Quaternions and Biquaternions for Symmetric Markov-Chain System Analysis

Home , Bicomplex number, Biquaternion, Definable real number, Octonion, Quaternion

15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3-7, 2007, copyright by EURASIP

QUATERNIONS AND BIQUATERNIONS FOR SYMMETRIC MARKOV-CHAIN SYSTEM ANALYSIS

Soo-Chang Pei, Jian-Jiun Ding

Department of Electrical Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd., 10617, Taipei, Taiwan, R.O.C TEL: 886-2-23635251-321, Fax: 886-2-23671909, Email: [email protected], [email protected]

ABSTRACT In this condition, j2 = 1, k2 = −1, ij = ji, ik = ki, and jk = kj. In this conference, we use quaternions, biquaternions, and The Clifford biquaternion and the complex Tessarine have their related algebras similar to them to model symmetric idempotent elements e1 and e2: Markov systems. With these algebras, the original N×N eI1 = (1+ ) / 2 , eI2 = (1− ) / 2 , (6) Markov system can be reduced into an (N/2)×(N/2) or 2 2 e1 = e1, e2 = e2, e1e2 = 0. (N/4)×(N/4) system. It makes the system easier for imple- Idempotent elements are helpful for fast computation. If q menting and analysis. In addition to Markov chains, the is defined in (3) and p is defined as proposed idea is also helpful for simplifying the complexi- pp=+⋅+⋅+⋅0 pipjpkij k ties of other symmetric systems whose interactions between , (7) two objects are determined by their distance. + pIiIjIkIIpiIpjIpkI+⋅+⋅+⋅ then p and q can be expressed as: 1. INTRODUCTION p = pe+ pe , qqeqe= + , (8) 11 2 2 11 2 2 The quaternion [1][2] is a generalization of the complex where algebra. Its application in signal and image processing were p110= pe=+⋅+⋅+⋅( p pij i p j p k k noticed in recent years [2][3][4]. The quaternion has been , p22= pe . (9) ++ppipjpk ⋅+⋅+⋅)/2 applied for filter design, feature extraction, pattern recogni- IiIjIkI tion, rotation analysis, differential equation analysis, and Similarly,

SVD decomposition, etc. In this paper, we propose another qqe11= and qqe22= . (10) application of the quaternion and its related algebras, i.e., Then, the product of p and q can be expressed as: the Markov chain analysis. pqpepeqeqepqepqe= ()()++=+. (11) 11 22 11 22 111 2 22 2. QUATERNIONS AND RELATED ALGEBRAS In addition to the quaternion and the Clifford biquaternion, there are also other types of 4-D and 8-D algebras. Quaternions [1][2] are generalizations of complex numbers. The algebras used in this paper are: A quaternion has 4 components: real, i-, j-, and k-parts: [2-D algebras] • Complex algebra. q = qr + qi ⋅i + q j ⋅ j + qk ⋅k , (1) • Tessarines (also called as split-complex numbers, and and i, j, k obey the rules as below: double complex numbers) [5][6]: 2 2 2 i = j = k = −1, ij⋅= k, jk⋅= i, ki⋅= j, 2 qa= rI+ Ia, where I = 1. (12) ji⋅=− k, kj⋅=− i, ik⋅=− j. (2) [4-D algebras] Quaternions can be used for the color image analysis, filter • Quaternions (defined in (1) and (2)) design, and 3-D object analysis. • Complex Tessarines (defined in (5)). The definition of the Clifford biquaternion [6][10] is • Reduced biquaternion: See [3][7]. It is also called the similar to that of the quaternion, but there are 8 elements: bicomplex number (C×C). Its definition is the same as the 2 qq=+⋅+⋅+⋅0 qiqjqkij k reduced biquaternion in (5) but I = −1. , (3) ++⋅+⋅+⋅qI q iI q jI q kI • 4-D purely hyperbolic complex numbers: IiIjIkI qq= ++ qIqJq +⋅ IJ, I2 = J2 = 1, IJ = JI. (13) where i, j, and k satisfy (2) and rI J IJ [8-D algebras] I2 = 1, iI⋅=⋅ Ii, jI⋅=⋅ I j, kI⋅=⋅ Ik. (4) • Octonions (standard definition [1][9]). It has 7 imaginary The Tessarine with complex coefficients [5] (we call it the parts and the multiplicative rule is shown in [6]. complex Tessarine) is defined in the similar form. • Clifford biquaternions (see (3)). qq=+⋅++⋅ri qiqIqiI I iI. (5) • 8-D Tessarines (4-D hyperbolic complex numbers with complex coefficients)

