of a general roots io = io (yo, 0o), i'1 = i'1 (71, 01), ..., iN-1 = iN-1 (yam, 0N-1) of -1 (see cloud of imaginary units on Fig. 4) in Clifford algebra to create multi-parameter and fractional Fourier-Clifford transforms (with eigenvalues e-io(Yo'0o),
g-i\(yu 01) e—N-1 (YN-1, 0N-1) )
Labunets, Rundblad-Ostheimer and Astola [13-15] used the classical and number theoretical quaternion Fourier and Fourier-Clifford transforms for fast invariant recognition of 2-D, 3-D and n-D color and hyperspectral images, defined on Euclidean and non-Euclidean spaces. These publications give useful interpretation of quaternion and Fourier-Clifford coefficients: they are relative quaternion- and Clifford-valued invariants of hyperspectral images with re-
Системный анализ System analysis
spect to Euclidean and non-Euclidean rotations and motions of physical and hyperspectral spaces. It removes the veil of mysticism and mystery from quaternion- and Clifford-valued Fourier coefficients. In the works of scientists F. Brackx, H. De Schepper, F. Sommen, and H. De Bie [1; 16-18] mathematical theory of Fourier-Clifford transforms accepted the final completeness, beauty and elegance.
The main subject of this subsection is the exponential map
2 m 00 m
exp(jt) = 1 + x + — + .... + —-+ ... = Y—-.
PW 2! m! to ml
Clearly for x = a e R, exp (x) = ea is the usual real exponent map on R.
Theorem 1. If x = X is a non-zero element inAlgpn,r then exp (aXe) = cos (a) Xe sin (a),
ip,q,r ^лр ^ wj ^ ле
_ " \2k — ck nnA X2k+
Therefore
Proof: Since (xe)2 = s then (xe)2k = sk and (xe)2k+1 = sk xe.
(ахеГ ^ „2к , ~ V1 S (X2k+1 —
m._0 ml f02kl 'f0(2k + l)l
- COS(a) + xc SIN (a) =
cos(a) + i • sin(a), xe = i, cosg(a) + e-sing(a), xe = e, cosh(a) + d • sinh(a), xe = d,
where cosg (a), sing (a) are the Galilean cosine and sine, respectively.
Theorem 2. For x = a + x e exp (ax) = exp (a(a +xe)) = eaa exp (axxe) = eaa (cos (a) + eaa sin (a)).
In general case exp (0x1) o exp (Px2) ^ exp (Px2) 0 exp (ax1)), exp (0x1) o exp (ax2) ^ ^ exp (a^2) o exp (ax1)). However, exp (axxe) o exp (pxe) = exp ((a + P)xe).
Discrete Fourier-Clifford Transforms
According to Theorem 1, for non-zero a e R and for a set {i (0k) | k = 0, 1, 2, ..., N - 1} exp (i (0k) a) = exp (i (0k) a) = cos a + i (0k) sin a.
For discrete values a = ak = 2nk / N (k = 0, 1, 2, ..., N - 1) we obtain Clifford-valued discrete harmonics
= = cos^fcij + i'(0Jsin (fknj
where each harmonic e£" = exp (-2ni (0k) kn / N) has its own imaginary unit i (0k) | k = 0, 1, 2, ... , N - 1. Due to the non-commutative property of the Clifford multiplication, there are two different types of Fourier-Clifford transforms (FCTs). These FCTs are the left- and right-sided FCTs (LS-FCT and RS-FCT), respectively
Definition 6. The direct discrete Fourier-Clifford transforms of a function f(n) : [0, N - 1] ^ ^ Algpqq,r are defined as
CF(*|9j = --Ug e^o f(„) = CF«*.....e->{f(«)},
Системный анализ System analysis
"-1 -4(e A
FC(*|e,) = " = FC(91'02.....
where CF(01,.. ,0^-1), FC(e'.....0«-1) are LS-FCT and RS-FCT.
Definition 7. The inverse quaternion Fourier-Clifford transforms are defined as
w-^JpWM-wSfcg-^.....
We see, that CF(e1, e2,...,eN-0, FC(e1, fe-.^jv-O and depend on N sets of parameters.
