Maximum Rate of Unitary-Weight, Single-Symbol Decodable STBCs

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1 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. XX, NO. XX, XXXX 1 Mximum Rte of Unitry-Weight, Single-Symbol Decodble STBCs Snjy Krmkr, K. Pvn Srinth nd B. Sundr Rjn, Senior Member, IEEE rxiv: v1 [cs.it] 13 Jn 011 Abstrct It is well known tht the Spce-time Block Codes (STBCs from Complex orthogonl designs (CODs re single-symbol decodble/symbol-by-symbol decodble (SSD. The weight mtrices of the squre CODs re ll unitry nd obtinble from the unitry mtrix representtions of Clifford Algebrs when the number of trnsmit ntenns n is power of. The rte of the squre CODs for n = hs been shown to be +1 complex symbols per chnnel use. However, SSD codes hving unitry-weight mtrices need not be CODs, n exmple being the Minimum-Decoding-Complexity STBCs from Qusi-Orthogonl Designs. In this pper, n chievble upper bound on the rte of ny unitry-weight SSD code is derived to be complex 1 symbols per chnnel use for ntenns, nd this upper bound is lrger thn tht of the CODs. By wy of code construction, the interreltionship between the weight mtrices of unitry-weight SSD codes is studied. Also, the coding gin of ll unitry-weight SSD codes is proved to be the sme for QAM constelltions nd conditions tht re necessry for unitry-weight SSD codes to chieve full trnsmit diversity nd optimum coding gin re presented. Index Terms Anticommuting mtrices, Complex orthogonl designs, Minimum-Decoding-Complexity codes, Qusiorthogonl designs, Spce-time block codes. I. INTRODUCTION Spce-Time Block Codes from Complex Orthogonl Designs [1] re populr becuse they offer full trnsmit diversity for ny rbitrry signl constelltion nd lso re single-symbol decodble. In fct, CODs re single-rel-symbol decodble for constelltions such s the rectngulr QAM, which cn be expressed s Crtesin product of two PAM constelltions, while for constelltions such s PSK, CODs re single-complex-symbol decodble. The weight mtrices, lso clled liner dispersion mtrices [] (refer Subsection II-A for definition of weight mtrices, of the squre CODs re ll unitry, nd detiled construction method to obtin these weight mtrices from irreducible mtrix representtions of Clifford Algebrs hs been presented in [3] for trnsmit ntenns. It hs lso been shown tht the mximum rte of the squre CODs for trnsmit ntenns is +1 complex This work ws prtly supported by the DRDO-IISc Progrm on Advnced Reserch in Mthemticl Engineering nd by the Council of Scientific & Industril Reserch (CSIR, Indi, through Reserch Grnt ((0365/04/EMR- II to B.S. Rjn. Snjy Krmkr is with the Electricl nd Computer Science Deprtment t the University of Colordo t Boulder. Prt of this work ws crried out when he ws t the Indin Institute of Science, Bnglore K. Pvn Srinth nd B. Sundr Rjn re with the Deprtment of Electricl Communiction Engineering, Indin Institute of Science, Bnglore emil:bsrjn@ece.iisc.ernet.in. Different prts of the content of this pper ppered in the Proc. of IEEE Interntionl Symposium on Informtion on Informtion Theory (ISIT 006, Settle, Wshington, July 09-14, 006. symbols per chnnel use. Although rectngulr CODs [4] offer higher rte, they re not dely efficient, mking squre CODs more ttrctive in prctice. In generl, single-complex-symbol decodble codes need not be CODs. Throughout this pper, unless otherwise mentioned, SSD codes refer to single-complex-symbol decodble codes. The Co-ordinte Interleved Orthogonl Designs (CIODs [5] hve been shown to be SSD codes, while offering full trnsmit diversity for specific complex constelltions only. However, the CIODs hve non-unitry-weight mtrices. SSD codes, tht include unitry-weight codes nd rectngulr designs, hve been reported in [6], [7] nd re populrly known s Minimum-Decoding-Complexity codes from Qusiorthogonl designs (MDCQODs. The rtes of both the CIODs nd the clss of codes reported in [7] for trnsmit ntenns hve been shown to be complex symbols per chnnel use. 1 In [7], the mximum rte of the MDCQODs hs been reported, nd this rte includes tht for rectngulr designs. However, the mximum rte of generl SSD codes hs not been reported so fr in the literture, to the best of our knowledge. In this pper, we mke the following contributions. We derive n upper bound on the rte of unitry-weight SSD codes for trnsmit ntenns. This upper bound is found to be 1 complex symbols per chnnel use. We give generl construction method to obtin codes tht meet this upper bound nd further show the interreltionship between the weight mtrices of generl unitryweight SSD codes. All known unitry-weight SSD codes including squre MDCQODs re specil cses of this construction. We prove tht ll unitry-weight SSD codes hve the sme coding gin nd specificlly for QAM constelltions, we provide the ngle of rottion tht ensures full trnsmit diversity nd optimum coding gin for ll unitry-weight SSD codes. The orgniztion of the pper is s follows. Section II gives the system model nd relevnt definitions. Section III introduces the notion of normliztion nd its use in the nlysis of unitry-weight SSD codes. Section IV provides the upper bound on the rte of unitry-weight SSD codes nd the structure of generl unitry-weight SSD codes. Diversity conditions, coding gin clcultions for QAM nd simultion results re given in Section V, Subsections V-A nd V-B, respectively. Discussions on the direction for future reserch constitute Section VI. Nottions: R nd C denote the field of rel nd complex numbers, respectively nd j represents 1. GL(n,C denotes the group of invertible mtrices of size n n with com-