JJG JJG q=+ ari ia + Ja J + iJa iJI + Ia + iIa iIJIiJI + JIa + iJIa , (b) If pqrs= − , then the interactions between p and q 2 2 2 where i = −1, I = J = 1, iI = Ii, iJ = Ji, and IJ = JI. (14) (denoted by hp,q) and r and s (hr,s)) has the relation of • Hamilton's biquaternions [1][3]: Its definition is the same hp,q = −hr,s as the biquaternion in (3) but I2 = −1. then • Tri-complex numbers x112[naxnbxn+=1] [ ][] − , (23) q=+ ari ia + Ia I + iIa iIJ + Ja + iJa iJK + Ka + iKa iK, x212[nbxnaxn+=1] [ ][] + , (24) where I2 = J2 = K2 = −1, Ii = iI, Ji = iJ, Ki = iK, IJ = JI, IK = KI, and JK = KJ. (15) then, we can define a function of x[nxnIxn] =+12[ ] [ ] (25) 3. 2×2 AND 4×4 SYMMETRIC SYSTEM where I2 = −1, iI = I i. (26) In this case, we can also prove that For many systems, the distance between two points will x nhxn+=⋅1 , where ha=+ι b. (27) determine the mutual effect. For example, in physics, the [ ] [ ] 2 gravitation between two objects is proportional to 1/R If x1[n] and x2[n] are complex functions, then the system is where R is the distance between two objects. Intensities of modeled by the following algebra: the electric and magnetic fields also depend on R. In social qq= 01+⋅++⋅ qiq 4ι qi 5ι . (28) science, the interaction between two cities can also be It is just the same as the reduced biquaternion algebra. modeled by a function of the distance between two points. Here, we suppose that there are N points in a Markov 1 3 system and the field in each point is denoted by xk[n], where k = 1, 2, …, N. Suppose that the interaction between two dots is determined by their distance: N xmkmk[1]nxna+=∑ [], , where afRmk,,= ( mk) , (16) 2 4 k =1