Fractional and Many-Parameter Fourier-Clifford Transforms
If Clifford harmonics have equivalent imaginary units ik (ek) = i (e), Vk = k = 0, 1, ... , N - 1, then Fourier-Clifford matrices contains commutative entries
-m2*kn
For this reason, CF and FC have the same real-valued eigenfunctionas ordinary DFT but with Clifford-valued eigenvalues {e2lw(e)s/4)}S=0 = {is(9)}S=0 [19-23]. Therefore,
-i(0)2„> о iV
ЛГ-1
s=0
2it-i(e)i/4
\Ш)(Ш\
N-1
Si'(e)|Ä,(*))<Ä,(n)|
«=0
Hence, we can define fractional and many-fractional Fourier-Clifford transforms. Definition 8. For single parameter a G Tor2n we introduce fractional Fourier-Clifford transforms (FrFCT) with classical and Bargmann (s(m) = m) structures as
(0i,e2.....вдчЭт^а _
m=0
= U- Diag(eHe»)s(m)a} IT1.
Definition 9. For N parameters (ao, ... , aN-1) G Tor2n we introduce many-fractional Fourier-Clifford transforms (MFFCT) with classical and Bargmann structures as
(Mî.....8»-i)T3" _
JV—1
m=0
= U-Diag(ei{e»')i('">a™)-U-1.
Due to the non-commutative property of quaternion multiplication, there are left- and right-sided transforms (LS-FrFCTs, LS-MPFCTs and RS-FrFCTs, RS-MPFCTs).
Definition 10. The direct discrete LS- and RS-FrFCTs, LS-and RS-MPFCTs of f(n) : [0, N -- 1] ^ Algp,q,r are defined as
I (в,
JV-l
.....e^>CFe(*)) = (в"02.....e-)CFa|f(«))= ^\hm(k)) (ei(K)s(m)a o(hm(n)\f(n))),
m=0
(вл.....^FC"*») = (0Л.....9h)FC" |f(«)) =£|hm{k))({hm{n) I f(«)> « em")s(m)a),
n=0
Системный анализ System analysis
n-1
|(0i
•02.....0ff_l)CFa(£)) = (e"02.....e-}CFa|f(H)) = (e^-W-)^ .(hm(n)\f(и))),
m=0
= (вл....."-^C" |f(«))= Y\hm(k)){{hm(n) I f(«))
Fast classical discrete Fourier transform is the following iteration procedure
F = n([v, ® (V e a2„_, (s-1))] • [v, ® f2 9 /2„j),
r=l
where
(10)
A2»-<sH = Diag*- (1, s2""'1, s2-1-2), ... , s^2"-1)) = = Diag2«-r (1 ei2n2r-1-1/N e;2n2r-1-2/# g!2n2r-1 -(2n-r- 1)/N) s = g/2n/N
Our fast many-parameter Fourier-Clifford transform is based on (10) and Clifford-valued phasors:
Diag2- (1, s2"-r'1/N, ... , el2n2r-1<2"-r-1)/N) ^ ^ Diag2«-r (1, e '1 (01)2n2r-1-1/N, ... , e'2n2r-1<2n-r-1)/N) = A2- (sp1),
where s^"r-1 = e i(0)2n2r 1/N. As result we obtain fast many-parameter Fourier-Clifford transform
Fc=n([v. ® (V © V «))] • ®:1® V] )•
Summary and Conclusion
In this paper, we have shown a new unified approach to the many-parametric Fourier-Clifford transforms (MPFCTs). Defined transforms depend on finite set of free parameters, which could be changed independently of one another. For each fixed value of parameter we get the unique orthogonal transform over Clifford algebra.
These transforms can be used for novel class of 6G Intelligent OFDM-telecommunication systems. We are going to use MPFCT to develop novel Intelligent OFDM-telecommunication systems. The new systems will use inverse MPQFT for modulation at the transmitter and direct MPFQT for demodulation at the receiver.
References
1. Brackx F., Schepper N., Sommen F. The Fourier transform in Clifford analysis. Advances in imaging and electron physics. 2009. Vol. 156. Pp. 55-201.
2. Labunets V. G., Ostheimer E. New many-parameter quaternion Fourier-Clifford-Hamilton transforms for intelligent OFDM TCS. Information technologies and nanotechnologies (ITNT-2020): conference proceedings. Samara: Samara University, 2020. Vol. 2. Pp. 591-601. EDN: OCRMSN. (In Russ.)
