Optimizing Quasi-Orthogonal STBC Through Group-Constrained Linear Transformation

Accepted for publication in IEE Proc. Comm unications 1 Optimizing Quasi-Orthogonal STBC Through Group-Constrained Linear Transformation Abstract — In this paper, we first derive the generi c algebraic structure of a Quasi-Orthogonal STBC (QO-STBC). Next we propose Group-Constrained Linear Transformation (GCLT) as a means to optimize the diversity and coding gains of a QO-STBC with s quare or rectangular QAM constellations. Compared with QO-STBC with constellation rotation (CR), we s how that QO-STBC with GCLT requires only half the number of symbols for joint detection, hence lowe r m aximum -likelihood decoding complexity. We also derive analytically the optimum GCLT parameters for QO-STBC with square QAM constellation. The optimized QO-STBCs with GCLT are able to achieve full transmit diversity, and have negligible performance loss compared with QO-STBCs with CR at the sam e code rate. Index Terms — Constellation Rotation, Group-Constrained Linear Transformation, Quasi-Orthogonality Constraint, Quasi-Orthogonal Space-Time Block Code. I. INTRODUCTION Orthogonal Space-Time Block Code (O-STBC) offers full transm it diversity with linear decoding complexity [1]. Unf ortunately, O-ST BC suffers from reduced code rate when complex constellations are necessitated by high transmission rate requirement, and when the required transm it diversity is greater than two. As a result, Quasi-Orthogonal STBCs (QO-STBC) were proposed. Some well known examples include the QO-STBC from [2], the ABBA code from [3] and th e transm it diversity scheme from [4,5]. W ith its quasi-orthogonal code structure, the maximum -likeli hood (ML) decoding of a QO-STBC can be performed by searching over (or joint detection of) only a subset of the total number of transm itted symbols, hence the decoding complexity of quasi-orthogonal STBC is lower than the general non-orthogonal STBC. Accepted for publication in IEE Proc. Comm unications 2 The first-generation QO-STBCs, however, could not ach ieve full transmit diversity. Fortunately, this problem was solved by the technique of Constellation Ro tation (CR) [6-10]. To date, full-rate full-diversity QO-STBC for four transmit antennas can be ML-decoded by the joint detection of at least two com plex symbols [6-10]. For eight transmit antennas, full-divers ity QO-STBC requires joint detection of at least two complex symbols at a code rate of 3/4 [8], or four com plex symbols at a code rate of 1 [5,10]. In this paper, we shall show that the number of sy mbols required for the joint detection of the existing full-diversity QO-STBCs with square or rectangular regular QAM constellations can actually be halved if, instead of CR, a novel “ Group-Constrained Linear Transformation (GCLT)” is used to optimize the original QO-STBCs. To explain the principles of the proposed GCLT, the generic algebraic structure of QO-STBC is first derived in this paper. We then examine the algebraic structure of ex isting QO-STBCs and study the impact of CR on their decoding complexity. Ne xt, we derive analytically the optim al GCLT parameters for a full-rate QO-STBC for four transm it antennas [2] and a rate-3/4 QO-STBC for eight transmit antennas [2] with square QAM constellation. While the optimum GCLT parameters for a full-rate QO-STBC for eight transmit antennas [3] is obtained by computer searc h. The bit error rate (BER) perform ance of QO-STBC designed using CR and GCLT are then compared. II. QO-STBC AND I TS S IGNAL M ODEL A. Signal Model for QO-STBC with QAM Constellation Suppose that there are N t transmit antennas, N r receive antennas, and an interval of T symbols during which the propagation channel is constant and known to the receiver. The transmitted signal can be written as a T × N t matrix C that governs the transm ission over the N t antennas during the T symbol intervals. It is assumed that the data sequence has been broken into blocks with K square or rectangular regular QAM symbols, x 1 , x 2 , …, x K , in each block for transmission over T symbol periods of tim e. The code rate of a QO-STBC is defined as R = K / T . If square or rectangul ar regular QAM constellation is used, every complex Accepted for publication in IEE Proc. Comm unications 3 QAM symbol can be treated as two i ndependent real PAM symbols. W ith this and the modeling approach in [11], a STBC C can be expressed as: 2 11 () ( ) KK q q Kq Kq p p qp ss s ++ == =+ = ∑∑ CA A A (1) where the transmitted symbols x q = s q + js K + q for 1 ≤ q ≤ K . The matrices A p of size T × N t , for 1 ≤ p ≤ 2 K , are called the “dispersion matrices”. To limit the total transmission power, they must conform to the power distribution constraint [11]: H tr ( ) / pp t TN K = AA ( 2 ) The received signal model can be modeled as [11]: t N ρ =+ rH s η   ( 3 ) where the normalization factor t N ρ is to ensure that the SNR ( ρ ) at the receiver is independent of the number of transmit antennas, and RR 11 1 II 11 RR II 2 ,, , rr rr K NN NN K s s s ⎡⎤ ⎡ ⎤ ⎡⎤ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ == = ⎢⎥ ⎢ ⎥ ⎢⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎢⎥ ⎢ ⎥ ⎢⎥ ⎣⎦ ⎣⎦ ⎣ ⎦ r η r η rs η r η r η #   ## # 11 1 2 1 12 ... ... ... ... rr r KK NK N K N ⎡⎤ ⎢⎥ = ⎢⎥ ⎢⎥ ⎣⎦ hh h H hh h #% # % # AA A AA A , RI R IR I ,. pp i pi pp i ⎡⎤ − ⎡ ⎤ == ⎢⎥ ⎢ ⎥ ⎢⎥ ⎣ ⎦ ⎣⎦ AA h h AA h A In the above equations, the superscript R and I denote the real part and imaginary part of a scalar, vector or matrix respectively. The r i and η i , for 1 ≤ i ≤ N r , are T × 1 column vectors which contain the received signal and the zero-mean unit-variance AWGN noise sam ples for the i th receive antenna over T symbol periods respectively, h i is a N t × 1 column vector that contains N t independent Rayleigh fl at fading coefficients between the j th transmit antenna and the i th receive antenna, h j , i , for 1 ≤ j ≤ N t . Accepted for publication in IEE Proc. Comm unications 4 B. Review of QO-STBC with Constellation Rotation In this paper, the rate-1 QO-STBC in [2] for four transm it antennas (herein called the code Q4), the 3/4-rate QO-STBC in [2] for eight transm it antennas (h erein called the code Q8) and the rate-1 QO-STBC in [3] for eight transm it antennas (herein called the code T8) will be used as representative code examples. First, the code matrix of the Q4 code, C Q4 , is shown in (4): 123 4 ** * * 21 4 3 ** * * 34 1 2 43 2 1 x xx x -x x -x x = -x -x x x x -x -x x ⎡⎤ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎣⎦ Q4 C ( 4 ) After appropriate CR, Q4 can achieve full diversity w ith joint detection of two com plex symbols for ML decoding [8,10]. The resultant code, called Q4_CR in this paper, has code matrix C Q4_CR as shown in (5). /4 /4 /4 * /4 * /4 * /4 * /4 /4 () () () () jj 12 3 4 ** j j 21 4 3 jj * * 34 1 2 jj 43 2 1 xx x e x e - x x - xe xe = -x e -x e x x xe - xe - x x ππ ππ ππ ππ ⎡⎤ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎣⎦ Q4_CR C ( 5 ) where the factor e j π /4 denotes the CR angle for the QAM symbols x 3 and x 4 . The ML decoding metrics for Q4_CR is shown in (6). It is derived based on the ML decoding metrics of Q4 from [2], but taking CR into account. We can see that the decoding decision for sym bols x 1 and x 4 is obtained by minimizing the m etric f 14 , similarly the decoding decision for symbols x 2 and x 3 is obtained by mini mizi ng t he met ri c f 23 . Clearly, decoding of x 1 and x 4 can be performed separately from the decoding of x 2 and x 3 . Since x 1 and x 4 (or x 2 and x 3 ) are each a complex sym bol, thei r ML decoding requires the joint detection of two complex symbols (i.e. four real symbols) in total. Accepted for publication in IEE Proc. Comm unications 5 ** * * 11 , 1 2 , 23 , 3 4 , 4 4 2 22 /4 * * * * 14 1 4 , 1 4 4 4, 1 3, 2 2, 3 1 , 4 11 */ 4 * * 1 4 1 , 4, 2, 3, 2 23 2 3 , () (, ) ( ) ( ) 2 R e ( ) 2R e ( ) (, ) ( r rr r r N j nr r r r r rn j rr rr nr xh r h r h r h r f x x h x x x e hr hr hr h r xx e h h h h fx x h π π == − ⎡⎤ ⎧ ⎫ −− − − + ⎢⎥ ⎪ ⎪ ⎪ ⎪ = + + −++− + ⎢⎥ ⎨ ⎬ ⎢⎥ ⎪ ⎪ ×− ⎢⎥ ⎪ ⎪ ⎩⎭ ⎣⎦ = ∑∑ ** * * 22 , 1 1 , 2 4 , 3 3 , 4 4 22 /4 * * * * 23 3 3 , 1 4 , 2 1 , 3 2 , 4 11 */ 4 * * 2 3 1 , 4, 2, 3, () )( ) 2 Re ( ) 2R e ( ) r rr r r N j rr r r rn j rr r r xh rh r h r h r xx x e h r h r h r h r xx e h h h h π π == − ⎡⎤ ⎧ ⎫ −+ −+ + ⎢⎥ ⎪ ⎪ ⎪ ⎪ ++ − −+ + + ⎢⎥ ⎨ ⎬ ⎢⎥ ⎪ ⎪ ×− + ⎢⎥ ⎪ ⎪ ⎩⎭ ⎣⎦ ∑∑ (6) where x 1 to x 4 in (6) are each non-rotated complex constellation symbols. We shall show in Section IV that by optim izing Q4 with the proposed GCLT instead of CR, the resultant code can be decoded with joint detection of only two real symbols, wh ile still achieving full transmit diversity gain and full code rate. III. Q UASI -O RTHOGONALITY C ONSTRAINT A. Algebraic Structure of QO-STBC In order to quantify the number of symbols required for joint detection, we now derive the algebraic structure of generic QO-STBC, ca lled the Quasi-Orthogonality (QO) C onstraint. The concept of QO-STBC is to divide the K transmitted sym bols of a codeword into G independent groups, such that sym bols in any group are orthogonal to all symbols in the other groups after appropriate m atched filtering, while strict orthogonality among the symbols within a group is not required. As a result, the received sym bols can be separated into G independent groups by simple linear processing, such that the ML decoding of different groups can be performed separately and in parallel, and the ML decoding of every group can be achieved by jointly detecting only K / G complex sym bols that are within the same group. Accepted for publication in IEE Proc. Comm unications 6 Definition 1 : A quasi-orthogonal design 2 1 () K pp p s = = ∑ CA is such that, when multiplied w ith the channel fading coefficients to obtain H as defined in (3), H T H is block-diagonal and consists of G smaller sub-matrices each with size (2 K / G ) × (2 K / G ). To derive the QO-Constraint, let us m ultiply a matched filter ( H T ) to the received signal r  in (3), and consider a snapshot of H T H as shown below: () () () TT T T 1 111 1 TT T T 1 T TT T T 1 TT T T 1 TT TT T T 11 1 r r r rrr r r RR R pN p pu qv uN u qN q pN uN qN vN vN v NN N ip p i ip u i ip q i i ii i == = ⎡⎤ ⎢⎥ ⎢⎥ ⎡⎤ ⎢⎥ ⎢⎥ = ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎣⎦ ⎢⎥ ⎢⎥ ⎢⎥ ⎣⎦ = ∑∑ ∑ hh hhh h hh HH hh hhh h hh hh h h h h h #" # " "" " ## # # # # " "" " #" # "" " " " " # AA AA A A AA AA AA A A AA AA AA AA () () () () () () () () () () () () TT 1 TT TT T T TT 11 1 1 TT TT T T TT 11 1 1 TT TT T T T 11 1 R RR R R RR R R RR R N pv i i NN N N iu p i iu u i iu q i iu v i ii i i NN N N iq p i iq u i iq q i iq v i ii i i NN N iv p i iv u i i v q i i ii i = == = = == = = == = ∑ ∑∑ ∑ ∑ ∑∑ ∑ ∑ ∑∑ ∑ h hh h h h h h h hh h h h h h h hh h h h h h # # # # # # AA AA AA AA AA AA AA AA AA AA AA AA A () T 1 R N vvi i = ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ∑ h # "" " " " " A (7) where 1 ≤ p, q, u, v ≤ 2 K . Assume that the symbols s p and s u are in the same group (hence they are not orthogonal), while the symbols s q and s v are in another group (hence they are orthogonal to s p and s u ), we writ e { p, u } ⊂ G ( p ) and { q, v } ⊄ G ( p ) where G ( p ) represents a set of symbol indi ces that are in the same group as s p , including s p ; similarly, { q, v } ⊂ G ( q ) and { p, u } ⊄ G ( q ). In order to achieve orthogonality among the symbols of different groups, e.g. between symbols s p and s q , the summation term s included in th e boxes in (7) are required to be zero. A way to achieve this is to make A p T A q and A q T A p (likewise A p T A v , A u T A q , A u T A v etc.) skew-symm etric, due to Lemma 1 as stated below. Accepted for publication in IEE Proc. Comm unications 7 Lemma 1 : If a matrix M of size v × v is skew-symm etric (i.e. M T = – M ), then v T Mv = 0 for any vector v of size v × 1. Proof of Lemma 1 : Let c = v T Mv . Since c is a scalar, c T = c . If M T = – M , then c + c T = v T Mv+v T M T v = 0. Hence c = 0 if M is skew-symm etric, and Lemma 1 is proved. ■ Theorem 1 : By ensuring that the dispersion matrices p A and q A of symbols s p and s q respectively meet the Quasi-Orthogonality (QO) Constraint specified in (8), their corresponding A p T A q and A q T A p will be skew-symm etric, and s p will be orthogonal to s q . HH for 1 , 2 and ( ) pq q p pq K q p =− ≤ ≤ ∉ AA AA G (8) Proof of Theorem 1 : From (8), () ( ) ( ) ( ) () ( ) ( ) () HH RI H RI RI H RI TT T T RR I I RR I I TT T T RI I R I R RI ( ) () () ( ) real pa rt : ima g part : pq q p pp qq qq pp pq p q q p q p pq p q q p q p jj j j =− ⇒+ + = − + + ⎧ += − − ⎪ ⇒ ⎨ ⎪ −= − ⎩ AA AA AA AA AA AA AA A A AA A A AA A A A A AA (9) Define () ( ) ( ) () ( ) ( ) TT HR R I I TT HR I I R Re Im pq p q p q pq p q p q =+ =− MA A A A A A NA A A A A A   . Then we know from (9) that M is skew-symm etric and N is symm etric. As a result, T pq − ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ MN NM AA is skew-symmetric. Sim ilar conclusion can be drawn on A q T A p . By Lemma 1 , we then know that H T H in (7) is always block-diagonal. Hence Theorem 1 i s p r o v e d . ■ Accepted for publication in IEE Proc. Comm unications 8 It can be shown that all the QO-STBCs proposed in the literature, such as [2-4], follow the algebraic structure specified in (8). With this algebraic stru cture, we shall now exam in e the effect of CR on the decoding complexity of a QO-STBC. B. Group Structure of QO-STBCs with CR We now examine the eight dispersion m atrices of the code Q4 (listed in Appendix A as matrices A 1 , A 2 , …, A 8 ) according to the derived QO-Constraint in (8). The fulfillment of QO-Constraint of A 1 , A 2 , …, A 8 is shown in Table 1(a). For example, Table 1(a) shows that A 1 is orthogonal to all the other dispersion matrices except A 4 . Likewise, each of A 2 , …, A 8 are orthogonal to all but one of the other dispersion matrices. By re-arranging the rows and columns of Table 1(a) to obtai n Table 1(b), it is clear that, the dispersion matrices A 1 , A 2 , …, A 8 of the code Q4 can be groupe d into four orthogonal groups, { A 1 , A 4 }; { A 2 , A 3 }; { A 5 , A 8 }; { A 6 , A 7 }, as depicted in Figure 1(a). Since there are onl y two non-orthogonal dispersion matrices (modulating two real symbols) in each group, the ML decoding of Q4 can be achieved by joint detection of two real symbols, instead of two complex symbols as reported in [2]. Table 2(a) examines the fulfillm ent of QO-Cons traint for the dispersion m atrices of the constellation-rotated Q4, i.e. Q4_CR, (listed in Appendix B as matrices A CR_1 , A CR_2 , …, A CR_8 ). By re-arranging the rows and columns of Table 2(a) to obtai n Table 2(b), it is clear th at the dispersion m atrices of Q4_CR can be grouped into only two orthogonal groups, { A CR_1 , A CR_4 , A CR_5 , A CR_8 }; { A CR_2 , A CR_3 , A CR_6 , A CR_7 }, as depicted in Figure 1(b). Since there ar e four non-orthogonal dispersion matrices (modulating four real symbols) in each group, it implies that in or der to achieve full diversity using CR, the ML decoder for Q4_CR needs to jointly decode four real symbols, rather than two real symbols before CR. It can similarly be shown that the rate-3/4 QO-ST BC for eight transmit antennas proposed in [2] requires joint detection of two real symbols before CR (denoted herein as the Q8 code), and four real symbols after Accepted for publication in IEE Proc. Comm unications 9 CR [8] (denoted herein as the Q8_CR code). Like wise the rate-1 QO-STBC for eight transmit antennas proposed in [3] requires joint detection of four real sy mbols before CR (denoted herein as the T8 code), and four real symbols after CR [10] (denoted herein as the T8_CR code). This is summ arized in Table 3. IV. G ROUP -C ONSTRAINED L INEAR T RANSFORMATION (GCLT) A. Definition of GCLT In order to optimize a QO-STBC to achieve full diversity and m aximum coding gain, while m aintaining the original symbol groupings and hence the decoding complexity, we propose the Group-Constrained Linear Transformation (GCLT) as defined in Proposition 1 . Proposition 1 : By linearly combining the dispersion matrices A within a group in accordance with (10) and (11) we can obtain a new se t of dispersion matrices A LT that will satisfy the QO-Constraint with the same symbol grouping structure as the original A matrices. Hence the transform ation rules (10) and (11) do not destroy the quasi-orthogonal structure, nor change the number of quasi-orthogonal groups, of a QO-STBC. _, () 1 2 qq v v vq qK α ∈ =≤ ≤ ∑ LT AA  G (10) __ 1 2 qq q cq K =≤ ≤ LT LT AA  ( 1 1 ) where α q,v is the GCLT parameters and are real constants. The scalar factor H __ tr( ) t qq q TN c K = LT LT AA  is to ensure that the dispersion matrices of the QO-STBC, af ter GCLT, satisfy the power distribution constraint in (2). Proof of Proposition 1 : Applying (8) with A p Æ A LT_ p and A q Æ A LT_ q gives Accepted for publication in IEE Proc. Comm unications 10 HH __ _ _ HH ,, , , ( ) () () ( ) HH ,, , , () ( ) ( ( ) 1 2 and ( ) () ( ) ( ) () () ( pq q p pp u u q q v v q q v v pp u u up v q v q up p q pu qv u v p q qv pu v u up v q v up p, q K p q ccc c cc cc αα α α αα α α ∈∈ ∈ ∈ ∈∈ ∈ ∈ +≤ ≤ ∉ =+ =+ ∑∑ ∑ ∑ ∑∑ ∑ LT LT LT LT AA A A AA A A AA AA GG G G GG G G G ) HH ,, () ( ) =0 as per QO Constraint ) () 0 q pq p u q v u v v u up v q cc αα ∈∈ =+ = ∑ ∑∑ AA AA      GG ( 1 2 ) Since the above expression is equal to zero, matrices { A LT } satisfy QO-Constraint (8) as matrices { A } do, hence Proposition 2 i s p r o v e n . ■ B. Optimization of GCLT Parameters The GCLT parameters, α , can be chosen such that certain performance criteria, such as the rank and determinant criteria in [12], optim ize the resultant { A LT }. To provide a systematic way to optimize the GCLT parameters in (10), Multi-dimensional Lattice Rotati on (MLR) technique in [13] can be employed. For simplicity, consider a QO-STBC with two real symbol s per group (such as code Q4). Assum e that the matrices A q and A v are in the same group, i.e. { q , v }= G ( q )= G ( v ), the GCLT of the dispersion matrices can be expressed as follows: () ,, _ MLR ,, _ qq qv qq q TT TT vq vv vv v αα αα ×× ⎡⎤ ⎛ ⎞ ⎡⎤ ⎡⎤ ⎡⎤ =⊗ = ⊗ ⎜⎟ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎜⎟ ⎢⎥ ⎣⎦ ⎣⎦ ⎣⎦ ⎣⎦ ⎝ ⎠ LT LT AA A II L AA A   ( 1 3 ) where ⊗ represents the Kronecker product, and I T × T is an identity matrix of size T × T , and L MLR is an orthogonal matrix as specified in [13]. For a two-dimensional case, L MLR maps four GCLT parameters into one variable θ using: MLR cos( ) si n( ) sin( ) cos( ) θ θ θ θ ⎡⎤ = ⎢⎥ − ⎣⎦ L ( 1 4 ) Hence, ,, , , cos( ) ; sin( ) qq vv qv vq α αθ α α θ == = − = in this case. This facilitates th e search or analysis of the optimum GCLT param eters. Denoting the Q4 code after GCLT as Q4_LT, we provide here an analytical Accepted for publication in IEE Proc. Comm unications 11 derivation for the optimization of its GCLT parameters. First, the determinant expression for the codeword distance matrix of Q4_LT is derived as follows: () () 2 222 2 14 23 58 67 22 2 2 14 2 3 58 6 7 () ( ) ( ) ( ) det () ( ) ( ) ( ) ⎡⎤ ∆+ ∆ + ∆ − ∆ + ∆ + ∆ + ∆ − ∆ × ⎢⎥ = ⎢⎥ ∆− ∆ + ∆ + ∆ + ∆ − ∆ + ∆ + ∆ ⎣⎦             ( 1 5 ) where cos sin qq v θ θ ∆= ∆ − ∆  and sin cos vq v θ θ ∆= ∆ + ∆  for {( , ) } {( 1 , 4), ( 2, 3), (5, 8), ( 6, 7 )} qv ∈ , and q ∆ represents the possible error in the real PAM symbol s q (remembering that a QAM sym bol x q is expressed in terms of two real PAM symbols, i.e. x q = s q + js K + q where K is the number of com plex symbols being transmitted in a STBC codeword). Since the symbol grouping of Q4_LT is such that s 1 and s 4 are in a group, and they are independent of (i.e. orthogonal to) the other symbols in the ML decoding ope ration, without loss of generality, it can be assum ed that only s 1 and s 4 have errors and the other symbols are erro r-free [6]. As a result, the worst-case (i.e. minimum ) determinant value in (15) can be sim plified to: 2 22 1 4 1 4 235678 4 22 14 4 22 14 1 4 4 22 11 4 4 det ( ) ( ) assum ing that = = = = = =0 () ( ) (c o s s i n ) (s i n c o s ) c os (2 ) 2 s i n (2 ) co s(2 ) θθ θ θ θθ θ ⎡⎤ =∆ + ∆ × ∆ − ∆ ∆∆ ∆∆ ∆∆ ⎣⎦ ⎡⎤ =∆ − ∆ ⎣⎦ ⎡⎤ =∆ − ∆ − ∆ + ∆ ⎣⎦ ⎡⎤ =∆ −∆ ∆ − ∆ ⎣⎦          ( 1 6 ) 4-QAM Constellation Consider first 4-QAM constellation. The I and Q com ponents of a 4-QAM symbol can be viewed as two independent 2-PAM symbols. Hence ∆ 1 , ∆ 4 ∈ {0, ± d min } where d min is the minimum Euclidean distance between two constellation points as shown in Figure 2, and ∆ 1 and ∆ 4 cannot be both zero in (16). To maximize the