2 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. XX, NO. XX, XXXX plex entries. For ny complex mtrix A, tr(a, A, A H nd det(a represent the trce, the Frobenius norm, the Hermitin nd the determinnt of A, respectively. I n nd O n represent the n n identity mtrix nd the zero mtrix, respectively. For complex rndom vrible X, X N C (0,N denotes tht X hs complex norml distribution with men 0 nd vrince N. For complex vrible x, x I nd x Q represent the rel nd the imginry prts of x, respectively, nd x denotes the bsolute vlue of x. For set S, S denotes the crdinlity of S. II. SYSTEM MODEL We consider Ryleigh qusi-sttic flt-fding MIMO chnnel with full chnnel stte informtion (CSI t the receiver nd no CSI t the trnsmitter. We ssume MIMO system with n trnsmit ntenns nd m receive ntenns. Since we re considering only squre STBCs in this pper, the number of time slots is lso n. The chnnel model is Y = SH +N, where S C n n is the codeword mtrix, trnsmitted over n chnnel uses, N C n m is complex white Gussin noise mtrix with i.i.d. entries N C (0,N 0, H C n m is the chnnel mtrix with the entries ssumed to be i.i.d. circulrly symmetric Gussin rndom vribles N C (0,1 nd Y C n m is the received mtrix. Definition 1: (STBC A spce-time block code S is set of complex mtrices clled codeword mtrices. For system with n trnsmit ntenns, codeword mtrix is T n mtrix, where T is the number of time slots (T = n in this pper nd the (i,j th entry of the codeword mtrix refers to the signl trnsmitted by the j th trnsmit ntenn in the i th time slot. Definition : (Code rte If there re k independent complex informtion symbols in the codeword which re trnsmitted over T chnnel uses, then, the code rte is defined to be k/t complex symbols per chnnel use. For instnce, for the Almouti code, k = nd T =. So, its code rte is 1 complex symbol per chnnel use. Definition 3: (Full-Diversity Code An STBC encoding symbols chosen from constelltion A is sid to offer fulldiversity iff for every possible codeword pir (S,Ŝ, with S Ŝ, the codeword difference mtrix S Ŝ is full-rnked [9]. In generl, whether code offers full-diversity or not depends on the constelltion tht it employs. A code cn offer full diversity for certin complex constelltion A but not for nother complex constelltion. The CODs re specil in this spect since they offer full-diversity for ny rbitrry complex constelltion. Definition 4: (Coding Gin The coding gin δ of n STBC is defined s δ = min S Ŝ,S Ŝ ( r λ i 1 r, where λ ( i,i = 1,,,r, re the non-zero eigenvlues of the H ( mtrix S Ŝ S Ŝ nd r is the minimum of the rnk of ( H ( S Ŝ S Ŝ for ll possible codeword pirs (S,Ŝ, S Ŝ. If the code offers full-diversity for constelltion A, then, the coding gin is δ 1 n min, where δ min is the minimum of ( H ( the determinnt of the mtrix S Ŝ S Ŝ mong ll possible codeword mtrix pirs (S,Ŝ, with S Ŝ. A. Single-Symbol Decodble Codes In this subsection, we formlly define nd clssify liner SSD codes. Any n n codeword mtrix S of liner dispersion STBC S with k complex informtion symbols x 1,x,,x k cn be expressed s S = k (x ii A ii +x iq A iq, (1 wherex i = x ii +jx iq, 1 i k, tke vlues from complex constelltion A. Then, S, i.e., the number of codewords, is A k. The set of n n complex mtrices {A ii,a iq },1 i k, clled weight mtrices defines. Notice tht in (1, ll thek weight mtrices re required to form linerly independent set over R, since we re trnsmitting k independent informtion symbols. Assuming tht perfect chnnel stte informtion (CSI is vilble t the receiver, the mximum likelihood (ML decision rule minimizes the metric, M(S tr((y SH H (Y SH = Y SH. ( Since there re A k different codewords, in generl, ML decoding requires A k computtions, one for ech codeword. Suppose the set of weight mtrices re chosen such tht the decoding metric ( could be decomposed s p M(S = f j (x (j 1q+1,x (j 1q+,,x (j 1q+q, j=1 which is sum of p positive terms, ech involving exctly q complex vribles only, where pq = k. Then, decoding requires p j=1 A q = p A q computtions nd the code is clled q-symbol decodble code [10]. The cse q = 1 corresponds to SSD codes tht include the well known CODs s proper subclss, nd hve been extensively studied [1], [3], [5], [6], [7], [8]. The codes corresponding to q =, re clled Double-Symbol-Decodble (DSD codes. The Qusi- Orthogonl Designs studied in [11], [1], [13] re proper subclsses of DSD codes. Definition 5: [1] A squre complex orthogonl design S for n trnsmit ntenns is set of codeword mtrices of size n n, with ech codeword mtrix S stisfying the following conditions: the entries of S re complex liner combintion of x 1,x,,x k nd their complex conjugtes x 1,x,,x k. (Orthonormlity: S H S = ( x x k I n holds for ny complex vlues for x i,i = 1,,,k.