Rm,k is the distance between m and. Fig. 2 A doubly symmetric 4 × 4 rectangular system. We find that in the system is symmetric, we can model The idea can be extended to any symmetric Markov sys- the system by the complex Tessarine. tem. For the system in Fig. 2, suppose that the values at the For example, for the system of Fig. 1 points 1, 2, 3, and 4 are 1 2 x1[n], x2[n], x3[n], and x4[n] (29) and the interactions between the points m, n are denoted by h . Therefore, m,n Fig. 1 A symmetric 2 × 2 system. ⎡ x11[1]nxn+ ⎤⎡⎤⎡⎤hhhh1,1 1,2 1,3 1,4 [] ⎢⎥ If the interaction is determined by the instance between two ⎢x [1]nxn+ ⎥⎢⎥hhhh [] dots, it can be modeled by the following Markov chain: ⎢ 22⎥⎢⎥= ⎢⎥2,1 2,2 2,3 2,4 . (30) ⎢x [1]nxn+ ⎥⎢⎥⎢⎥hhhh [] x naxnbxn+=1 + , (17) 333,1 3,2 3,3 1,4 112[ ] [] [] ⎢ ⎥⎢⎥⎢⎥ hhhh ⎣x44[1]nxn+ ⎦⎣⎦⎣⎦⎢⎥4,1 4,2 4,3 4,4 [] x212[nbxnaxn+=1 ] [] + []. (18) First, note that Then, we can define a function as hm,n = hn,m. (31) x[]nxnIxn=+12 [] [], (19) Moreover, since interactions are determined by distances, 2 where I = 1. If we define h1,1 = h2,2 = h3,3 = h4,4 = a h1,2 = h2,1 = h3,4 = h4,3 = b, haIb=+ , (20) h1,3 = h2,4 = h3,1 = h4,2 = c, h1,4 = h2,3 = h3,2 = h4,1 = d. (32) then Then (30) can be rewritten as: x [1]nxn+ abcd [] h⋅= x[] n ax12 [] n + bx [] n + I() ax 21 [] n + bx [] n , (21) ⎡ 11⎤⎡⎤⎡⎤ ⎢ ⎥⎢⎥⎢⎥ i.e., . (22) x22[1]nxn+ badc [] x[]nhxn+=⋅1 [] ⎢ ⎥⎢⎥= ⎢⎥ (33) If x1[n] and x2[n] are complex functions, then (22) is a ⎢x33[1]nxn+ ⎥⎢⎥⎢⎥cdab [] complex Tessarine operation. That is, the system 1 can be ⎢ ⎥⎢⎥⎢⎥ ⎣x44[1]nxn+ ⎦⎣⎦⎣⎦dcba [] modeled by a complex Tessarine operation. If x [n] and 1 If we set that x2[n] are quaternion functions, Fig. 1 can be modeled by a Clifford biquaternion operation. yn11[ ] =+ xn[ ] Ixn 2[ ] , yn23[]=+ xn [] Ixn 4[ ] , (34) If the effect of the system is determined by the only the haIb= + , hcId= + . distance but also the direction, that is, 1 2 Then (33) can be rewritten as (a) The interaction between JJGthe pointsJJG p and q and that be- ⎡ yn1121[1]+ ⎤⎡ h h ⎤⎡ yn [] ⎤ tween r and s are the same pq and rs have the same dis- ⎢ ⎥⎢= ⎥⎢ ⎥. (35) tance and direction. ⎣ yn2212[1]+ ⎦⎣ h h ⎦⎣ yn [] ⎦

That is, after converting into the complex Tessarine form, Table 1 2×2 and 4×4 symmetric system simplification by quater- the 4 × 4 system become a 2×2 system. nion and octonion. Moreover, notice that (35) is also a symmetric system. original system algebra reduced system We can further reduce it into the 1 × 1 form. We can apply 2×2 symmetric complex 1-to-1 system the 8-D Tessarines algebra. We can set that system Tessarine 2×2 directional reduced 1-to-1 system yn[ ] =+ x123[ n] Jxn[ ] + Ixn[ ] + IJxn 4[ ] , (36) symmetric system biquaternion Then 2×2 symmetric Tessarine 1-to-1 system yn+=1 hyn where h=+ a Jb + Ic + IJd . (37) [ ] [ ] system, real input That is, after applying the complex Tessarines, the 4 × 4 2×2 symmetric sys- Clifford 1-to-1 system symmetric rectangular system becomes a 2 × 2 symmetric tem, quaternion input biquaternion system. After applying the 8-D Tessarines n, the 4 × 4 2×2 directional complex 1-to-1 system symmetric rectangular system becomes a one-input and symmetric system, algebra one-output system. real input In the case where the interactions between two dots are 2×2 directional Hamilton's 1-to-1 system affected by the direction: symmetric system, biquaternions hm,k = −hk,m if m ≠ k, (38) quaternion input then 4×4 symmetric 8-D 1-to-1 system

⎡⎤x11[1]nxn+ ⎡⎤abcd−−− ⎡⎤ [] system Tessarines ⎢⎥⎢⎥ ⎢⎥ 4×4 directional complex 2×2 system x22[1]nxn+ ba−− f c [] ⎢⎥= ⎢⎥ ⎢⎥. (39) symmetric system Tessarines with offsets ⎢⎥x [1]nxn+ ⎢⎥cf a− b ⎢⎥ [] ⎢⎥33⎢⎥ ⎢⎥ 4×4 directional octonion-V 1-to-1 system ⎣⎦x44[1]nxn+ ⎣⎦dc b a ⎣⎦ [] symmetric system, d with offsets

