Unitary Differential Space-Time Modulation with Joint Modulation

Accepted by IEEE Trans . Vehicular Technology 1 Unitary Differential Space-Time Modulation with Joint Modulation C. Yuen, Y. L. Guan, T. T. Tjhung Abstract – We develop two new designs of unitary differential space-time modulation (DSTM) with low decod ing complex ity. Thei r decoder c an be sepa rated into a few parallel decoders, each of which has a decodi ng search space of less t han N if the DSTM codebook contains N codewords. Both designs are based on the concept of joint modulati on, which means that several information symbols are jointly modulated, unlike the conventional sy mbol-by-symbol modulation. The first design is based on Orthogonal Space-Time Block Code (O-STBC) with joint constellation constructed from spherical codes. The second design is based on Quasi- Orthogonal S pace-Time Block Code (QO-STBC) wit h specially designed pair-wise constellation sets. Both the proposed unitary DSTM schemes have considerably lower deco ding complexity than many prior DSTM schem es, in cludin g thos e bas ed on Group Codes and Sp(2) which gen erally have a decoding search space of N for a co debook size of N codewords, and much better decoding performance than the existing O-STBC DSTM scheme. Between two designs, the proposed DSTM based on O-STBC ge nerally has better decoding performance, while the proposed DSTM based on QO-STBC has lower de coding complexity when 8 transmit antennas. I. I NTRODU CTI ON Modulation techniques designed for multiple tr an smit antennas , called space-time modulation or transmit diversity can be used to reduce fa ding effects e ffectively. Ear ly transmit diversit y schemes were designed for coheren t detection, w ith cha nnel estimates assumed available at the receiver. However, the co mplexity and co st of channel esti ma tion grow with the number of transmit and receive antennas. Therefore, transmit diversity schemes that do not require channel estimation are desirable. To this end, several different ial space-time modul atio n (DSTM) schemes have been propo sed [1-7]. The DSTM schemes in [1-6] generally ha ve a decodin g search space of N for a DSTM codeboo k with N codewords due to the lack of orthogonality or quasi-ort hogonality in the code structure. This leads to an exponential increase in decoding complexity with sp ectral efficiency. For instance, in order to provide a spectral efficiency of 1.5bps/Hz, the codebook of the DSTM in [1,2] for four transmit antennas has N Accepted by IEEE Trans . Vehicular Technology 2 = 2 6 = 64 codewords, he nce its optimal decode r needs to sear ch over a spac e of 64. This is in creased to 2 8 = 256 if the spectral efficiency is increased to 2bps/Hz. On the other hand, the scheme in [7] is single-symbol decodable as it is design ed based on the square Orthogonal Space-Time Block Code (O- STBC). Hence for the sa me four trans mit antennas and spectral eff iciency of 1.5b ps/Hz, the dec oding search space per decoder o f O-STBC DSTM could be as low as 4, or 3 N where N = 64 is the codeb ook size. Howe ver, such r educt ion in deco ding comple xity is obtained with a sacrifice in the decoding performance, an d its maximum achievable co de rate is limited to ¾ for four antennas and ½ for eight antennas. To trade d ecoding comple xity for pe rfor mance, a non-unit ary DSTM scheme based on the Quasi-Orthogonal ST BC (QO-STBC) and a unitary DSTM sch eme based on unitary non-linear STBC have been proposed in [15] and [16] respectively. Both support full rate (code rate = 1 s ymbol/channel use) for fo ur trans mit antennas, and both ar e pair- wise decoda ble with a deco ding sear ch space i n between