m inimum determinant value in (16) base d on the rank and determ inant criteria in [12], the following four cases of ( ∆ 1 , ∆ 4 ) and their resultant determinant values as per (16) are considered: Accepted for publication in IEE Proc. Comm unications 12 Case 1: ( ∆ 1 , ∆ 4 ) = ± (0, d min ) Î [ ] 4 8 1m i n det cos(2 ) d θ = ( 1 7 ) Case 2: ( ∆ 1 , ∆ 4 ) = ± ( d min , 0) Î [ ] 4 8 2m i n det c os( 2 ) d θ = ( 1 8 ) Case 3: ( ∆ 1 , ∆ 4 ) = ± ( d min , d min ) Î [ ] 4 8 3m i n det 2 sin(2 ) d θ =− ( 1 9 ) Case 4: ( ∆ 1 , ∆ 4 ) = ± ( d min , - d min ) Î [ ] 4 8 4m i n det 2 sin( 2 ) d θ = ( 2 0 ) Note that det 1 =det 2 and det 3 =det 4 . In order to maximize the sm aller value between det 1 and det 3 , we equate det 1 and det 3 to get: 10 cos( 2 ) 2 sin (2 ) tan(2 ) 1 / 2 11 tan ( ) 13.28 22 opt opt opt opt θ θ θ θ − = ⇒= ⇒= = ( 2 1 ) So the optimum GCLT param eters for Q4_LT are: 0 ,, 0 ,, cos( ) cos( 13.28 ) sin( ) sin( 13.28 ) qq vv o p t q v v q opt αα θ αα θ == = =− = = ( 2 2 ) where {( , )} {( 1 , 4), ( 2 , 3), (5, 8), ( 6 , 7 )} qv ∈ , and the minimum determinant value of the codeword distance matrix is 8 min 0.64 d . Compared with Q4_CR, which has a minim um determinant value 8 min d [10], Q4_LT has a slightly smaller minim um determ inant value (which will be shown later to give less than 0.5dB loss in coding gain), but the ML decoding of Q4_LT requires th e joint detection of half the number of sym bols as required by Q4_CR. M-ary QAM Constellation We now derive the optimum GCLT parameters of Q4_LT for larger square QAM constellations. Consider the M -ary square-QAM constellation, where the I and Q components of a symbol can be viewed as two independent M -ary PAM symbols. The following four cases of ( ∆ 1 , ∆ 4 ) and their resultant determinant values as per (16) are considered: Accepted for publication in IEE Proc. Comm unications 13 Case 1: ( ∆ 1 , ∆ 4 ) = ± (0, nd min ) Î 4 82 5m i n det cos(2 ) dn θ ⎡ ⎤ = ⎣ ⎦ ( 2 3 ) Case 2: ( ∆ 1 , ∆ 4 ) = ± ( md min , 0) Î 4 82 6m i n det cos(2 ) dm θ ⎡ ⎤ = ⎣ ⎦ ( 2 4 ) Case 3: ( ∆ 1 , ∆ 4 ) = ± ( md min , nd min ) Î 4 82 2 7m i n det ( ) cos( 2 ) 2 sin(2 ) dm n m n θ θ ⎡ ⎤ =− − ⎣ ⎦ (25) Case 4: ( ∆ 1 , ∆ 4 ) = ± ( md min , - nd min ) Î 4 82 2 8m i n det ( ) cos( 2 ) 2 sin( 2 ) dm n m n θ θ ⎡ ⎤ =− + ⎣ ⎦ (26) where d min represents the minimum Euclidean distance between the PAM constellation points, m and n are integers such that 1 ≤ m , n ≤ 1 M − , and M is the cardinality of the QAM constellation. To maximize the sm aller value of det 5 to det 8 for all valid values of m and n , consider first the sm allest value of m = n = 1. For this case, det 5 to det 8 are identical to det 1 to det 4 , hence the optimum θ value for (23) to (26) is the same as that for (21), i.e. θ opt = ½ tan -1 (½), and the corresponding det 5 to det 8 values are identical and equal to 8 min 0.64 d . Next, consider m , n > 1. In this case, it can be shown that with θ = ½ tan -1 (½), () 88 5m i n det 0.64 nd = () 88 6m i n det 0.64 md = ( ) 4 22 8 7m i n det ( ) 0.64 mm n n d ⎡⎤ =− − ⎣⎦ ( ) 4 22 8 8m i n det ( ) 0.64 mm n n d ⎡⎤ =+ − ⎣⎦ which are all greater than or equal to 8 min 0.64 d for m , n > 1. Hence the worst-case (i.e. m inimum) determinant value for Q4_LT with M -ary square-QAM constellation occurs when m = n = 1, and it is optimized when θ = θ opt = ½ tan -1 (½). Therefore, we have shown that the optimum GCLT param eters derived in (22) apply to all QAM size. Accepted for publication in IEE Proc. Comm unications 14 To show the above result graphically, the dete rminant values of Q4_LT with square 16-QAM constellation is plotted as a function of θ in Figure 3. For the square 16-QAM, the values of m and n can each take values of 1, 2 or 3. A few combinations of m and n are shown in Figure 3 as illustration. We can see that at θ opt , all the determinant values corresponding to all possible values of m and n are greater than or equal to the determinant values corresponding to the case of m = n =1, and the optimum determinant values corresponding to m = n =1 occurs at θ opt . Denoted as A LT_1 , A LT_2 , …, A LT_8 , the dispersion matrices of Q 4_LT obtained based on the optimum GCLT parameters derived in (22) are shown in Appendix C. As shown in Figure 1( c), Q4_LT has exactly the same symbol grouping structure as Q4. It can similarly be shown that the optimum GCLT pa rameters for Q8 with M-ary QAM constellation are: 0 ,, 0 ,, cos( 1 3.28 ) sin( 13.28 ) qq vv qv vq αα αα == =− = ( 2 7 ) wh er e { ( , )} {( 1 , 10 ), ( 2, 11 ), ( 3 , 12 ), ( 4, 7 ) , (5, 8) , ( 6, 9 )} qv ∈ . The resultant code, denoted as Q8_LT, has minimum determinant value of 16 min 4 0.4096( ) 3 d , as compared with 16 min 4 () 3 d for Q8_CR. In [13], the rate-1 QO-STBC for eight transm it antennas T8 requires a joint detection of four real symbols. Its L MLR matrix corresponding to (13): MLR 13 , 1 4 (, , ) ik ii k ik θ ≤≤ + ≤ ≤ = ∏ LG ( 2 8 ) where G ( i , k , θ ik ) is a 4 × 4 matrix with entries at ( i , i ) and ( k , k ) equal to cos( θ ik ), entry at ( i , k ) equals to sin( θ ik ), and entry at ( k , i ) equals to -sin( θ ik ), one on the remaining diagonal positions