New amicable orthogonal designs AODs(8; 1; 1; 1; 2; 2; 2), AODs(8; 1; 1; 4; 1; 2; 2), AODs(8; 1; 2;2; 2; 2; 4), AODs(8; 1; 2; 2; 1; 2; 4), AODs(8; 1; 1; 2; 1; 2; 4), AODs(8; 1; 2; 4; 2; 2; 2), AODs(8; 1; 1; 4; 1; 1; 2; 2), AODs(8; 2; 2; 2; 2; 2; 2; 2; 2) and AODs(8; 1; 1; 1; 2; 1; 2; 2; 2) are found by applying a new theorem or by an exhaustive search. Also some previously undecided cases of amicable pairs are demonstrated to be non-existent after a complete search of the equivalence classes for orthogonal designs

Seberry, Jennifer

Tran, Le Chung

Wang, Y.

Wysocki, Beata J

Wysocki, Tadeusz A

Xia, Tianbing

Zhao, Y.

English

Research Online

University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering and Information Sciences 1-1-2006 Amicable Orthogonal Designs of Order 8 for Complex Space-Time Block Codes Y. Zhao University of Wollongong, yue_zhao@uow.edu.au Jennifer Seberry University of Wollongong, jennie@uow.edu.au Tianbing Xia University of Wollongong, txia@uow.edu.au Y. Wang University of Wollongong Beata J. Wysocki University of Wollongong, bjw@uow.edu.au See next page for additional authors Follow this and additional works at: https://ro.uow.edu.au/infopapers  Part of the Physical Sciences and Mathematics Commons Recommended Citation Zhao, Y.; Seberry, Jennifer; Xia, Tianbing; Wang, Y.; Wysocki, Beata J.; Wysocki, Tadeusz A.; and Tran, Le Chung: Amicable Orthogonal Designs of Order 8 for Complex Space-Time Block Codes 2006. https://ro.uow.edu.au/infopapers/358 Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: research-pubs@uow.edu.au Amicable Orthogonal Designs of Order 8 for Complex Space-Time Block Codes Abstract New amicable orthogonal designs AODs(8; 1; 1; 1; 2; 2; 2), AODs(8; 1; 1; 4; 1; 2; 2), AODs(8; 1; 2;2; 2; 2; 4), AODs(8; 1; 2; 2; 1; 2; 4), AODs(8; 1; 1; 2; 1; 2; 4), AODs(8; 1; 2; 4; 2; 2; 2), AODs(8; 1; 1; 4; 1; 1; 2; 2), AODs(8; 2; 2; 2; 2; 2; 2; 2; 2) and AODs(8; 1; 1; 1; 2; 1; 2; 2; 2) are found by applying a new theorem or by an exhaustive search. Also some previously undecided cases of amicable pairs are demonstrated to be non-existent after a complete search of the equivalence classes for orthogonal designs. Keywords 2000 Mathematics subject classification (Amer. Math. Soc.): 05B30, 15A36. Disciplines Physical Sciences and Mathematics Publication Details This article was originally published as Zhao, Y, Seberry, J, Xia, T, Wang, Y, Wysocki, BJ, Wysocki, TA and Tran, LC, Amicable Orthogonal Designs of Order 8 for Complex Space-Time Block Codes, Australasian Journal of Combinatorics, 34, 2006, 137-144. Authors Y. Zhao, Jennifer Seberry, Tianbing Xia, Y. Wang, Beata J. Wysocki, Tadeusz A. Wysocki, and Le Chung Tran This journal article is available at Research Online: https://ro.uow.edu.au/infopapers/358 Amicable Orthogonal Designs of Order 8 for ComplexSpace-Time Block CodesYing Zhao1, Jennifer Seberry1, Tianbing