N and 3 N per deco der. In this paper, we propose two new unita ry DSTM schemes based on the concept of j oint modulation . Their decoding can be performed by two or three parallel decoders, with a search space of less than N per decoder. The first design is based on unitary matrices derived from O-STBC with joint constellation constructed from spherical codes . The second design is based on unitary matrices derived from double-symbol-decodable QO-ST BC [8 -13] with a pairwise constellation set. Compared with the sche me from [15], which is also based on QO-STBC, ou r proposed schemes are unitary DSTM designs, while that i n [15] is not. The DSTM encoder of [1 5] also has higher computational co mplexity as it needs to sol ve a se t of linear equations for every codeword to be transmitted. Compared with the schemes from [16], our proposed designs can be extended to pairwise- decodable QO-STBCs of any number of transmit antenn as; it is also able to support a wide range of spectral efficiencies; while those in [16] are not known to exhibit such flexiblity. Accepted by IEEE Trans . Vehicular Technology 3 II. R EVIEW OF U NITARY DSTM A. Unitary DSTM Si gnal Model Consider a MIM O communication sy stem with N T transmit and N R receive ante nnas. Let H t be the N R × N T channel gain matrix at a time t . Let C t be the N T × P codeword transmitt ed at a time t . Then, th e received si gnal matrix R t can be written as tt t t =+ RH C N ( 1 ) where N t is the additive white Gaussian noise. At the start of the transmission, we transmit a known codew ord C 0 , which is a unitary matrix. The codeword C t transmitted at a time t is differentially encoded by 1 tt t − = CC U ( 2 ) where U t is a unitary matrix of size N T × N T (such that U t U t H = I ), called the code matrix , that c ontains information of the transmitted data. If we assu me that the channel remains unchanged during two consecutive code periods, i.e. H t = H t -1 , the received signal R t at a time t can be expressed [7] as 11 1 1 () t t t tt t ttt t tt −− − − =+ = − + = + RH C U N R N U NR U N  (3) where 1 tt t t − =− + NN U N  is an equivalent additive white Gau ssian noise. The corresponding decision metric for (3) is, {} {} () ( ) { } H H 11 1 ˆ arg min tr arg max R e tr t t tt t t t t t t t t −− − ∈ ∈ =− − = U U U R RU R RU R RU U U (4) where U denotes the set of all possibl e code matrices. B. Diversity and Co ding Gain The design criteria of unitary DSTM scheme have b een formulated in [1] and found to be the same as those for coherent space-time coding. The transmit diversity level that can be achieved is given b y: () Min rank kl kl ⎡− ⎤ ∀ ≠ ⎣⎦ UU . (5) In order to achieve full transmit diversity, the minimum rank in (5) has to be equal to N T and the DSTM code is said to be of full rank. Fo r a full-rank unitary DSTM code, its coding gain is defined in [ 1, 7] a s () () () 1/ H Min det T N Tk l k l Nk l ⎡⎤ ×− − ∀ ≠ ⎢⎥ ⎣⎦ UU UU . (6) Accepted by IEEE Trans . Vehicular Technology 4 In order to achieve optimum decoding performance, the coding gain has to be maximized. III. A N EW U NITARY DSTM S CHE ME B ASE D ON O-STBC In this section, we shall develop a new unita ry DSTM scheme using the well-known square O- STBC. For simplicity, we will use th e rate-3/4 O-STBC for four transm it antennas describ ed in [17] as an example. The propose d unitary DSTM techniqu e is applicable to any square O-STBC for any number of transm it antenn as. A. Orthogonal Space-Time Bl ock Code The 4×4 co dewor d of rate-3 /4 O-STBC in [17] ( herein denoted as C O4 ) is shown in (7): 12 3 * 13 2 * 23 1 32 1 0 0 0 0 * * ** cc c cc c = cc c cc c ⎡⎤ − ⎢⎥ ⎢⎥ ⎢⎥ −− ⎢⎥ − ⎢⎥ ⎣⎦ O4 C ; H 000 00 