3 SANJAY et l.: MAXIMUM RATE OF UNITARY-WEIGHT, SINGLE-SYMBOL DECODABLE STBCS 3 A set of necessry nd sufficient conditions for S to be COD is [1], [3] A H iia ii = A H iqa iq = I n, i = 1,,,k; (3 for 1 i j k, nd A H ii A jq +A H jq A ii = O n, A H ii A ji +A H ji A ii = O n, A H iq A jq +A H jq A iq = O n, (4 (4b (4c A H ii A iq +A H iq A ii = O n, i = 1,,,k. (5 STBCs obtined from CODs [1], [3] re SSD like the well known Almouti code [8], nd stisfy (3, (4 nd (5. For S to be SSD, it is not necessry tht it stisfies (3 nd (5, i.e., it is sufficient tht it stisfies only (4 - this result ws shown in [5]. Since then, different clsses of SSD codes tht re not CODs hve been studied by severl uthors, [5], [6], [7]. SSD codes cn be systemticlly clssified s follows. 1 Liner STBCs stisfying (3, (4 nd (5 re CODs. Liner STBCs stisfying (4 re clled SSD codes. These my or my not stisfy (3 nd (5. 3 Liner STBCs stisfying (3 nd (4 nd not stisfying (5 re clled Unitry-Weight SSD codes. 4 Liner STBCs stisfying (4 nd not stisfying (3 re clled Non-Unitry-weight SSD codes. These my or my not stisfy (5. The codes discussed in [5], which re clled CIODs, constitute n exmple clss of Non-unitry-weight SSD codes. The clsses of codes studied in [7] re unitry-weight SSD codes. The clsses of codes studied in [6], clled Minimum Decoding Complexity codes from Qusi-Orthogonl Designs (MDCQOD codes, include some unitry-weight SSD codes s well s non unitry-weight SSD codes. The notion of SSD codes hve been extended to coding for MIMO-OFDM systems in [14], [15] nd recently, lowdecoding complexity codes clled -group nd 4-group decodble codes [16], [17], [18], [19] nd SSD codes [0] in prticulr hve been studied for use in coopertive networks s distributed STBCs. III. UNITARY-WEIGHT SSD CODES In this section, we nlyze the structure of the weight mtrices of unitry-weight SSD codes. We mke use of the following lemm in our nlysis. Lemm 1: Let S = {S S = k x iia ii +x iq A iq,x i A} be unitry-weight STBC nd consider the STBC S U {US S S}, whereu is ny unitry mtrix. Then,S U is SSD iff S is SSD. Further, both the codes hve the sme coding gin for the constelltion A. Proof: The proof is strightforwrd. For the STBC S U, the weight mtrices re UA 1I, UA iq, i = 1,,,k. It is esy to verify tht if the mtrices A ii, A iq, i = 1,,,k stisfy (4, then, the mtrices UA 1I, UA iq, i = 1,,,k lso stisfy (4 nd vice-vers. Further, for ny pir of distinct codeword mtrices S nd Ŝ, the eigenvlues of (S ŜH (S ŝ re the sme s tht of (US UŜH (US UŜ, mking the coding gin the sme for both the STBCs. The STBCs S nd S U re sid to be equivlent. To simplify our nlysis of unitry-weight codes, we mke use of normliztion s described below. Let S be unitry-weight STBC nd let its codeword mtrix S be expressed s S = k (x ii A ii +x iq A iq. Consider the code S N {A H 1IS S S}. Clerly, from Lemm 1, S N is equivlent to S. The weight mtrices of S N re A ii = A H A iq = A H 1I A ii, 1I A iq. With this, codeword mtrix S N of S N cn be written s k S N = x 1I I n +x 1Q A 1Q + (x ii A ii +x iq A iq. i= We cll the code S N to be the normlized code of S. In generl, ny unitry-weight SSD code with one of its weight mtrices being the identity mtrix is clled normlized unitry-weight SSD code. Studying unitry-weight SSD codes becomes simpler by studying normlized unitry-weight SSD codes. Also, n upper bound on the rte of unitry-weight SSD codes is the sme s tht of the normlized unitryweight SSD codes. For the normlized unitry-weight SSD code trnsmitting k symbols in n chnnel uses, the conditions presented in (3 nd (4 cn be rewritten s A H ii = A ii (equivlently, A ii = I n, (6 A H iq = A iq (equivlently, A iq = I n, (7 A H 1Q A ii = A ii A 1Q, (8 A H 1Q A iq = A iq A 1Q, (9 for i =,,k, nd A ii A ji = A ji A ii, (10 A iq A jq = A jq A iq, (11 A ii A jq = A jq A ii, (1 for i j k. So, every weight mtrix except A 1I = I n nd A 1Q should squre to I n. Shown below is the grouping of weight mtrices (We will lter show tht A iq = ±A 1Q A ii, i =, k. A 1I = I n A I A ki A 1Q A Q A kq Except I n, the elements in the first row should mutully nticommute nd lso squre to I n. From (8 nd (9, it is cler tht if A 1Q = I n, then, A 1Q should nticommute with ll the weight mtrices except A 1I = I n. So, the upper bound on the rte of unitry-weight SSD code is determined by the number of mutully nticommuting unitry mtrices. The following section dels with determining the upper bound. IV. AN UPPER BOUND ON THE RATE OF UNITARY-WEIGHT SSD CODES In this section, we determine the upper bound on the rte of unitry-weight SSD codes nd lso give generl construction