Xia1, Yejing Wang1,Beata J. Wysocki2, Tadeusz A. Wysocki2, Le Chung Tran21School of Information Technology and Computer ScienceUniversity of Wollongong, Wollongong, NSW, 2522, Australia2School of Electrical, Computer and Telecommunications EngineeringUniversity of Wollongong, Wollongong, NSW, 2522, AustraliaCorresponding Author’s Email: yz03@uow.edu.auAbstractNew amicable orthogonal designs AODs(8; 1, 1, 1; 2, 2, 2), AODs(8; 1, 1, 4; 1, 2, 2), AODs(8; 1, 2,2; 2, 2, 4), AODs(8; 1, 2, 2; 1, 2, 4), AODs(8; 1, 1, 2; 1, 2, 4), AODs(8; 1, 2, 4; 2, 2, 2), AODs(8; 1, 1, 4;1, 1, 2, 2), AODs(8; 2, 2, 2, 2; 2, 2, 2, 2) and AODs(8; 1, 1, 1, 2; 1, 2, 2, 2) are found by applying a newtheorem or by an exhaustive search. Also some previously undecided cases of amicable pairs aredemonstrated to be non-existent after a complete search of the equivalence classes for orthogonaldesigns.2000 Mathematics subject classification (Amer. Math. Soc.): 05B30, 15A36.1 MotivationComplex orthogonal space-time block codes(STBCs) based on Amicable Orthogonal Designs(AODs)[1] are known for relatively simple receiver structure and minimum processing delay when dealingwith complex signal constellations. The simplest complex STBCs is an Alamouti code [8] for twotransmit antennas, which is based on an amicable orthogonal design of order 2. The complex STBCsfor more than two transmit antennas cannot achieve rate one [7], but they can still provide fulldiversity. To date, only the complex STBCs based on full(i.e., no zeros) designs of order two and fourhave been proposed [5][8]. The known complex STBCs for higher number of transmit antennas, e.g.eight, have zeros in the code matrices that results in the waste of several time slots when no usefulinformation is being transmitted from a given antenna. Moreover, those zeros make implementationof the transmitter rather difficult at high symbol rates, as it requires the transmitter to be switchedon and off frequently. It has been shown that by utilizing some amicable pairs, especially the newAODs(8; 2, 2, 2, 2; 2, 2, 2, 2) which is the only full design with 4 variables of order 8, the new complexcodes have much better error performance than the conventional ones [9][10].2 Introduction and Basic DefinitionsDefinition 2.1 Let x1, x2, · · · , xt be commuting indeterminates. An orthogonal design X of ordern and type (s1, s2, · · · , st) denoted OD(n; s1, · · · , st), where si are positive integers, is a matrix oforder n with entries from {0,±x1, · · · , ± xt}, such thatXXT = (t∑i=1six2i )In,where XT denotes the transpose of X and In is the identity matrix of order n.Alternatively, each row of X has si entries of the type ±xi and the rows are pairwise-orthogonal underthe Euclidean inner product [1]. The above description of X applies to the columns of X as well.Definition 2.2 Let X be an OD(n;u1, · · · , us) on the variables {x1, · · · , xs} and Y an OD(n; v1, · · · , vt)on