0 00 0 000 α α α α ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ O4 O4 CC (7) where c i , 1 ≤ i ≤ 3, represents the complex inform ation symbol to be transmitted and 3 2 1 i i =c α = ∑ ( 8 ) The determinant of the code word distance m atrix o f C O4 can be eas ily shown t o be: 4 3 1 det 2 i i = ⎛⎞ =∆ ⎜⎟ ⎝⎠ ∑ ( 9 ) where ∆ i , 1 ≤ i ≤ 3, represents the p ossible error in the i th transmitted constellation symbol. O-STBC can always ac hieve full di versity as ( 9) can neve r become zer o as long as ∆ i is not zero for all i . B. New Unitary DSTM Scheme Based on O-STBC with Joint Modulation C O4 in (7) can be used as the u nitary code matrix of a unitary DSTM scheme as long as α is equal to 1. The DSTM approach taken in [7] is to select the symbols c i from a PSK constellation (that has a constant power) to conform to (7). In order to achieve a better performance, we let multiple symbols be jointly modulated in order to conform to (7). Specifically , we separate the data into two groups with three real symbols per group, and jointly modulate these three real symbols. This suggests that RI R 11 2 ,, cc c cou ld be jointly mapped to a tri-symbol { a k , b k , c k }, while IRI 23 3 ,, cc c are mapped to another tri- Accepted by IEEE Trans . Vehicular Technology 5 symb ol { a l , b l , c l } , s u c h t h a t | a k | 2 + | b k | 2 + | c k | 2 = | a l | 2 + | b l | 2 + | c l | 2 = 0.5 f or al l val ues of k and l . To maximize the coding gain, the symbol-pairs should further be designed to maximize the minimum value of det in (9). In short, in this sch eme we use C O4 with code symb ols drawn fro m a special j oint constellation set M consisting of complex-valued symbol-pairs { a k , b k , c k } that satisfy the following criteria: (i) Power Criterion: 0.5 222 kkk a+ b c += ( 1 0 ) (ii) Performance Criterion: { } 222 maxi mize Mi n kl kl kl ab c ∆+ ∆ + ∆ where kl k l aa a ∆= − , kl k l bb b ∆=− , and kl k l cc c ∆ =− for all k ≠ l . T he syst ematic desig n of M will be elaborated in the next section. The spectral efficiency , Eff , of the resultant unitary DSTM schem e based on rate-3/4 O-STBC is Eff = 2(l og 2 L )/ N T bps/Hz w here L is the total number of symbol-pairs { a k , b k , c k } in M . For example, consider the case of four tr ansmit antennas ( N T = 4) and a tar get spectral effici ency of Eff = 1.5 bp s/Hz. From the Eff expression ab ove, the requi red conste llation size is L = 8 = 2 3 . Therefore, in t he encoder of the proposed DSTM scheme, a constellation set M with 8 tri-symbol’s will first h ave to be designed according to (10). Every first three informat ion bits will be mapped to a tri-symbol { a k , b k , c k } drawn from M to constitute the code symbols { RI R 11 2 ,, cc c } in C O4 in (7), while every next three information bits will be mapped to another symbol-pair { a l , b l , c l } drawn from M to constitute the code symbols { IRI 23 3 ,, cc c } in C O4 . In the decoder of the proposed DSTM scheme, with U t in (4) s et to C O4 in (7), the decision metrics in ( 4) can be sim pli fied to: { } ( ) { } ( ) { } {} ( ) {} () {} RIR 11 2 IRI 23 3 RI R H R H I 11 2 1 1 1 1 1, 2 {, ,} IRI H R H I 23 3 1 3 3 1 2,3 {, ,} ˆˆ ˆ , , arg max Re tr Re t r ˆˆ ˆ , , arg max R e tr Re tr tt i i tt i cc c tt tt i i i cc c cc c c j c cc c c j c −− = ∈ −− = ∈ ⎡⎤ =+ ⎣⎦ ⎡⎤ =+ ⎣⎦ ∑ ∑ RR A RR B RR A RR B M M (11) where A k and B k are the dispersio n matrices of C O4 [17]. The above shows that the proposed DSTM scheme can be decoded by join tly detecting the three real sy mbols { RI R 11 2 ,, cc c }, then the other three re al sy mbols { IRI 23 3 ,, cc c }. Hence th e search spac e is the Accepted by IEEE Trans . Vehicular