4 4 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. XX, NO. XX, XXXX scheme to obtin codes meeting the upper bound. To do so, we mke use of the following lemms regrding mtrices of size n n. Lemm : [1] Consider n n mtrices with complex entries. 1 If n = n 0, with n 0 odd, then there re l elements of GL(n,C tht nticommute pirwise if nd only if l +1. If n = nd mtrices F 1,,F nticommute pirwise, then the set of products F i1 F i F is with 1 i 1 < < i s long with I n forms bsis for the dimensionl spce of ll n n mtrices over C. In ech cse Fi is scled identity mtrix. Proof: Avilble in [1]. Let F 1,,F be nticommuting, nti-hermitin, unitry mtrices (so tht Fi = I n, i = 1,,,. The following two lemms re pplicble for such mtrices. Lemm 3: The product F i1 F i F is with 1 i 1 < < i s squres to ( 1 s(s+1 I n. Proof: (F i1 F i F is (F i1 F i F is = ( 1 s 1 (Fi 1 F i F is (F i F i3 F is = ( 1 s 1 ( 1 s (Fi 1 Fi F is (F i3 F i4 F is = ( 1 [(s 1+(s + 1] (Fi 1 Fi Fi s = ( 1 s(s 1 ( 1 s I n = ( 1 s(s+1 I n. This proves the lemm. Lemm 4: Let Ω 1 = {F i1,f i,,f is } nd Ω = {F j1,f j,,f jr } with 1 i 1 < < i s nd 1 j 1 < < j r. Let Ω 1 Ω = p. Then the product mtrix F i1 F i F is commutes with F j1 F j F jr if exctly one of the following is stisfied, nd nticommutes otherwise. 1 r,s nd p re ll odd. The product rs is even nd p is even (including 0. Proof: If F jk Ω 1 Ω, we note tht (F i1 F i F is F jk = ( 1 s 1 F jk (F i1 F i F is, nd if F jk / Ω 1 Ω, (F i1 F i F is F jk = ( 1 s F jk (F i1 F i F is. Now, (F i1 F i F is (F j1 F j F jr = ( 1 p(s 1 ( 1 (r ps (F j1 F j F jr (F i1 F i F is = ( 1 rs p (F j1 F j F jr (F i1 F i F is. cse 1. Since r,s nd p re ll odd, ( 1 rs p = 1. cse. The product rs is even nd p is even (including 0. So, ( 1 rs p = 1. From Lemm, the mximum number of pirwise nticommuting mtrices of size is + 1. Hence, the mximum possible vlue of k, i.e., the number of complex informtion symbols, is +, since we lso consider I n s weight mtrix. In order to provide n chievble upper bound on the rte of unitry-weight SSD codes, we first ssume tht the cse k = + is possibility. Denoting the + 1 nticommuting mtrices by F 1, F,,F +1, we note from Lemm tht the set {F λ1,λ i {0,1},i = 1,,} is bsis for C over C. Therefore, {F λ1,jfλ1,λ i {0,1},i = 1,,} is bsis for C over R. It cn be checked by pplying Lemm 4 tht the only product mtrix tht nticommutes with F 1,F,, nd F is cf 1 F F, where c C. So, it must be tht F +1 = cf 1 F F, c C. For our construction, we need nticommuting, nti- Hermitin, unitry mtrices (so tht they squre to I n. An excellent tretment of irreducible mtrix representtions of Clifford lgebrs is given in [3] nd the sme pper lso presents n lgorithm to obtin +1 pirwise nticommuting mtrices tht ll squre to I n (n =. In fct, these mtrices re precisely the weight mtrices (except I n of squre CODs. As mentioned before, we denote them by F 1, F,,F +1, with F +1 = cf 1 F F, nd { ±j if (F1 F c = F = I n, ±1 otherwise. It must be noted tht the mtrices obtined from [3] re not unique, i.e., these re not the only set of mutully nticommuting, nti-hermitin, unitry mtrices of size. It cn be noted by pplying Lemm 3 tht (F 1 F F = I n when is odd. We re now redy to prove the min result of the pper. Theorem 1: The rte k of unitry-weight SSD code is upper bounded s k = Proof: We prove the theorem 1. in three prts s follows. Clim 1: k +. To prove this, let us first suppose tht k = +, in which cse, we hve the following grouping scheme. I n F 1 F F +1 A 1Q A Q A 3Q A (+Q Let A iq = j=1 i,jf λ1,j 1 F λ,j F λ,j, λ m,j {0,1}, m = 1,,,, i = 1,,, + nd i,j C. This is possible becuse of Lemm. Considering A Q, since A Q nticommutes with F, F 3, nd F +1, every individul term of A Q must nticommute with F, F 3, nd F +1. So, we look for ll possible cndidtes from the set {F λ1,λ i {0,1},i = 1,,} which nticommute with F, F 3, nd F +1. By pplying Lemm 4, the only possible choice is F 1. Since the weight mtrices re required to be independent over R nd in view of the condition in (7, there is no vlid possibility for A Q. As result, unitry-weight SSD code with + independent complex symbols does not exist. Clim : k +1. To prove this, we ssume tht k = + 1 is possibility, in which cse, we hve the following grouping of weight mtrices.