Specially, if d = f, and x1[n], x2[n], x3[n], and x4[n] are real, = f 4×4 symmetric sys- tri-complex 1-to-1 system ⎡⎤x11[1]nxn+ ⎡⎤abcd−−− ⎡⎤ [] ⎢⎥⎢⎥ ⎢⎥ tem, real input numbers x22[1]nxn+ ba−− d c [] ⎢⎥= ⎢⎥ ⎢⎥ (40) 4×4 directional complex 2×2 system ⎢⎥x33[1]nxn+ ⎢⎥cd a− b ⎢⎥ [] symmetric system, algebra with offsets ⎢⎥⎢⎥ ⎢⎥ real input ⎣⎦x44[1]nxn+ ⎣⎦dc b a ⎣⎦ [] 4×4 directional quaternion 1-to-1 system We can compare (39) with the quaternion multiplication. If symmetric system, d with offsets zizjzkzri++ j + k = f, real input , (41) For the directional symmetric rectangular system in =+++()()xriix jx j kx kri y +++ iy jy j ky k (40), in the case where d ≠ f, the system can be represented then by a Clifford octonion system: zx⎡⎤yyy y ⎡⎤rrri j k ⎡⎤ yn[1]+ hh yn [] δ ⎢⎥⎢⎥ ⎢⎥ ⎡ 111⎤⎡⎤⎡⎤⎡⎤13 zxyy y− y ⎢ ⎥⎢⎥⎢⎥=+⎢⎥ , (47) ⎢⎥ii= ⎢⎥ir k j ⎢⎥ yn[1]+ hh yn [] δ ⎢⎥. (42) ⎣ 222⎦⎣⎦⎣⎦⎣⎦32 ⎢⎥zxjjyyyyjkri− ⎢⎥ ⎢⎥⎢⎥ ⎢⎥ where yn=+ xn Ixn, yn=+ xn Ixn, yy− yy 11[ ] [ ] 4[ ] 22[] [] 3[ ] ⎣⎦zxkk⎣⎦⎢⎥kj ir ⎣⎦ haId= + , haIf= + , hbIc=+ , I2 = −1, Since 1 2 3

δ132= Re(hyn) Re( [ ]) , δ231= Re(hyn) Im( [ ]) . (48) ⎡⎤x13[1]2[]ncxn++ ⎡⎤abcd−− ⎡⎤ xn 1 [] ⎢⎥⎢⎥ ⎢⎥In (43) and (47), the inputs are real is considered. x21[nbxn+− 1]2 []−−−ba d c xn 2 [] ⎢⎥= ⎢⎥ ⎢⎥, (43) When xk[n] are complex, (44) can be rewritten as ⎢⎥⎢⎥ ⎢⎥ x33[1]nxn+ cda− b [] yn=+ x n Ixn + Jxn + Kxn, (49) ⎢⎥⎢⎥ ⎢⎥[ ] 12[ ] [ ][][] 3 4 ⎣⎦x41[ndxn+− 1]2 []⎣⎦−dc b a ⎣⎦ xn 4 [] where the tri-complex numbers (defined in (15) is applied. we find that, if In this case, (43) can still be applied and (44) is rewritten as haIbJcKd= −+− , yn[]=+ x12 [] n ixn [] + jxn 3 [] + kxn 4 [], (44) h=− a ib + jc − kd , then δ =−−2(cynIbynKdynjr[ ] [ ][] r) . (50) yn[][]+=1 ynh +δ , (45) The relations between the quaternion-related algebras and the symmetric system are listed in Table 1. where δ =−−2 cyjr[] n iby [] n kdy r [] n , (46) ()4. GENERALIZE SYMMETRIC SYSTEM ANALYSIS yr[n] and yj[n] mean the real and j- parts of y[n]. That is, a directional symmetric rectangular system can be repre- The results in Section 3 can be generalized for all sym- sented by the quaternion. metric and doubly symmetric systems.