and zero elsewhere. Accepted for publication in IEE Proc. Comm unications 15 G ( i , k , θ ik ) basically models a counter-clockwise rotation by θ degree with respect to the ( i , k ) plane. For example, for i = 2, k = 3, the G matrix becom es: 23 23 23 23 23 10 0 0 0 cos( ) sin( ) 0 (2 , 3 , ) 0s i n ( ) c o s ( ) 0 00 0 1 G θθ θ θθ ⎡⎤ ⎢⎥ ⎢⎥ ⎢⎥ = ⎢⎥ − ⎢⎥ ⎢⎥ ⎣⎦ ( 2 9 ) The GCLT parameters are related to L MLR by the following relationship: ,, , , ,, , , MLR ,, , , ,, , , p p pq pm pn q p qq qm qn m p mq mm mn n p nq nm nn αα α α αα α α αα α α αα α α ⎡⎤ ⎢⎥ ⎢⎥ = ⎢⎥ ⎢⎥ ⎢⎥ ⎣⎦ L ( 3 0 ) where {( p , q , m , n )} ∈ {(1, 4, 6, 7), (2, 3, 5, 8), (9, 12, 14, 15) , (10, 11, 13, 16)} for T8. The mathematical derivation of optimum GCLT param eters for T8 is difficult because of the large dimension involved. Hence we rely on computer search to find the optim ization solution. The best solution found is: θ 12 = -45.66 0 , θ 23 = 9.43 0 , θ 34 = -46.11 0 , θ 14 = 37.78 0 , θ 13 = 9.13 0 , θ 24 = 44.24 0 . The diversity product ζ , defined in (31), is a good indicator of the decoding performance of a STBC [8]. () 1/ 2 min 1 Det 2 T t N ζ = ( 3 1 ) The diversity product of Q4_CR, Q4_LT, Q8_CR, Q 8_LT, T8_CR and T8_LT with 4-QAM are listed in Table 3. We can see that Q4_LT, Q8_LT and T8_LT achieve a lower diversity product (hence coding gain) than Q4_CR, Q8_CR and T8_CR, but Q4_LT, Q8_LT a nd T8_LT only need to join tly decode half of the symbols as required by Q4_CR, Q8_CR and T8_CR respec tively. Furthermore, it will be shown later on that, despite their reduced coding gains, Q4_LT, Q8_LT a nd T8_LT have negligible performance loss compared to Q4_CR, Q8_CR and T8_CR respectively. The reduction in diversity product (and hence c oding gain) of the QO-STBC with GCLT over QO-STBC with CR can be explained as follows: since GCLT onl y restrict the symbols in a group to be linearly Accepted for publication in IEE Proc. Comm unications 16 transformed, while CR combines sym bols across diffe rent group, hence CR has a higher degree of freedom when performing the minim um determinant optim izati on, and hence it achieves a higher diversity product. However, it should be noted that the higher diversity product achieved by CR comes at the expense of an increased decoding complexity. C. ML Decoding The ML decoding metrics of Q4_LT are shown in (32). {} {} 4 2 22 1 1 4 , 14 1 4 14 1 4 14 1 4 11 4 2 22 2 2 3 , 23 2 3 23 2 3 23 2 3 11 3 ( , ) ( ) ( ab b + a ) 2 R e ( ab )( b + a ) ( ab ) ( b + a ) ( , ) ( ) ( ab b + a ) 2 R e ( ab ) ( b + a ) ( ab ) ( b + a ) ( r r N nr rn N nr rn f s s h ss s s ss s s ss s s f s s h ss s s ss s s ss s s fs αβ γ χδ ϕ == == ⎡ ⎤ =− + + − + + − ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ =− + + − + + − ⎢ ⎥ ⎣ ⎦ ∑∑ ∑∑ { } { } 4 2 22 58 , 5 8 5 8 5 8 5 8 5 8 5 8 11 4 2 22 4 6 7 , 67 6 7 67 6 7 67 6 7 11 , ) ( )( a b b +a ) 2 Re (a b ) ( b +a ) ( a b )( b +a ) ( , ) ( ) ( ab b + a ) 2 R e ( ab ) ( b + a ) ( ab ) ( b + a ) r r N nr rn N nr rn s h ss s s j ss j s s ss s s f s s h ss s s j ss j s s ss s s αβ γ χ δϕ == == ⎡ ⎤ =− + + − + + − ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ =− + + − + + − ⎢ ⎥ ⎣ ⎦ ∑∑ ∑∑ (32) where ** * * ** * * * * 1 , 12 , 2 3 , 3 4 , 4 4 , 13 , 2 2 , 3 1 , 4 1 , 4 , 2 , 3 , ,, 2 R e ( ) , rr r r rr r r r r r r h r hr h r hr hr h r hr h r h h hh αβ γ = − − − − = −++− = − ** * * * * * * * * 2 , 1 1 ,2 4 ,3 3 ,4 3 , 1 4 ,2 1 ,3 2 ,4 1 , 4 , 2 , 3 , ,, 2 R e ( ) , rr r r r r r r r r r r hr h r hr h r h r hr h r hr h h hh χδ ϕ =− + − + =− − + + = − + a = cos(13.28 0 ), b = sin(13.28 0 ), and each s i is a real symbol. Each of the decoding metrics shown above depends only on two real symbols, in contrast to (6) which shows that each decoding metr ic of Q4_CR relies on two complex symbols. Hence the proposed GCLT scheme can achieve a significant reduction of decoding complexity com pared with the constellation rotation scheme. This complexity reduction is all the m ore significant for the larger QAM constellation size. Although only Q4, Q8 from [2] and T8 from [3] are us ed as exam ples in this paper, the approach described in this paper can be used to achieve the sa me reduction in decoding complexity (i.e. halving of the number of sym bols required for ML joint detection) for the other QO-STBCs reported in the literature [5-8,10] too. Accepted for publication in IEE Proc. Comm unications 17 D. Decoding Performance The BER performance of the Q4, Q4_CR and Q4_LT c odes are compared in this section, using the O-STBC from [1] as performance be nchmark. Since Q4, Q4_CR and Q4_LT are full-rate codes, while the O-STBC G4C is a half-rate code, 16-QAM constellation is used for the latter while 4-QAM constellation is used for the QO-STBCs in order to achieve the same spectra l efficiency of 2 bits/s/Hz for all codes. In Figure 4, it is observed that both Q4_CR [8] and Q4_LT (construc ted in this paper) achieve full transmit diversity as they have the same BER slope as the G4C. Q4_CR and Q4_LT also have lower BER than the G4C because they are full-rate codes with smaller QAM dimensi on and hence larger Euclidean distance. Although Q4_CR has