5 SANJAY et l.: MAXIMUM RATE OF UNITARY-WEIGHT, SINGLE-SYMBOL DECODABLE STBCS 5 I n F 1 F F A 1Q A Q A 3Q A (+1Q Considering A Q, ech of the terms tht A Q is liner combintion of should nticommute with F, F 3, nd F. The only possibilities from the set {F λ1,λ i {0,1},i = 1,,} re F 1 nd F 1 F F = cf +1,c = ±j or ±1. Therefore, A Q =,1 F 1 +, F 1 F F. Next, considering A 3Q, the only elements nticommuting with F 1, F 3, nd F re F nd F 1 F F. Therefore, A 3Q = 3,1 F + 3, F 1 F F. Since A Q should lso nticommute with A 3Q, either, = 0 or 3, = 0. So, either A Q = ±F 1 nd A 3Q = 3,1 F + 3, F 1 F F or A Q =,1 F 1 +, F 1 F F nd A 3Q = ±F. In either cse, the ssignment violtes the rule tht the weight mtrices re linerly independent over R. As result, we cn t hve ny vlid elements s the weight mtrices nd hence, k +1. Clim 3: k =. Consider the following grouping scheme of weight mtrices. I n F l F 1 m 1 F i m 1,i l F i m F i In the bove grouping scheme, m = j if is odd, nd m = 1 if is even. It cn be noted tht A iq = A 1Q A ii, i =,3,,. Clerly, the weight mtrices re linerly independent over R nd stisfy (6-(1. Hence, n SSD code trnsmitting complex symbols in chnnel uses exists. This completes the proof. We observe tht for trnsmit ntenns, k 3. So, the rte of unitry-weight SSD code for trnsmit ntenns cn be t most 1 complex symbol per chnnel use, which is lso the rte of the well-known Almouti code, which is single-rel-symbol decodble nd offers full diversity for ll complex constelltions. So, the unitry-weight SSD code for trnsmit ntenns offers no dvntge compred to the Almouti code. So, in the subsequent nlysis, we only consider codes for trnsmit ntenns, > 1. Theorem : Any mximl rte, normlized unitry-weight SSD code must stisfy the following in ddition to stisfying (6-(1. A 1Q = A H 1Q(equivlently,A 1Q = I n, A ii A 1Q = A 1Q A ii, A iq = ±A ii A 1Q, for i =,3,,. Proof: Consider the following grouping of weight mtrices. I n F 1 F F 1 A 1Q A Q A 3Q A (Q A 1Q cn hve two possibilities. Either A 1Q = I n or A 1Q I n. 1 Let A 1Q = I n. We prove tht this is not possibility. If it were true, i.e., A 1Q = I n, then A 1Q should nticommute withf 1,F, ndf 1, s lso seen in (8. The only mtrices from the set {F λ1,λ i {0,1},i = 1,,} tht nticommute with F 1, F, nd F 1 re F nd F 1 F F. So, let A 1Q = 1,1 F + 1, F 1 F F, with 1,1 +c 1, = 1, (13 where c = 1 if (F 1 F F = I n ( is odd nd c = 1 if (F 1 F F = I n ( is even. Next, considering A Q, the only mtrices from the set {F λ1,λ i {0,1},i = 1,,} tht nticommute with F, F 3, nd F 1 re F 1, F, F 1 F F nd F F 3 F 1. As result, A Q =,1 F 1 +, F +,3 F F 3 F 1 +,4 F 1 F F. Further, since A Q = I n, we hve A Q = (,1 +, c,3 + c,4 I n +,1,3 F 1 F F 1 +,,3 F F 3 F + c,3,4 F 1 F, with c s mentioned before. Since I n, F F 3 F, F 1 F F 1 nd F 1 F re linerly independent over C, either,1 =, =,4 = 0 or,3 = 0. Suppose,1 =, =,4 = 0, A Q =,3 F F 3 F 1. Since A 1Q nticommutes with A Q, we see tht A Q cnnot be,3 F F 3 F 1. This is becuse both F nd F 1 F F commute with F F 3 F 1. Therefore, A Q =,1 F 1 + A, F +,4 F 1 F F. By similr rgument, A 3Q = 3,1 F + 3, F + 3,4 F 1 F F. Considering tht A 1Q, A Q nd A 3Q nticommute pirwise, we must hve 1,1, +c 1,,4 = 0, (14 1,1 3, +c 1, 3,4 = 0, (15, 3, +c,4 3,4 = 0. (16 From (14, (15 nd (16, we obtin 1,1 = c,4 = c 3,4 =,. 1,, 3,,4 Hence, 1,1 + c 1, = 0, which contrdicts (13. So, A 1Q I n nd A 1Q cnnot be nti-hermitin. Let A 1Q I n. In this cse, we first look for possibilities for A iq, i =,,. As rgued before, either A Q =,1 F 1 +, F +,4 F 1 F F or A Q = mf F 3 F 1, with m = ±1 if is even nd m = ±j if is odd. Assuming tht A Q =,1 F 1 +, F +,4 F 1 F F, we hve A 3Q = 3,1 F + 3, F + 3,4 F 1 F F, A 4Q = 4,1 F 3 + 4, F + 4,4 F 1 F F. Since A Q = A 3Q = A 4Q = I n,,1 +, +c,4 = 1, 3,1 + 3, +c 3,4 = 1, 4,1 + 4, +c 4,4 = 1, (17