For the example in Fig. 3, there are 6 dots and form a 5 hexagon. We also suppose that the interactions between two points (denoted by hm,n) are determined by their dis- 1 3 tance. Suppose that the system is symmetric respect to the dash line L1. That is, h1,3 = h2,4, h1,4 = h2,3, h1,5 = h2,6, h1,6 = h2,5, L1 h = h , h = h , h = h . (51) 2 3,5 4,6 3,6 4,5 k,n n,k 4 Then the system can be expressed as 6 ⎡⎤x11[]nxn+1 ⎡⎤abcde f ⎡⎤[] L ⎢⎥ ⎢⎥ 2 x nxn+1 ⎢⎥badcf e ⎢⎥22[]⎢⎥ ⎢⎥[] ⎢⎥ ⎢⎥ Fig. 3 A symmetric hexagon system. x33[]nxn+1 ⎢⎥cdapgl [] ⎢⎥= ⎢⎥ ⎢⎥ (52) 1 3 5 7 ⎢⎥x44[]nxn+1 ⎢⎥dcpalg ⎢⎥[] ⎢⎥⎢⎥ ⎢⎥ x55[]nxn+1 efglam [] ⎢⎥⎢⎥ ⎢⎥ L1 ⎣⎦⎢⎥x66[]nxn+1 ⎣⎦⎢⎥felgma ⎣⎦⎢⎥[] It is a 6 × 6 system. If we set that 2 4 L2 6 8 y nxnIxn=+ , y nxnIxn=+ , (53) 11[ ] [ ] 2[ ] 23[ ] [ ] 4[ ] Fig. 4 A symmetric system consists of 8 points yn35[ ] =+ xn[ ] Ixn 6[ ] , haIb1 =+ , hcId2 =+ ,