slightly better performance (due to a larger mi nim um determ inant value as shown in Table 3) than Q4_LT, their performance difference is less than 0.5dB . Q4_LT, however, needs jo int detection of only two real symbols; hence it has a significantly lower decoding com plexity than Q4_CR, which requires the joint detection of two complex symbols. Similar observations hold for the case with two receive antennas. The comparisons between Q8_CR and Q8_LT, T 8_CR and T8_LT with 4-QAM and one receive antennas is shown in Figure 5. It can be seen agai n that QO-STBC optimized w ith GCLT has only less than 0.5dB loss in decoding performance, but the number of sym bols required for joint detection is halved as shown in Table 3. It should be noted that we adopt the STBC signal model in [11] in which the dispersion matrices are used to modulate the real and imaginary parts of a complex symbols , as opposed to the STBC signal model in [5] in which the dispersion matrices are used to modulate the complex symbols and their corresponding conjugate symbols . As a result, we are able to get an insight into the decoding complexity of QO-STBC with GCLT versus QO-STBC with CR [5]. Accepted for publication in IEE Proc. Comm unications 18 V. C ONCLUSION In this paper, we first derive the generic alge braic structure of QO-STBC, called Quasi-Orthogonality (QO) Constraint. It can be shown that all existing QO-STBCs are unified under this algebraic structure. Based on the derived QO Constraint, we find that th e constellation rotation (CR) technique, which is commonly used to im prove the decoding performan ce of a QO-STBC, actually increases the decoding complexity of the resultant QO-STBC, as the num ber of symbols required for joint detection in ML decoding is doubled after CR is applied. Hence we propose Gr oup-Constrained Linear Transformation (GCLT) as a means to improve the decoding perform ance of a QO- STBC with QAM constellation without increasing the number of symbols required for joint detection. The optimum GCLT param eters for achieving maximum diversity and coding gains are derived analytically fo r square QAM constellations. Simulation results show that QO-STBC with GCLT can achieve full diversity at less than 0.5 dB loss in coding gain compared to QO-STBC with CR. Accepted for publication in IEE Proc. Comm unications 19 R EFERENCES [1] V. Tarokh, H. Jafarkhani, and A. R. Calderba nk, “Space-time block codes from orthogonal designs”, IEEE Trans. on Information Theory, vol. 45, pp. 1456-1467, Jul. 1999. [2] H. Jafarkhani, “A quasi-orthogonal space-time block code”, IEEE Trans. on Communications, vol. 49, pp. 1-4, Jan. 2001. [3] O. Tirkkonen, A. Boariu, and A. Hottinen, “Minim al non-orthogonality rate 1 space-time block code for 3+ tx antennas”, IEEE ISSSTA 2000, vol. 2, pp. 429-432. [4] C. B. Papadias and G. J. Foschini, “Capacity-a pproaching space-time codes for systems em ploying four transmitter antennas”, IEEE Trans. on Information Theory , vol. 49, pp. 726- 733, Mar. 2003. [5] N. Sharma and C. B. Papadias, “Full rate full diversity linear quasi-orthogonal space-time codes for any transmit antennas”, EURASIP Journal on Applied Signal Processing , vol. 9, pp. 1246-1256, Aug. 2004. [6] O. Tirkkonen, “Optimizing space-time bl ock codes by constellation rotations”, Finnish Wireless Communications Workshop 2001, pp. 59-60. [7] N. Sharma and C. B. Papadi as, “Improved quasi-orthogonal codes through constellation rotation”, IEEE Trans. on Communications , vol. 51, pp. 332- 335, Mar. 2003. [8] W. Su and X. Xia, “Signal constellations fo r quasi-orthogonal space-time block codes with full diversity”, IEEE Trans. on Information Theory , vol. 50, pp. 2331 – 2347, Oct. 2004. [9] D. Wang and X. Xia, “Optimal Diversity Produc t Rotations for Quasi-Orthogonal STBC with MPSK Symbols”, to appear in IEEE Communications Letters . [10] C. Yuen, Y. L. Guan, and T. T. Tjhung, “Full-ra te full-diversity STBC with constellation rotation”, IEEE VTC-Spring 2003 , vol. 1, pp. 296 –300. [11] B. Hassibi and B. M. Hochwald, “High-rate codes that are linear in space and time”, IEEE Trans. on Information Theory , vol. 48, pp. 1804 –1824, Jul. 2002. Accepted for publication in IEE Proc. Comm unications 20 [12] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: perform ance criterion and code construction”, IEEE Trans. on Information Theory , vol. 44, pp. 744-765, Mar. 1998. [13] V. M. DaSilva and E. S. Sousa, “Fading-resist ant modulation using several transmitter antennas”, IEEE Trans. on Communications , vol. 45, pp. 1236-1244, Oct. 1997. Accepted for publication in IEE Proc. Comm unications 21 A PPENDIX A Dispersion Matrices of Q4 [2]: 1 1000 010 0 0010 0001 ⎡⎤ ⎢⎥ ⎢⎥ = ⎢⎥ ⎢⎥ ⎣⎦ A , 010 0 1 000 0001 00 1 0 ⎡⎤ ⎢⎥ − ⎢⎥ = ⎢⎥ ⎢⎥ − ⎣⎦ 2 A , 00 1 0 00 0 1 10 0 0 01 0 0 ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ − ⎢ ⎥ − ⎣ ⎦ 3 A , 00 01 00 1 0 01 0 0 10 00 ⎡ ⎤ ⎢ ⎥ − ⎢ ⎥ = ⎢ ⎥ − ⎢ ⎥ ⎣ ⎦ 4 A , 5 00 0 00 0 00 0 00 0 j j j j ⎡⎤ ⎢⎥ − ⎢⎥ = ⎢⎥ − ⎢⎥ ⎣⎦ A , 6 00 0 00 0 00 0 00 0 j j j j ⎡⎤ ⎢⎥ ⎢⎥ = ⎢⎥ − ⎢⎥ − ⎣⎦ A , 7 00 0 000 000 00 0 j j j j ⎡ ⎤ ⎢ ⎥ − ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ − ⎣ ⎦ A , 8 000 00 0 00 0 000 j j j j ⎡⎤ ⎢⎥ ⎢⎥ = ⎢⎥ ⎢⎥ ⎣⎦ A . A PPENDIX B Dispersion Matrices of Q4_CR [8]: CR_1 1000 0100 0010 0001 ⎡⎤ ⎢⎥ ⎢⎥ = ⎢⎥ ⎢⎥ ⎣⎦ A , CR_ 010 0 1 000 0001 00 1 0 ⎡⎤ ⎢⎥ − ⎢⎥ = ⎢⎥ ⎢⎥ − ⎣⎦ 2 A , CR_ 00 1 0 00 0 1 1 10 0 0 2 01 0 0 j j j j + ⎡⎤ ⎢⎥ − ⎢⎥ = ⎢⎥ −+ ⎢⎥ −− ⎣⎦ 3 A , CR_ 00 0 1 00 1 0 1 01 0 0 2 10 0 0 j j j j + ⎡ ⎤ ⎢ ⎥ −+ ⎢ ⎥ = ⎢ ⎥ −+ ⎢ ⎥ + ⎣ ⎦ 4 A , CR_ 5 00 0 00 0 00 0 00 0 j j j j ⎡⎤ ⎢⎥ − ⎢⎥ = ⎢⎥ − ⎢⎥ ⎣⎦ A , CR_ 6 00 0 00 0 00 0 00 0 j j j j ⎡⎤ ⎢⎥ ⎢⎥ = ⎢⎥ − ⎢⎥ − ⎣⎦ A , CR_ 7 00 1 0 000 1 10 0 0 2 01 0 0 j j j j j + ⎡⎤ ⎢⎥ −+ ⎢⎥ = ⎢⎥ − ⎢⎥ −− ⎣⎦ A , CR_ 8 000 1 00 1 0 01 0 0 2 10 0 0 j j j j j + ⎡ ⎤ ⎢ ⎥ − ⎢ ⎥ = ⎢ ⎥ − ⎢ ⎥ + ⎣ ⎦ A . Accepted for publication in IEE Proc. Comm unications 22 A PPENDIX C Dispersion Matrices of Q4_LT: LT_ 1 00 00 00 00 ab ab ba ba ⎡⎤ ⎢⎥ − ⎢⎥ = ⎢⎥ − ⎢⎥ ⎣⎦ A , LT_ 00 00 00 00 ab ab ba ba ⎡⎤ ⎢⎥ − ⎢⎥ = ⎢⎥ − ⎢⎥ −− ⎣⎦ 2 A , LT_ 00 00 00 00 ba ba ab ab − ⎡⎤ ⎢⎥ ⎢⎥ = ⎢⎥ −− ⎢⎥ − ⎣⎦ 3 A , LT_ 00 00 00 00 ba ba ab ab − ⎡⎤ ⎢⎥ −− ⎢⎥ = ⎢⎥ −− ⎢⎥ − ⎣⎦ 4 A , LT_ 5 00 00 00 00 j aj b ja jb jb ja j bj a ⎡⎤ ⎢⎥ − ⎢⎥ = ⎢⎥ − ⎢⎥ ⎣⎦ A , LT_ 6 00 00 00 00 ja jb j aj b j bj a jb ja ⎡ ⎤ ⎢ ⎥ − ⎢ ⎥ = ⎢ ⎥ − ⎢ ⎥ −− ⎣ ⎦ A , LT_ 7 00 00 00 00 jb ja j bj a j aj b ja jb − ⎡⎤ ⎢⎥ −− ⎢⎥ = ⎢⎥ ⎢⎥ − ⎣⎦ A , LT_ 8 00 00 00 00 j bj a jb ja ja jb j aj b − ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ − ⎣ ⎦ A . where a = cos(13.28 0 ) and b = sin(13.28 0 ). Accepted for publication in IEE Proc. Comm unications 23 L IST OF I LLUSTRATIONS Figure 1 Grouping structure of dispersion matrices of Q4 [2], Q4 with CR (Q4_CR) [8] and Q4 with Figure 2 QAM-constellation and its minimum Euclidean distance dmin Figure 3 Determinant values versus angle of optimization Figure 4 Simulation results of QO-STBCs for four transm it antennas with spectral efficiency of 2 bits/sec/Hz Figure 5 Simulation results of QO-STBCs for eight transmit and one receive Table 1 Non-fulfillment of Quasi-Orthogonality Constraint in (8) for the code Q4 Table 2 Non-fulfillment of Quasi-Orthogonality Constraint in (8) for the code Q4_CR Table 3 Comparison of QO-STBCs with CR and GCLT A 1 A 4 Group 1 Group 2 Group 3 Group 4 A 2 A 3 A 5 A 8 A 6 A 7 A CR1 A CR4 A CR5 A CR8 Group 1 Group 2 A CR2 A CR3 A CR6 A CR7 Q4_CR Q4 A LT6 A LT7 Group 1 Group 2 Group 3 Group 4 Q4_LT A LT5 A LT8 A LT1 A LT4 A LT2 A LT3 ( a ) ( b ) ( c ) Figure 1 Grouping structure of dispersion matrices of Q4 [2], Q4 with CR (Q4_CR) [8] and Q4 with GCLT (Q4_LT) Accepted for publication in IEE Proc. Comm unications 24 d min d min I Q Figure 2 QAM-constellation and its minimum Euclidean distance d min 0 5 10 15 20 25 30 35 40 45 0 0. 005 0. 01 0. 015 0. 02 0. 025 0. 03 0. 035 0. 04 0. 045 0. 05 det 5 , d et 6 m =1 n= 1 det 7 , d et 8 m =1 n= 1 det 7 m = 2 n= 1 det 6 m = 3 n= 1 det 7 m = 3 n= 1 det 5 m = 3 n= 2 det 7 m = 3 n= 2 θ opt θ det ermi nant v alue Figure 3 Determinant values versus angle of optimization Accepted for publication in IEE Proc. Comm unications 25 4 6 8 10 12 14 16 18 20 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 BER SN R 2 R x A n ten n as 1 R x A nt e nna G4C ( 16 QA M , 1rx ) Q 4 ( 4 Q AM, 1 r x) Q4 CR (4QA M, 1 rx ) Q4 LT (4QA M, 1 rx ) Q4 CR (4QA M, 2 rx ) Q4 LT (4QA M, 2 rx ) Figure 4 Simulation results of QO-STBCs for four transm it antennas with spectral efficiency of 2 bits/sec/Hz 4 6 8 10 12 14 16 18 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 Ra te -1 Q8 C R (4 QA M) Ra te -1 Q8 L T (4 QA M) Ra te -3 /4 T8 C R (4 QA M) Ra te -3 /4 T8 L T (4 QA M) BER S NR Spectr al Eff i cien cy : 1. 5 bps/ H z Spectr al Eff i cien cy : 2 bps/ H z Figure 5 Simulation results of QO-STBCs fo r eight transmit and one receive antenna Accepted for publication in IEE Proc. Comm unications 26 Table 1 Non-fulfillment of Quasi-Orthogonality Constraint in (8) for the code Q4 [2] (a) QO-Constraint fulfillment f or Q4 A 1 A 2 A 3 A 4 A 5 A 6 A 7 A 8 A 1 X X A 2 X X A 3 X X A 4 X X A 5 X X A 6 X X A 7 X X A 8 X X X : QO-Constraint is not fulfilled (b) Re-arrange rows and columns of (a) A 1 A 4 A 2 A 3 A 5 A 8 A 6 A 7 A 1 X X A 4 X X A 2 X X A 3 X X A 5 X X A 8 X X A 6 X X A 7 X X X : QO-Constraint is not fulfilled Table 2 Non-fulfillment of Quasi-Orthogonality Constraint in (8) for the code Q4_CR [8] (a) QO-Constraint fulfillment f or Q4_CR A CR1 A CR2 A CR3 A CR4 A CR5 A CR6 A CR7 A CR8 A CR1 X X A CR2 X X A CR3 X X X X A CR4 X X X X A CR5 X X X A CR6 X X X A CR7 X X X A CR8 X X X X : QO-Constraint is not fulfilled Accepted for publication in IEE Proc. Comm unications 27 (b) Rearrange rows and columns of (a) A CR1 A CR4 A CR5 A CR8 A CR2 A CR3 A CR6 A CR7 A CR1 X X A CR4 X X X X A CR5 X X X A CR8 X X X A CR2 X X A CR3 X X X X A CR6 X X X A CR7 X X X X : QO-Constraint is not fulfilled Table 3 Comparison of QO-STBCs with CR and GCLT No. of real symbols for ML joint detection Diversity product ζ for 4-QAM Q4 [2] 2 Non full diversity Q4_CR [8] 4 0.3536 Rate-1 QO-STBC for four transmit antennas Q4_LT 2 0.3344 Q8 [2] 2 Non full diversity Q8_CR [8] 4 0.2887 Rate-3/4 QO-STBC for eight transmit antennas Q8_LT 2 0.2730 T8 [3] 4 Non full diversity T8_CR [10] 8 0.2187 Rate-1 QO-STBC for eight transmit antennas T8_LT 4 0.1531