6 6 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. XX, NO. XX, XXXX A 1I = I n A I = F 1 A 3I = F 1F A ( = F 1F 1 A 1Q = ±m 1 F i A Q = F 1A 1Q A 3Q = ±m 1 i=3 F i A (Q = ±m i= F i with { 1 if (F1 F c = F = I n ( is odd, 1 if (F 1 F F = I n ( is even. Further, considering tht A Q, A 3Q nd A 4Q nticommute with ech other, we hve the following equlities., 3, +c,4 3,4 = 0,, 4, +c,4 4,4 = 0, 3, 4, +c 3,4 4,4 = 0. with c s mentioned before. From the bove set of equtions, we obtin,,4 = c 3,4 3, = c 4,4 4, = 4, 4,4. So, 4, + c 4,4 = 0 p 4, = 4,4, with p = ±j if c = 1 nd p = ±1 if c = 1. So, from (17, we obtin, 4,1 = ±1. By Similr rgument, we obtin,,1 = ±1,p, =,4, 3,1 = ±1,p 3, = 3,4. Therefore, A Q = ±F 1 +, (F +pf 1 F F, A 3Q = ±F + 3, (F +pf 1 F F, A 4Q = ±F 3 + 4, (F +pf 1 F F. It is esy to see tht the bove ssignment of mtrices violtes the liner independence of the mtrices A ii,a iq, i =,3,4 over R. Therefore, the ssumption tht A Q =,1 F 1 +A, F +,4 F 1 F F is not vlid. So, let where, A Q = ±mf F 3 F 1 = ±m m = { j if is odd, 1 if is even.,i A ii, Now, the only possibility is tht A 3Q = ±mf 1 F 3 F 1 = ±m,i 3 A ii. Similrly, by ssigning ±m,i j A ii to A jq,j = 4,,, we see tht the conditions in (10, (11 nd (1 re stisfied nd from the discussion mde bove, this is the only ssignment possible. Now, we only need to find vlid ssignment for A 1Q. Firstly, we note tht A ii A iq = ±A ji A jq, i j. (18 From Lemm 1, multiplying ll the weight mtrices by A I (i.e., F 1 will result in nother unitry-weight SSD code with the weight mtrices grouped s shown t the top of the pge, fter interchnging the first nd the second columns. It should be noted in the bove grouping scheme tht the elements in the first row except I n re ll mutully nticommuting mtrices nd ll of them lso nticommute with F 1 F. So, F 1, F 1 F, F 1 F 3,, F 1 F 1 nd F 1 F re pirwise nticommuting mtrices. Hence, insted of F 1, F,, F 1, if we were to chose F 1, F 1 F, F 1 F 3, nd F 1 F 1 s the 1 nticommuting mtrices, we would end up with the weight mtrices s shown in the tble t the top of the pge. So, A iia iq = ±A jia jq, i j. (19 From (18 nd (19, A 1Q = ±mf 1 F F 1. This further implies tht A 1Q must be unitry, Hermitin mtrix, becuse of the choice of m. So, the weight mtrices of the normlized unitry-weight SSD code for trnsmit ntenns re I n F 1 F 1 ±m 1 F i ±m 1 i= F i ±m F i This completes the proof of the theorem. For 4 trnsmit ntenns, by pplying the procedure outlined in [3], we obtin the following pirwise nticommuting, nti- Hermitin mtrices. j F 1 = 0 j j 0, F = j F 3 = , F 4 = 0 j 0 0 j j 0 0 j 0. For constructing mximl rte, unitry weight SSD code for 4 trnsmit ntenns, we only need 3 pirwise nticommuting, nti-hermitin mtrices. Hence, choosing F 1, F nd F 3 nd pplying the construction method described bove, we obtin mximl rte, unitry-weight SSD code for 4 trnsmit ntenns, codeword mtrix S of which is shown in (0, t the top of the next pge. In generl, for trnsmit ntenns, we need unitry, nti-hermitin, pirwise nticommuting mtrices. If there re exctly 1 pirwise nticommuting mtrices of size, then, ny mtrix mong them is scled product of the other. So, the following observtions cn be mde bout ny mximl rte, normlized unitry-weight SSD code. 1 EitherA ii,i =,3,, re 1 mong+1 pirwise nticommuting mtrices nd A iq,i =,3,, re exctly 1 pirwise nticommuting mtrices, or A iq,i =,3,, re 1 mong +1 pirwise nticommuting mtrices nd A ii,i =,3,, re exctly 1 pirwise nticommuting mtrices. A 1Q is Hermitin mtrix nd A 1Q = i= A ii if A ii,i =,3,, re 1 mong + 1 pirwise nticommuting mtrices, or A 1Q = i= A iq ifa iq,i =,3,, re 1 mong+1 pirwise nticommuting mtrices.