h1,3 = h2,4, h1,4 = h2,3, h1,5 = h2,6, h1,6 = h2,5, h1,7 = h2,8, heIf3 =+ , hgIl4 =+, haIp5 = + , haIm6 =+ , 2 h = h , h = h , h = h , h = h , h = h , where I = 1 and iI = Ii (i.e., the reduced biquaternion alge- 1,8 2,7 3,5 4,6 3,6 4,5 3,7 4,8 3,8 4,7 h = h , h = h . (60) bra is applied), then (52) can be expressed as 5,7 6,8 5,8 6,7 Therefore, the system can be expressed as ⎡⎤yn11231[ +1] ⎡⎤ h h h ⎡⎤ yn[ ] ⎢⎥ ⎢⎥ ⎡x11[nxn+1]⎤⎡⎤⎡⎤abcde fgl [ ] yn+=1 ⎢⎥ h h h yn , (54) ⎢⎥22542[]⎢⎥ ⎢⎥[] ⎢ ⎥⎢⎥⎢⎥ x22[]nxn+1 badc f elg [] ⎢⎥yn+1 ⎢⎥ h h h ⎢⎥ yn ⎢ ⎥⎢⎥⎢⎥ ⎣⎦33463[]⎣⎦ ⎣⎦[] ⎢ ⎥⎢⎥ x33[]nxn+1 ⎢⎥cdamoprt [] and the system is reduced into a 3 × 3 system by the re- ⎢ ⎥⎢⎥⎢⎥ ⎢⎥x44[]nxn+1 dcmapotr ⎢[]⎥ duced biquaternion algebra. = ⎢⎥ (61) ⎢⎥⎢⎥ ⎢⎥ Furthermore, if the hexagon is also symmetric respect to x55[]nxn+1 efopauvw [] ⎢⎥⎢⎥ ⎢⎥ L2 (L2 is perpendicular to L1), i.e., in addition to (51), ⎢⎥x66[]nxn+1 ⎢⎥fepouawv ⎢[]⎥ h = h , h = h , h = h , h = h , h = h , (55) 1,2 3,4 1,5 3,5 1,6 3,6 2,5 4,5 2,6 4,6 ⎢⎥x nxn+1 ⎢⎥glrtvwaα ⎢⎥ then the system can be expressed as the following operation: ⎢⎥77[]⎢⎥ ⎢[]⎥ ⎢⎥x88[]nxn+1 ⎣⎦⎢⎥lgtrwvα a ⎢[]⎥ ⎡⎤x11[]nxn+1 ⎡⎤abcde f ⎡⎤[] ⎣ ⎦⎣⎦ ⎢⎥ ⎢⎥ x nxn+1 ⎢⎥badc f e Then, we can apply the complex Tessarines algebra to sim- ⎢⎥22[]⎢⎥ ⎢⎥[] plify the system. We set that ⎢⎥ ⎢⎥ x33[]nxn+1 ⎢⎥cdabe f [] yn=+ xn Ixn, yn=+ xn Ixn, (62) ⎢⎥= ⎢⎥ ⎢⎥. (56) 11[ ] [ ] 2[ ] 23[] [] 4[ ] ⎢⎥x44[]nxn+1 dcbafe ⎢⎥[] ⎢⎥ yn35[ ] =+ xn[ ] Ixn 6[ ] , yn47[]=+ xn [] Ixn 8[ ] , ⎢⎥⎢⎥ ⎢⎥ x55[]nxn+1 efefam [] ⎢⎥⎢⎥ ⎢⎥ haIb1 = + , hcId2 = + , heIf3 =+ , hgIl4 = + , ⎢⎥x66[]nxn+1 ⎢⎥fefema ⎢⎥[] ⎣⎦⎣⎦ ⎣⎦ haIm5 = + , hoIp6 = + , hrIt7 =+ , haIu8 = + , Then, we can use the 8-D Tessarines algebra in (14) and set hvIw9 = + , haI10 = + α , (63) z11[ n] =+ x[ n] Ix 2[ n] + Jx 3[ n] + IJx 4[ n] , (57) and (64) is simplified as zn=+ xn Ixn, 25[ ] [ ] 6[ ] ⎡ yn11[ +1]⎤⎡⎤⎡⎤hhh123 h 4 yn[ ] s =+aIbJcIJd + + , s = eIf+ , (58) ⎢ ⎥⎢⎥⎢⎥ 1 2 yn[]+1 hhhh256 7 yn[] ⎢ 22⎥⎢⎥= ⎢⎥, (64) s3 =+eIfJeIJf + + , s4 = aIm+ , ⎢ ⎥⎢⎥ yn33[]+1 ⎢⎥hhhh368 9 yn[] and (56) can be rewritten as ⎢ ⎥⎢⎥⎢⎥ ⎣⎢yn44[]+1 ⎦⎣⎦⎥⎢⎥⎣⎦hhhh47910 yn[] ⎡⎤z11[1]nzn+ ⎡⎤ss12 ⎡⎤ [] ⎢⎥= ⎢⎥ ⎢⎥. (59) and the original 8×8 system is reduced into the 4×4 system. ⎣⎦z22[1]nzn+ ⎣⎦ss34 ⎣⎦ [] Moreover, if the system in Fig. 4 is also symmetric respect We then give another example in Fig. 4, which is an 8×8 to L2, then it can be simplified by the 8-D Tessarines: system. Suppose that the system is symmetric respect to L ⎡ zn11[1]+ ⎤⎡⎤⎡⎤ss12 zn [] 1 = , (65) and the interaction between n and k is denoted by hn,k. Then ⎢ ⎥⎢⎥⎢⎥ ⎣zn22[1]+ ⎦⎣⎦⎣⎦ss23 zn []

©2007 EURASIP 1340 15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3-7, 2007, copyright by EURASIP where (a) channe1 1 c(t, τ) zn11[]=+ xn [] Ixn 2 [] + Jxn 7[ ][] + IJxn 8, (66) x(t) x1(t) z23[] n=+ x [] n Ix 4 [] n + Jx 5[ n][] + IJx 6 n , n(t, τ) n(t, τ) s1 =+aIbJgIJl + + s2 =+cIdJeIJf + + , (67) y(t) y1(t) channe1 2 c(t, τ) s3 =+aImJoIJp + + . In summary, for a Markov chain system, if the interac- (b) z(t) z1(t) tions between two points are determined by their distance, c1(t, τ) as in (16), then we can use the quaternion or the octonion algebras to simplify the system. The rules are: Fig. 5 The symmetric 2-channel communication system. (A) If the system consists of N points and is symmetric respect to a line L1, then we can use the complex Tessarine 6. CONCLUSIONS algebra to simplify the system into the