Technology 6 square root of those reported in [1- 6]. In gene ral, this DST M achieves a decoding se arch space of N per decoder for a codebook with N codewords. C. Design of Joint Constellation Set from Spherical Code To design the joint constellation set M of tri-symbol to meet the c ond itions in (10), we note from (10)(i) that th e constellation points mu st lie on the surface of a 3-dime nsional sp here, and fro m (10)(ii) that the constellation points must be spaced as far apart as p ossible (i.e. the minim um distance betw een them is maximized). This falls under the realm of spherical code . Spherical code (or spheri cal packing ) dea ls with th e problem of distributing n points on a sphere in d dimensi on suc h that the mini mum dist ance (or e quivale ntly the mini mal a ngle) bet ween a ny pair of point s is ma ximiz ed, an d the ma ximu m dist ance i s call ed the covering radius . For the above DSTM example of four transmit antennas wi th spectral efficiency of 1.5bps/Hz, we need a spherical code with 3 dimensions and eight points (i.e. ei ght sets of tri-symbol). A list of optimal spherical code has been found in [18]. In Tab le 1 we list the dimension d , number of points n , and mini mum an gula r sepa rati on θ min of a few spherical codes. The exact config urations of the spherical code with dimension 3 an d 16 points are shown in Appendix A for illustrati on. Table 1 Minimal separation for some optimal spherical codes Dimen sion d Points n Minimum separation θ min (degree) 3 8 74.85 85 3 16 52.2444 4 64 42 .3062 IV. A N EW U NITARY DSTM S CHEME B ASED ON QO-STBC The O-STBC DSTM schem e proposed in Section II I does not have fu ll rate due to the complex O- STBC used. We are in terested to know how a DSTM scheme based on QO-STBC, which is known to support a higher code rate than O-STBC, will perfo rm. Of course, for a fair comparison, both DSTM Accepted by IEEE Trans . Vehicular Technology 7 schemes need to h ave the same or si milar decoding complexity, which i n this paper means a decodi ng search space less than N per decoder for a codebook with N codewords. In this section, we propose anot her new unitary DSTM scheme based on rate-1 double-sy mbol- decodable QO-STBC. The proposed technique is appl icable to any double-sy mbol-decodable square QO-STBC, such as those desi gned in [8-11]. Here we will use the QO-STBC in [8] for four transmit antenna s as an example . A. Quasi-Orthogonal Space-Time Block Code The 4 × 4 codeword of th e QO-STBC in [8] (herein denoted as C Q4 ) is shown in (12): ** 123 4 ** 21 4 3 ** 34 1 2 ** 43 2 1 c- c - c c cc- c - c = c- c c - c cc c c ⎡⎤ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎢⎥ ⎣⎦ Q4 C ; H 00 00 00 00 α β αβ βα β α ⎡ ⎤ ⎢ ⎥ − ⎢ ⎥ = ⎢ −⎥ ⎢ ⎥ ⎣ ⎦ Q4 Q4 CC (12) where c i , 1 ≤ i ≤ 4, represents the complex inform ation symbol to be transmitted and 4 2 1 i i =c α = ∑ ; ** 14 2 3 =2 × R e ( c × c c × c ) β − . (13) Following [9-13], the m ini mum d et er mi na nt val ue of C Q4 is obtained by assum ing half of the codeword errors to be zero, i.e . 2 min 23 det ( ) ( ) assuming = =0 22 14 14 ⎡⎤ =∆ + ∆ × ∆ − ∆ ∆ ∆ ⎣⎦ (14) In order to achieve full transmit diversity and opti mum coding gain, the value of det min in (14 ) h as to be non-zero and maximized. B. New Unit ary DSTM Sch eme Based on QO-STBC with Joint Modulation To use the C Q4 in (12) as the unitary code matrix of a unitary DSTM scheme, its α and β values in (13) must be equal to 1 and 0 respectively, i.e. C Q4 must be unitary. However, unlike O-STBC, generally β = 0 cannot be achieved in QO-STBC if c 1 to c 4 are conventional independent PS K or QAM symb ols. To have β = 0, we can see from (13) that Re( c 1 c 4 * ) must be equal t o Re( c 2 c 3 * ). This suggests that c 1 and c 4 should be jointly map ped to a symbol-pair { a k , b k }, while c 2 and c 3 should be mapp ed to another sy mbol-pair { a l , b l }, such that Re( a k b k * ) = Re( a l b l * ) for all k and l . To further achieve α = 1, Accepted by IEEE Trans . Vehicular Technology 8 Eq.