3. Labunets V. G., Ostheimer E. Many-parameter quaternion Fourier transforms for intelligent OFDM telecommunication system. Advances in artificial systems for medicine and education III. 2020. Vol. 1126. Pp. 76-92. DOI: https://doi.org/10.1007/978-3-030-39162-1_8.
Системный анализ System analysis
4. Labunets V. G., Ostheimer E. Intelligent OFDM telecommunication systems based on many-parameter complex or quaternion Fourier transforms. Advances in intelligent systems and computing. 2020. Vol. 1127. Pp. 129-144. DOI: https://doi.org/10.1007/978-3-030-39216-1_13. EDN: PONTQO.
5. Hestenes D., Sobczyk G. Clifford algebra to geometric calculus. Dordrecht: Springer Netherlands, 1984. 342 p. ISBN: 90-277-2561-6.
6. Labunets V. G. Quaternion number theoretical transform. Devices and Methods of Experimental Investigations in Automation. Dnepropetrovsk: Dnepropetrovsk University Press,
1981. Pp. 28-33. (In Russ.)
7. Sommen F. Product and an exponential function in hypercomplex function theory. Applicable analysis. 1981. Vol. 12, iss. 1. Pp. 13-26. DOI: https://doi.org/10.1080/00036818108839345.
8. Sommen F. Hypercomplex Fourier and Laplace transforms. Illinois journal of mathematics.
1982. Vol. 26, iss. 2. Pp. 332-352. DOI: https://doi.org/10.1215/ijm/1256046802.
9. Ernst R., Bodenhausen G., Wokaun A. Principles of nuclear magnetic resonance in one and two dimensions. Oxford: Clarendon Press, 1990. 640 p. ISBN: 978-0-19-855647-3.
10. Delsuc M. A. Spectral representation of 2D NMR spectra by hypercomplex numbers. Journal of magnetic resonance. 1969. Vol. 77, iss. 1. Pp. 119-124.
11. Rundblad-Labunets E., Labunets V., Astola J., Egiazarian K. Fast fractional Fourier-Clifford transforms. Second International Workshop on Transforms and Filter Banks. Tampere, 1999. Pp. 376-405.
12. Sangwine S. J., Ell T. A. The discrete Fourier transform of a colour image. In: Blackedge J. M., Turner M. J. (eds.). Image Processing II: Mathematical methods, algorithms and applications. Chichester, 1998. Pp. 430-441.
13. Labunets-Rundblad E. Fast Fourier-Clifford transforms design and application in invariant recognition: PhD thesis. Tampere: Tampere University of Technology, 2000. 262 p.
14. Rundblad E., Labunets V., Egiazarian K., Astola J. Fast invariant recognition of color images based on Fourier-Clifford number theoretical transform. Image and Signal Processing for Remote Sensing VI. Barcelona: Europt, 2000. Pp. 284-292.
15. Labunets V. G., Kohk E. V., Ostheimer E. Algebraic models and methods of image computer processing. Part 1. Multiplet models of multichannel images. Computer optics. 2018. Vol. 42, iss. 1. Pp. 84-96. DOI: https://doi.org/10.18287/2412-6179-2018-42-1-84-95.
16. Brackx F., De Schepper N., Sommen F. The two-dimensional Clifford-Fourier transform. Journal of mathematical imaging and vision. 2006. Vol. 26, iss. 1. Pp. 5-18. DOI: https://doi. org/10.1007/s10851-006-3605-y.
17. De Bie H., De Schepper N. The fractional Clifford-Fourier transform. Complex analysis and operator theory. 2012. Vol. 6, iss. 5. Pp. 1047-1067. DOI: https://doi.org/10.1007/ s11785-012-0229-7.
18. De Bie H., De Schepper N., Sommen F. The class of Clifford-Fourier transforms. Journal of Fourier analysis and applications. 2011. Vol. 17. Pp. 1198-1231. DOI: https://doi. org/10.1007/s00041-011-9177-2.
19. McBride A. C., Kerr F. H. On Namias' fractional Fourier transforms. IMA Journal of Applied Mathematics. 1987. Vol. 39, iss. 2. Pp. 159-175. DOI: https://doi.org/10.1093/imam-at/39.2.159.