7 SANJAY et l.: MAXIMUM RATE OF UNITARY-WEIGHT, SINGLE-SYMBOL DECODABLE STBCS 7 S = x 1I +jx I x 3I jx 4Q x 4I +jx 3Q x Q jx 1Q x 3I jx 4Q x 1I jx I x Q +jx 1Q x 4I +jx 3Q x 4I +jx 3Q x Q jx 1Q x 1I jx I x 3I +jx 4Q x Q +jx 1Q x 4I +jx 3Q x 3I +jx 4Q x 1I +jx I. (0 V. DIVERSITY AND CODING GAIN OF UNITARY-WEIGHT SSD CODES = n j=1 ( x ii +( 1 k i+s j x iq, We hve seen in Lemm 1 tht the coding gin of unitryweight SSD codes does not chnge when normlized. In this section, we obtin common expression for the coding gin of ll unitry-weight SSD codes nd identify the conditions on QAM constelltions tht will llow unitry weight SSD codes to hve full trnsmit diversity nd high coding gin. Let S nd S be two distinct codewords of ny normlized unitry-weight SSD code S N. Let with S = S = x ii A ii +x iq A iq, x iia ii +x iqa iq, A 1I = I n, A ii A iq = ±A ji A jq = ±A 1Q, i j, A H 1Q = A 1Q,A H ii = A ii,a H iq = A iq, i. Let S S S, x i x i x i, x ii x ii x ii nd x iq x iq x iq. Then, ( S H S = = = ( H x ii A ii + x iq A iq ( x mi A mi + x mq A mq m=1 ( x ii + x iq I n ± x ii x iq A ii A iq ( x ii + x iq I n ± x ii x iq A 1Q. Since A 1Q is unitry nd Hermitin, the eigenvlues of A 1Q re ±1 nd A 1Q is unitrily digonlizble. Let A 1Q = EΛE H, where E is unitry nd Λ is digonl mtrix with the digonl entries being ±1. Therefore, ( ( S H S = x ii + x iq EE H ± x ii x iq EΛE H ( ( = E x ii + x iq I n ± x ii x iq Λ E H nd det ( ( S H S ( ( = det x ii + x iq I n ± x ii x iq Λ n = ( x ii + x iq +( 1k i+s j x ii x iq j=1 where, nd s j = { 0 if the (j,j th entry of Λ is 1, 1 if the (j,j th entry of Λ is -1, k i = { 0 if AiI A iq = A 1Q, 1 if A ii A iq = A 1Q. The minimum of the determinnt, denoted by min, of ( S H S for ll possible non-zero S is given s n ( min = min xii +( 1 ki+sj x iq. S 0 j=1 Since the expression inside the brcket in the bove eqution is product of the sum of squres of rel numbers, its minimum occurs when ll but one mong x i,i = 1,, re zeros. So, n (. min = min xii +( 1 ki+sj x iq (1 x i 0 j=1 Let m be the lgebric multiplicity of 1 s the eigenvlue of A 1Q nd n m be tht of -1. We mke use of the following lemm to conclude tht m = n m. Lemm 5: Let F 1, F, nd F be unitry, pirwise nticommuting mtrices. Then, the product mtrix F λ1,λ i {0,1},i = 1,,, with the exception of I, is trceless. Proof: It is well known tht tr(ab = tr(ba for ny two mtrices A nd B. Let A nd B be two invertible, n n nticommuting mtrices. So, AB = BA. ABA 1 = B. tr(aba 1 = tr(b. tr(a 1 AB = tr(b tr(b = tr(b. tr(b = 0. ( Similrly, it cn be shown tht tr(a = 0. By pplying Lemm 4, it cn be seen tht ny product mtrix F λ 1 1 Fλ F λ, nticommutes with some other product mtrix from the set {F λ1,λ i {0,1},i = 1,,3,,}. Therefore, from the result obtined in (, we cn sy tht every product mtrix F λ1 except I is trceless. Since A 1Q is scled product of 1 mtrices mong unitry, pirwise nticommuting mtrices, A 1Q is trceless.