(N+N1)/2 × (N+N1)/2 ≈ N/2 × N/2 (68) In this paper, we introduce the way that uses the quaternion, system, where N1 is the number of points on L1. the biquaternion, and their related algebras to simplify the (B) If the system consists of N points and is symmetric re- symmetric Markov chain system analysis. If an N × N sys- spect to both the line L1 and the line L2, where L2 is perpen- tem is symmetric respect to one axis or two perpendicular dicular to L1, then we can use the 8-D Tessarine algebra to axes, we can use a proper algebra to reduce the system into simplify the system into the an (N/2) × (N/2) or (N/4) × (N/4) system, respectively. With (N+N1+ N2 +δ)/4 × (N+N1+ N2 +δ)//4 ≈ N/4 × N/4 (69) the simplification, the computation requirement is reduced, system, where N1 is the number of points on L1 and N2 is the system analysis problem can be solved in s simpler way, the number of points on L2. If there is a point at the inter- and many of the system properties can be analyzed by the section of L1 and L2, then δ = 1. In other conditions, δ = 0. quaternion and the related algebras. (C) For the case where the symmetric system is directional, we can use the complex Tessarine algebra or the conven- 7. REFERENCES tional quaternion to simplify the system (but in many conditions the offset terms should be included). The number of [1] W. R. Hamilton, “Elements of Quaternions”, Longmans, offset terms are near to Green and Co., London, 1866. 3N/4. (70) [2] T. A. Ell, “Quaternion-Fourier Transforms for Analysis (D) When the inputs and the responses are real, then the of Two-Dimensional Linear Time-Invariant Partial Dif- symmetric system can be simplified by the complex-II al- ferential Systems,” Proceedings of the 32nd Conferences gebra and the doubly symmetric system can be simplified on Decision and Control, p. 1830-1841, Dec. 1993. by the 4-D purely hyperbolic complex numbers. [3] Schütte, H. D. and Wenzel, J., “Hypercomplex numbers in digital signal processing,” ISCAS, vol. 2, pp. 1557- 5. COMMUNICATION SYSTEM SIMPLIFICATION 1560, 1990. [4] S. C. Pei, J. H. Chang, and J. J. Ding, “Quaternion ma- Then, we show how to use the above concept for the trix singular value decomposition and its applications for signal processing application of symmetric channel com- color image processing,” International Conference on munication. For the system in Fig. 5(a), the signals trans- Image Processing 2003, vol. 1, pp. 805-808, Sept. 2003. mitted along channels 1 and 2. However, channels 1 and 2 [5] J. Cockle, “On Certain Functions Resembling interfered with each other. Then, the relations between Quaternions and on a New Imaginary in Algebra”, {x1(t), y1(t)} and {x(t), y(t)} is London-Dublin-Edinburgh Philosophical Magazine, series 3, vol. 33, pp. 435-439, 1848. x ()tctxntyd=+ [ (,τ ) (ττττ ) (, ) ( )] , 1 ∫ [6] Wikipedia, http://en.wikipedia.org/wiki/ yt()=+ [ ct (,τ ) y (ττττ ) nt (, ) x ( )] d , (71) [7] V. S. Dimitrov, T. V. Cooklev, and B. D. Donevsky, 1 ∫ “On the multiplication of reduced biquaternions and ap- Where c(t, τ) is the channel transmission response and n(t, plications,” Information Processing Letters, vol. 43, pp. τ) is the interference from another channel. Note that it is in 161-164, 1992. fact a special case of (17) and (18). We can use the com- [8] T. A. Ell and S. J. Sangwine, “Decomposition of 2D plex Tessarine algebra to simplify it into the one-channel hypercomplex Fourier transforms into pairs of complex system as Fig. 5(b), where Fourier transforms”, EUSIPCO 2000, pp. 151-154. [9] John Baez, “The octonions”, Bull. Amer. Math. Soc., zt11()= ct (,τ ) z (ττ ) d, ct1(,τ )=+ ct (,ττ ) nt (, ) (72) ∫ vol. 39, pp. 145-205, 2002. zt()=+ xt () Iyt (), zt11()=+ xt () Iyt 1 (). (73) [10] W. K. Clifford, “Preliminary sketch of biquaternions,” Since the system is simplified into the 1-channel case, the Proc. London Math. Soc., pp. 381-396, 1873. analysis becomes easier and many of the conventional system analysis techniques can be used.