Системный анализ System analysis
20. Ozaktas H., Zalevsky Z., Kutay A. The fractional Fourier transform. Chichester: Wiley, 2001. 532 p. ISBN: 0-47-196346-1.
21. Carracedo C. M., Alix M. S. The theory of fractional powers of operators. London; New York: Elsevier, 2001. 365 p. ISBN: 978-0-44-488797-9.
22. Sharma D., Saxena R., Singh N. Hybrid encryption-compression scheme based on multiple parameter discrete fractional Fourier transform with eigen vector decomposition algorithm. International journal of computer network and information security. 2014. Vol. 10, no. 10. Pp. 1-12. DOI: https://doi.org/10.5815/ijcnis.2014.10.01.
23. Pariyal P. S., Koyani D. M., Gandhi D. M. et al. Comparison based analysis of different FFT architectures. International journal of image, graphics and signal processing. 2016. Vol. 6, no. 8. Pp. 41-47. DOI: https://doi.org/10.5815/ijigsp.2016.06.05.
Источники
1. Brackx F., Schepper N., Sommen F. The Fourier transform in Clifford analysis // Advances in imaging and electron physics. 2009. Vol. 156. Р. 55-201.
2. Labunets V. G., Ostheimer E. New many-parameter quaternion Fourier-Clifford-Hamilton transforms for intelligent OFDM TCS // Информационные технологии и нанотехнологии (ИТНТ-2020): сб. тр. по материалам VI Междунар. конф. и молодежной школы (Самара, 26-29 мая 2020 г.): в 4 т. Самара: Самарский университет, 2020. Т. 2. С. 591601. EDN: OCRMSN.
3. Labunets V. G., Ostheimer E. Many-parameter quaternion Fourier transforms for intelligent OFDM telecommunication system // Advances in artificial systems for medicine and education III. 2020. Vol. 1126. P. 76-92. DOI: https://doi.org/10.1007/978-3-030-39162-1_8.
4. Labunets V. G., Ostheimer E. Intelligent OFDM telecommunication systems based on many-parameter complex or quaternion Fourier transforms. Advances in intelligent systems and computing. 2020. Vol. 1127. P. 129-144. DOI: https://doi.org/10.1007/978-3-030-39216-1_13. EDN: PONTQO.
5. Hestenes D., Sobczyk G. Clifford algebra to geometric calculus. Dordrecht: Springer Netherlands, 1984. 342 p. ISBN: 90-277-2561-6.
6. Лабунец В. Г. Теоретико-числовое преобразование кватернионов // Приборы и методы автоматизации экспериментальных исследований: сб. науч. тр. Днепропетровск: ДГУ, 1981. С. 28-33.
7. Sommen F. Product and an exponential function in hypercomplex function theory //Applicable analysis. 1981. Vol. 12, iss. 1. P. 13-26. DOI: https://doi.org/10.1080/00036818108839345.
8. Sommen F. Hypercomplex Fourier and Laplace transforms // Illinois journal of mathematics. 1982. Vol. 26, iss. 2. P. 332-352. DOI: https://doi.org/10.1215/ijm/1256046802.
9. Ernst R., Bodenhausen G., Wokaun A. Principles of nuclear magnetic resonance in one and two dimensions. Oxford: Clarendon Press, 1990. 640 p. ISBN: 978-0-19-855647-3.
10. Delsuc M. A. Spectral representation of 2D NMR spectra by hypercomplex numbers // Journal of magnetic resonance. 1969. Vol. 77, iss. 1. P. 119-124.
11. Rundblad-Labunets E., Labunets V., Astola J., Egiazarian K. Fast fractional Fourier-Clifford transforms // Second International Workshop on Transforms and Filter Banks. Tampere, 1999. P. 376-405.
Системный анализ System analysis
12. Sangwine S. J., Ell T. A. The discrete Fourier transform of a colour image // Image Processing II: Mathematical methods, algorithms and applications / ed. by J. M. Blackedge, M. J. Turner. Chichester, 1998. P. 430-441.
13. Labunets-Rundblad E. Fast Fourier-Clifford transforms design and application in invariant recognition: PhD thesis. Tampere: Tampere University of Technology, 2000. 262 p.
14. Rundblad E., Labunets V., Egiazarian K., Astola J. Fast invariant recognition of color images based on Fourier-Clifford number theoretical transform // Image and Signal Processing for Remote Sensing VI. Barcelona: Europt, 2000. P. 284-292.