8 8 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. XX, NO. XX, XXXX Constelltion: 4-QAM 16-QAM 64-QAM CIOD MDCQOD [6] MDCQOD [7] New Design TABLE I COMPARISON OF THE MINIMUM DETERMINANTS OF A FEW SSD CODES FOR 4 TRANSMIT ANTENNAS Hence, m = n m. So, (1 becomes min = min x x i 0( ii x n. iq (3 From the bove expression, it is cler tht for mximl-rte, unitry-weight SSD codes to offer full trnsmit diversity, the difference set A { b,b A}, where A is the constelltion employed, should not hve ny points tht lie on lines tht re t ±45 degrees in the complex plne from the origin. Further, since the nlysis leding up to the expression in (3 is not specific to ny prticulr unitry-weight SSD code, we cn infer tht for ny prticulr constelltion A, ll mximl-rte, unitry-weight SSD codes hve the sme coding gin. CER QAM Rect. 8 QAM Sq. derived 8 QAM New Design Rect. 8 QAM CIOD Rect. 8 QAM MDCQOD Rect. 8 QAM New Design Sq. derived 8 QAM CIOD Sq. derived 8 QAM MDCQOD Sq. derived 8 QAM New Design 4 QAM CIOD 4 QAM MDCQOD 4 QAM A. Diversity, coding gin clcultions for QAM In this subsection we show tht ll mximl-rte, unitryweight SSD codes hve the sme coding gin s the CIODs [5] for QAM constelltions. Let y i, i = 1,,, be the informtion symbols tht tke vlues from constelltion A 1. Consider the following unitry rottion. [ ] [ ] 1 xii [ 1 ] = yii x iq 1 1, i = 1,,,. y iq The bove opertion is equivlent to rottingy i byπ/4 rdins to obtin x i. Then, from (3, we hve min = min y i 0 ( y ii y iq n. (4 The bove expression is the sme s the one for CIOD, obtined in [5]. It is to be noted tht (4 holds even when the ngle of rottion is π/4 rdins. In order to mximize min, the minimum of the product y ii y iq, clled the product distnce, must be mximized. This hs been done for QAM in [5], by rotting QAM constelltions by n ngle of ± 1 tn 1. So, y i, i = 1,,,, should tke vlues from rotted QAM constelltion, with the ngle of rottion being ± 1 tn 1. So, the originl informtion symbols x i, i = 1,,, should tke vlues from rotted QAM constelltion, the ngle of rottion being± π 4 ±1 tn 1. Since the coding gin for CIOD hs been mximized in [5] by using ± 1 tn 1 rdin rotted QAM constelltion, the coding gin for ll unitry-weight SSD codes when the symbols tke vlues from ± π 4 ± 1 tn 1 rdin rotted QAM constelltion is lso mximized. Tble I gives comprison of the minimum determinnts for the CIOD, MDCQOD nd the unitry-weight SSD code presented in (0, ll the codes designed for 4 trnsmit ntenns. In the clcultions, ll the codes hve the Fig SNR in db Comprison of the CER performnce of the SSD codes nd the CIOD sme verge energy but the constelltion energy hs been llowed to increse with the increse in constelltion size. As nlyticlly shown, the minimum determinnts re the sme for ll the three codes. B. Simultion results for our SSD codes In this subsection, we provide some simultion results for 4 trnsmit ntenns. All simultions re done ssuming qusi-sttic Ryleigh fding chnnel. The number of receive ntenns is 1. Fig. 1 shows the codeword error rte (CER performnces of the CIOD for 4 trnsmit ntenns, the MDCQOD for 4 trnsmit ntenns [6], nd the new design whose codeword mtrix is s in (0, t bits nd 3 bits per chnnel use (bpcu. For trnsmission t bpcu, the constelltion employed is the 1 tn 1 rdin rotted 4-QAM for the CIOD nd the π tn 1 rdin rotted 4-QAM for the MDCQOD nd the new design. For trnsmission t 3 bpcu, the constelltions employed re the rotted rectngulr 8-QAM nd the rotted squre-derived 8-QAM, the ngle of rottion being the sme s in the cse of 4-QAM. A squred-derived 8-QAM constelltion is obtined by removing the signl point with the highest energy from 9-QAM. Specificlly, it is the set { 1 j, 1+j, 1+3j, 1 j, 1+j, 1+3j, 3 j, 3+j}. The simultion results support the fct tht for QAM constelltions,

9 SANJAY et l.: MAXIMUM RATE OF UNITARY-WEIGHT, SINGLE-SYMBOL DECODABLE STBCS 9 the coding gin of the SSD codes is the sme s tht of the CIOD. VI. DISCUSSION AND CONCLUDING REMARKS In this pper, we hve provided n chievble upper bound on the rte of unitry-weight SSD codes for trnsmit ntenns. The upper bound hs been shown to be 1 complex symbols per chnnel use. We lso hve completely chrcterized the structure of the weight mtrices of the codes meeting the upper bound. We hve further shown tht ll unitry-weight SSD codes tht meet the upper bound hve the sme coding gin s tht of the CIODs nd we hve lso identified the ngle of rottion for QAM constelltions tht llow the codes to hve optimum coding gin. The nlysis done in this pper throws open the following questions. 1 Wht is the upper bound on the rte of squre, nonunitry-weight SSD codes? Further, wht re the conditions on the signl constelltion tht llow non-unitryweight SSD codes to chieve full-diversity nd optimum coding gin? Wht is the upper bound on the rte of rectngulr SSD codes, the clss of which the rectngulr MDCQODs presented in [6] nd [7] re subclss? Further, the nlysis in this pper cn be used to study the rtes of multi-symbol decodble codes, the upper bounds of which hs never been reported in literture. These questions provide some directions for future reserch. ACKNOWLEDGEMENT This work ws prtly supported by the DRDO-IISc Progrm on Advnced Reserch in Mthemticl Engineering nd by the Council of Scientific & Industril Reserch (CSIR, Indi, through Reserch Grnt ((0365/04/EMR-II to B.S. Rjn. We thnk X.-G.Xi for sending the preprint of [6]. We would lso like to thnk the nonymous Reviewers for pointing out [1], which helped in simplifying the proof for the upper bound on the rte of unitry-weight SSD codes. REFERENCES [1] V. Trokh, H. Jfrkhni nd A. R. Clderbnk, Spce-Time block codes from orthogonl designs, IEEE Trns. Inf. 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