the variables {y1, · · · , yt}. It is said that X and Y are amicable orthogonal designs AODs(n;u1, · · · , us;v1, · · · , vt) if XY T = Y XT .In the complex multiple antenna communication systems, each transmitted complex symbol containsboth real and imaginary part. Hence, the concept of complex orthogonal designs used for STBCs isdifferent from the definition introduced in Geramita-Geramita [2].Definition 2.3 A complex orthogonal design for STBC (CODSTBC, throughout this paper referredas COD for short) Z of order n is an n × n matrix on the complex indeterminates s1, · · · , st, withentries chosen from 0,±s1, · · · ,±st, their conjugates ±s∗1, · · · ,±s∗t , or their product with i =√−1such thatZHZ = (t∑i=1|si|2)In, (1)where ZH denotes the conjugate transpose of Z and In is the identity matrix of order n.Given an orthogonal design over s variables, we can get new designs of the same order but differenttypes by setting variables equal to one another or zero. The concept of “Equating and Killing variables”was first stated as Lemma 4.4 in [1].Definition 2.4 Let X be an OD(n; u1, · · · , us) on the variables {x1, · · · , xs}, then there existsOD(n;u1, · · · , ui + uj , · · · , us) and OD(n; u1, · · · , ui−1, ui+1, · · · , us) on s− 1 variables.The description of Equating and Killing variables applies as well to amicable orthogonal designs.Example 2.1 Given an AODs(8; 1, 1, 2, 2; 1, 1, 2, 2), if we set u′3=u3 +u4 for one of the amicable pair,then we get a new AODs(8; 1, 1, 4; 1, 1, 2, 2) with same order but different type. We can also get newdesigns AODs(8; 1, 1, 2; 1, 1, 2, 2), AODs(8; 1, 2, 2; 1, 1, 2, 2) by killing variables.The existence and non-existence results for AODs are established by using the theory of quadraticforms. For future reference, a detailed description about the linkage between quadratic forms andamicable orthogonal designs can be found in Street’s Thesis [3].3 Theorems and ExamplesIn this section, two previously known theorems will be quoted for constructing AODs. The firstcontribution is due to Wolfe [4] and the second one can be found in Street’s thesis [3]. An example ofStreet’s theorem is given followed by a new construction theorem.Theorem 3.1 (Wolfe, 1975). If there exist a pair of AODs (n; a1, ..., as; b1, ..., bt) and a pair of AODs(m; c1, ..., cu; d1, ..., dv), then there exists a pair of AODs (mn; b1c1, ..., b1cu−1, a1cu, ..., ascu; b1d1, ..., b1dv,b2cu, ..., btcu).The observation of Wolfe’s theorem leads to the following corollary.Corollary 3.1 If there exist a pair of amicable orthogonal designs AODs(n; a1, ..., as; b1, ..., bt), thenthere exists a pair of amicable orthogonal designs of typea)AODs(2n; a1, ..., as, as; b1, ..., bt, as),b)AODs(2n; a1, a1, 2a2..., 2as; 2b1, ..., 2bt),c)AODs(2n; a1, a1, a2..., as; b1, ..., bt).Proof. Let X =∑si=1 Aixi and Y =∑tj=1 Bjyj are the amicable designs in order n.a) Let one pair be