15. Labunets V. G., Kohk E. V., Ostheimer E. Algebraic models and methods of image computer processing. Part 1. Multiplet models of multichannel //. Computer optics. 2018. Vol. 42, iss. 1. P. 84-96. DOI: https://doi.org/10.18287/2412-6179-2018-42-1-84-95.
16. Brackx F., De Schepper N., Sommen F. The two-dimensional Clifford-Fourier transform // Journal of mathematical imaging and vision. 2006. Vol. 26, iss. 1. P. 5-18. DOI: https://doi. org/10.1007/s10851-006-3605-y.
17. De Bie H., De Schepper N. The fractional Clifford-Fourier transform // Complex analysis and operator theory. 2012. Vol. 6, iss. 5. P. 1047-1067. DOI: https://doi.org/10.1007/ s11785-012-0229-7.
18. De Bie H., De Schepper N., Sommen F. The class of Clifford-Fourier transforms // Journal of Fourier analysis and applications. 2011. Vol. 17. P. 1198-1231. DOI: https://doi. org/10.1007/s00041-011-9177-2.
19. McBride A. C., Kerr F. H. On Namias' fractional Fourier transforms // IMA Journal of Applied Mathematics. 1987. Vol. 39, iss. 2. P. 159-175. DOI: https://doi.org/10.1093/imam-at/39.2.159.
20. Ozaktas H., Zalevsky Z., Kutay A. The fractional Fourier transform. Chichester: Wiley, 2001. 532 p. ISBN: 0-47-196346-1.
21. Carracedo C. M., Alix M. S. The theory of fractional powers of operators. London; New York: Elsevier, 2001. 365 p. ISBN: 978-0-44-488797-9.
22. Sharma D., Saxena R., Singh N. Hybrid encryption-compression scheme based on multiple parameter discrete fractional Fourier transform with eigen vector decomposition algorithm // International journal of computer network and information security. 2014. Vol. 10, no. 10. P. 1-12. DOI: https://doi.org/10.5815/ijcnis.2014.10.01.
23. Pariyal P. S., Koyani D. M., Gandhi D. M. et al. Comparison based analysis of different FFT architectures // International journal of image, graphics and signal processing. 2016. Vol. 6, no. 8. P. 41-47. DOI: https://doi.org/10.5815/ijigsp.2016.06.05.
Information about the authors
Valery G. Labunets, Doctor of Technical Sciences, Professor of Chess Art and Computer Mathematics Dept. Ural State University of Economics, 620144, Russia, Ekaterinburg, 8 Mar-ta/Narodnaya Volya Str., 62/45. E-mail: vlabunets05@yahoo.com.
Системный анализ System analysis
Viktor P. Chasovskikh, Doctor of Technical Sciences, Professor of Chess Art and Computer Mathematics Dept. Ural State University of Economics, 620144, Russia, Ekaterinburg, 8 Mar-ta/Narodnaya Volya Str., 62/45. E-mail: u2007u@yandex.ru.
Evgeny N. Starikov, Candidate of Economic Sciences, Associate Professor, Acting Head of Chess Art and Computer Mathematics Dept. Ural State University of Economics, 620144, Russia, Ekaterinburg, 8 Marta/Narodnaya Volya Str., 62/45. E-mail: starikov_en@usue.ru.
Информация об авторах
Лабунец Валерий Григорьевич, доктор технических наук, профессор кафедры шахматного искусства и компьютерной математики. Уральский государственный экономический университет, 620144, РФ, г. Екатеринбург, ул. 8 Марта/Народной Воли, 62/45. E-mail: vlabunets05@yahoo.com.
Часовских Виктор Петрович, доктор технических наук, профессор кафедры шахматного искусства и компьютерной математики. Уральский государственный экономический университет, 620144, РФ, г. Екатеринбург, ул. 8 Марта/Народной Воли, 62/45. E-mail: u2007u@yandex.ru.
Стариков Евгений Николаевич, кандидат экономических наук, доцент, и. о. заведующего кафедрой шахматного искусства и компьютерной математики. Уральский государственный экономический университет, 620144, РФ, г. Екатеринбург, ул. 8 Марта/Народной Воли, 62/45. E-mail: starikov_en@usue.ru.