AODs(2; 1, 1; 1, 1) in Theorem 2.1.b) Let weighing matrices M = 0 1− 0, N = 1 11 − and construct the matricesP = (A1 ⊗ I2)p1 + (A1 ⊗M)p2 +s∑i=2(Ai ⊗N)pi+1 (2)andQ =t∑j=1(Bj ⊗N)qj (3)c) Same as b), only set N = 1 00 − .Here ⊗ denotes the Kronecker product and the pi’s and qi’s are distinct commuting indeterminates.It’s obvious that all the coefficient matrices Pi’s and Qi’s satisfy the conditions in (2) because theweighing matrices M and N have the following properties: M = −M t, N = N t, and MN t = NM t.¤Theorem 3.2 (Street, 1981). Suppose (A,B) and (C, D) are both AODs (n; a1, ..., as; b1, ..., bt).Suppose further that there exists a weighing matrix W (n, k) such thatAWT = WCT , BWT = −WDTThen there exist AODs (2n; k, a1, ..., as; k, b1, ..., bt).Example 3.1 Given W (4, 4), AODs(4; 1, 1, 1; 1, 1, 1), we use Theorem 3.2 to construct AODs(8; 1, 1, 1, 4;1, 1, 1, 4). We haveA =x1 0 x3 x20 x1 −x2 x3−x3 x2 x1 0−x2 −x3 0 x1; B =y1 0 y2 y30 y1 y3 −y2y2 y3 −y1 0y3 −y2 0 −y1, C =x1 x2 x3 0−x2 x1 0 x3−x3 0 x1 −x20 −x3 x2 x1;D =−y3 −y2 −y1 0−y2 y3 0 −y1−y1 0 y3 y20 −y1 y2 −y3and W =1 1 1 11 − 1 −1 1 − −1 − − 1.The above matrices satisfy the conditions of Theorem 3.2 and give AODs(8; 1, 1, 1, 4; 1, 1, 1, 4). Suchan amicable pair can be used to construct COD(8; 1, 1, 1, 4; 1, 1, 1, 4) for 8 transmitter antennas com-munication system. The pair of AODs is given below (we use x̄1 for −x1, x̄2 for −x2 and so on).X :x1 0 x3 x2 x4 x4 x4 x40 x1 x̄2 x3 x4 x̄4 x4 x̄4x̄3 x2 x1 0 x4 x4 x̄4 x̄4x̄2 x̄3 0 x1 x4 x̄4 x̄4 x4x̄4 x̄4 x̄4 x̄4 x1 x2 x3 0x̄4 x4 x̄4 x4 x̄2 x1 0 x3x̄4 x̄4 x4 x4 x̄3 0 x1 x̄2x̄4 x4 x4 x̄4 0 x̄3 x2 x1; Y :y1 0 y2 y3 y4 y4 y4 y40 y1 y3 ȳ2 y4 ȳ4 y4 ȳ4y2 y3 ȳ1 0 y4 y4 ȳ4 ȳ4y3 ȳ2 0 ȳ1 y4 ȳ4 ȳ4 y4y4 y4 y4 y4 ȳ3 ȳ2 ȳ1 0y4 ȳ4 y4 ȳ4 ȳ2 y3 0 ȳ1y4 y4 ȳ4 ȳ4 ȳ1 0 y3 y2y4 ȳ4 ȳ4 y4 0 ȳ1 y2 ȳ3Let {Xi} and {Yi} be coefficient matrices and {si = sRi + isIi }4i=1 be a set of complex symbols. thenthe corresponding complex orthogonal space-time block code, by using equation Z =∑4i=1 XisRi +i∑4i=1 YisIi , iss1 0 sR3 + isI2 sR2 + isI3s42s42s42s420 s1 −sR2 + isI3 sR3 − isI2 s42 −s42s42− s42−sR3 + isI2 sR2 + isI3 s∗1 0 s42s42− s42− s42−sR2 + isI3 −sR3 − isI2 0 s∗1 s42 −s42− s42s42− s∗42− s∗42− s∗42− s∗42sR1 − isI3 s∗2 sR3 − isI1 0− s∗42s∗42− s∗42s∗42−s2 sR1 + isI3 0 sR3 − isI1− s∗42− s∗42s∗42s∗42−sR3 − isI1 0 sR1 + isI3 −s∗2− s∗42s∗42s∗42− s∗420 −sR3 − isI1 s2 sR1 − isI3We now give a new theorem and all six new amicable designs derived from this theorem can be foundin Table 4.3.Theorem 3.3 If there exist a pair of amicable orthogonal designs AODs(n; a1, ..., as; b1, ..., bt), thenthere exists a pair of amicable orthogonal designs AODs(2n; e1a1, e2a2..., esas; k1b1, k2b2, ..., ktbt) forevery ei, kj ∈ {1, 2}.Proof. Let X =∑si=1 Aixi and Y =∑tj=1 Bjyj are the amicable designs in order n.Let C1 = 1 00 1 and C2 = 1 11 −, they have the following properties: C1Ct1 = I2, C2Ct2 = 2I2,and C1Ct2 = C2Ct1. Construct the matricesP =s∑i=1(Ai ⊗ Cei)pi (4)andQ =t∑j=1(Bj ⊗ Ckj )qj (5)¤Some new amicable orthogonal designs including AODs(8; 2, 2, 2, 2; 2, 2, 2, 2) and AODs(8; 1, 2, 2, 2; 1, 1,1, 2) of order 8 are found as part of a complete search of the equivalence classes for orthogonal designs.A detailed description about the search method can be found in [11].Example 3.2 For the AODs(8; 2, 2, 2, 2; 2, 2, 2, 2) given in [11], the corresponding COD(8; 2, 2, 2, 2; 2, 2,2, 2) is a full(i.e., no zeros) complex orthogonal design which results in no wasted time slots and efficientimplementation.s1√2s1√2s2√2s2√2s3√2s4√2s3√2s4√2s1√2− s1√2s2√2− s2√2s∗4√2− s∗3√2s∗4√2− s∗3√2s∗2√2s∗2√2− s∗1√2− s∗1√2s3√2s4√2− s3√2− s4√2s∗2√2− s∗2√2− s∗1√2s∗1√2s∗4√2− s∗3√2− s∗4√2s∗3√2−sR4 +isI3√2−sR3 +isI4√2−sR4 +isI3√2−sR3 +isI4√2sR2 −isI1√2sR2 −isI1√2sR1 −isI2√2sR1 −isI2√2−sR3 −isI4√2sR4 +isI3√2−sR3 −isI4√2sR4 +isI3√2sR2 −isI1√2−sR2 +isI1√2sR1 −isI2√2−sR1 +isI2√2−sR4 +isI3√2−sR3 +isI4√2sR4 −isI3√2sR3 −isI4√2sR1 +isI2√2sR1 +isI2√2−sR2 −isI1√2−sR2 −isI1√2−sR3 −isI4√2sR4 +isI3√2sR3 +isI4√2−sR4 −isI3√2sR1 +isI2√2−sR1 −isI2√2−sR2 −isI1√2sR2 +isI1√24 Amicable Pairs with Order EightGiven an AODs(8; u1, · · · , us; v1, · · · , vt), then the maximum achievable value for s + t is 8 accordingto Corollary 5.32 in [1]. We summarize all existence and non-existence results of AODs which areof 4-4-variables type, 3-4-variables type or 3-3-variables type in Table 4.1, Table 4.2 and Table 4.3respectively. In these tables? denotes such an AODs is newly constructed (remains undecided in Street’s paper [6]),× denotes such an AODs does not exist,⊗ denotes such an AODs is newly found non-existent (remains undecided in Street’s paper [6]),U denotes such an AODs remains unknown,¦ denotes such an AODs can be constructed using Equating and Killing variables concept,C denotes such an AODs can be constructed using Corollary 3.1,T1 denotes such an AODs can be constructed using Theorem 3.1 (Wolfe),T2 denotes such an AODs can be constructed using Theorem 3.2 (Street),T3 denotes such an AODs can be constructed using Theorem 3.3.S denotes such an AODs is found by exhaustive search.1111 1114 1122 2222 1112 1124 1222 1113 1223 1115 1123 11331111 C ⊗ ⊗ × × × × × × × × ×1114 T2 U × × × × × × × × ×1122 T1 × × × × × × × × ×2222 ?S × × × × × × × ×1112 T1 × ?S × × × × ×1124 × × × × × × ×1222 U × × × × ×1113 U × × × ×1223 × × × ×1115 × × ×1123 ⊗ ×1133 ×Table 4.1 4-4-variables type of AODs1111 1114 1122 2222 1112 1124 1222 1113 1223 1123 1115 1133111 ¦ ¦ U × ¦ × U U × × × ×114 ⊗ ¦ ?¦ × U × U U × × × ×122 ⊗ U ¦ × ¦ × U U × × × ×224 × × U C × C U × × × × ×112 ¦ U ¦ × ¦ × ?S × × ⊗ × ×124 ⊗ ¦ U × U × U × × ⊗ × ×222 ⊗ U ¦ ?¦ U U U × × ⊗ × ×113 × × × × ¦ × U U × ⊗ × ×134 × × × × × × × × × × × ×223 × × × × × × U × × ⊗ × ×123 ⊗ U ¦ × × × × U × ⊗ × ×115 ⊗ ¦ U × × × × × U ⊗ × ×133 × × × × × × × × × T1 × ×116 × × × × × × × × × × × ×125 × × × × × × × × × × × ×233 × × × × × × × × × × × ×Table 4.2 3-4-variables type of AODs111 114 122 224 112 124 222 113 223 134 123 115 133 116 125 233111 ¦ ¦ T3 × T3 ¦ ?T3 ¦ U × U ¦ U × × ×114 T3 ?T3 T3 T3 T3 T3 U U × ¦ ¦ U × × U122 T3 ?T3 T3 ?T3 T3 ¦ U × ¦ U U × × U224 T3 T3 T3 T3 × U ¦ U × U ¦ ¦ C112 T3 ?T3 T3 ¦ ?¦ × ¦ U ¦ × × U124 T3 ?T3 U U × U ¦ ¦ × × U222 T3 U U U ¦ U U U U U113 ¦ U × U U ¦ × × ×223 U × U U ¦ × × U134 × × × × × × ×123 ¦ U ¦ × × U115 ¦ ¦ × × ×133 ¦ × × ×116 × × ×125 × ×233 UTable 4.3 3-3-variables type of AODsIt is worth mentioning that amicable orthogonal designs are also important in the construction of or-thogonal designs. Using our new designs AODs(8; 2, 3; 1, 1, 1, 3) given in [11], AODs(8; 1, 1, 2; 1, 2, 2, 2)and Theorem 5.97 in [1] gives us the following new orthogonal designs in order 32: OD(32; 1, 1, 2, 2, 2, 2,3, 6, 6), OD(32; 2, 2, 2, 3, 7, 9), OD(32; 1, 1, 1, 1, 2, 2, 2, 3, 6) and OD(32; 1, 1, 1, 2, 2, 2, 3, 9).References[1] A.V.Geramita and J.Seberry, Orthogonal Designs,Quadratic Forms and Hadamard Matrices, ser.Lecture Notes in Pure and Applied Mathematics. Marcel Dekker, New York and Basel, vol.43,1979.[2] A.V.Geramita and J.M.Geramita, Complex orthogonal designs, J.Combinatorial Theory,ser.A,vol.25, pp.211-225, 1978.[3] D.J.Street, Cyclotomy and designs, Ph.D.Dissertation, University of Sydney, N.S.W, 1981.[4] W.W.Wolfe, Orthogonal designs-amicable orthogonal designs-some algebraic and combinatorialtechniques, Ph.D.Dissertation, Queen’s University, Kingston, Ontario,1975.[5] V.Tarokh, H.Jafarkhani, and A.R.Calderbank, Space-time block codes from orthogonal designs,IEEE Trans.Inform.Theory, vol.45, pp.1456-1467, July 1999.[6] D.J.Street, Amicable orthogonal designs of order eight, Journal of Australian Mathematical So-ciety(A), 33(1982), pp.23-29.[7] X.B.Liang, Orthogonal designs with maximal rates, IEEE Trans. Inform.Theory, Vol.49, No.10,pp.2468-2503, Oct.2003.[8] S.M.Alamouti, A simple transmit diversity scheme for wireless communications, IEEE J. Select.Areas Commun., Vol.16, pp.1451-1458, Oct.1998.[9] L.C.Tran, J.Seberry, B.J.Wysocki, T.A.Wysocki, T.Xia and Y.Zhao, Two new complex orthog-onal space-time codes for 8 transmit antennas, IEE Electronics Lett., Vol.40, No.1, pp.55-56,Jan.2004.[10] J.Seberry, L.C.Tran, Y.Wang, B.J.Wysocki, T.A.Wysocki, T.Xia and Y.Zhao, New complex or-thogonal space-time block codes of order eight, Signal Processing for Telecommunications andMultimedia, Ser.Multimedia Systems and Applications., Vol.27, pp.173-182, Springer, New York,2004.[11] Y.Zhao, Y.Wang and J.Seberry, On amicable orthogonal designs of order 8, Australasian. J.Combin., (to appear).

Amicable Orthogonal Designs of Order 8 for Complex Space-Time Block Codes

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Amicable Orthogonal Designs of Order 8 for Complex Space-Time Block Codes

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