(13) shows that these s ymbol-pairs must further satisfy | a k | 2 + | b k | 2 + | a l | 2 + | b l | 2 = 1, or | a k | 2 + | b k | 2 = | a l | 2 + | b l | 2 = 0.5 if all symbol-pairs are required to ha ve equal power. In ad dition, to maximize the codin g gain, t he symb ol-pai rs sho uld furt her be desig ned to ma ximi ze the val ue of det min in (14). Summari zing the above, we conclude that to ma ke the QO-STB C codeword C Q4 unitary, its code symbols must be drawn from a special pairwise/joint constellation set M wh ich consists of complex- valued sy mbol-pairs { a k , b k } that satisfy the following criteria: (i) Unitary Criterion: * Re( ) kk ab v = (ii) Power Criterion: 0.5 22 kk a+ b = (15) (iii) Performance Criterion: { } 2 22 maxi mize Mi n kl kl kl kl ab ab ⎡ ⎤ ∆+ ∆ × ∆− ∆ ⎣ ⎦ where v can be any constant value, and kl k l aa a ∆ =− , kl k l bb b ∆ =− for all k ≠ l . The spectr al efficiency , Eff , of the resultant unitary DSTM sche me ba sed o n full -rate QO- STBC i s Eff = 2(log 2 L )/ N T bps/Hz where L is the total n umber of sym bol-pairs { a k , b k } in M . For example, cons ider a system with four transmit ante nnas ( N T = 4) and a target spe ctral effici ency of Eff = 2 bps/Hz. From the equation of Eff , the required constellation size is L = 16 = 2 4 . In the encoder of th e proposed DSTM scheme, a pairwise constell ation set M with 16 symbol-pairs will first have to be designed acco rding to (15). Four in formation bits will be mapped to a symbol-pair { a k , b k } in M to constitute the code symbols { c 1 , c 4 } in C Q4 in (12), while another four information bits will be mapped to another sy mbol-pair { a l , b l } in M to constitute the code sym bols { c 2 , c 3 } in C Q4 . With such pairwise constellation design, the resultant C Q4 will be unitary and can now be used as the unitary DSTM code ma trix U t . In the decoder of th e prop osed DSTM sc heme, with U t in (4) set to C Q4 in (12), the de cisio n metr ics in ( 4) can be sim plified to : {} ( ) { } ( ) { } {} () {} () {} 14 23 HR H I 14 1 1 1, 4 1, 4 {, } HR H I 23 1 1 2,3 2,3 {, } ˆˆ , a r g m a x R et r R et r ˆˆ , a r g m a x R et r R et r tt i i tt i i ii cc t tii t t ii ii cc cc c j c cc c j c −− == ∈ −− == ∈ ⎡⎤ =+ ⎣⎦ ⎡⎤ =+ ⎣⎦ ∑∑ ∑∑ RR A RR B RR A RR B M M (16) where A k and B k are the dispersio n matrices of C Q4 [9]. As show n in (16), th e pro posed DSTM sc he me can be decoded by the joint detection of two complex symbols ( c 1 and c 4 , or c 2 and c 3 ), and the two decision metrics c an be computed separately. So Accepted by IEEE Trans . Vehicular Technology 9 the proposed unitary DSTM scheme is double-symbol decodable, just like its coherent counterpart i n [8], and it has a lower decoding co mplexity than the DSTM schemes in [1 - 6], which generally req uire a larger joint detection search space dimension. Thi s will be elaborated in S ection V, when we present perfor mance results of our pro posed code s. C. Design of Specific Joint Constellation Set In this section, we propose a pairwise constellation set M that satisfies all three requirements in (15) with good scalability in spectral ef ficiency. The proposed constellation set M is: {} () {} () exp 2 / 2 , for 1 / 2 0 0 , f o r / 2 exp 2( / 2) / 2 k kk k k kk k aj k M ab k L b a ab L k L bj k L M π πθ ⎧ =⎡ ⎤ ⎪ ⎣⎦ =≤ ≤ ⎨ = ⎪ ⎩ = ⎧ ⎪ =< ≤ ⎨ =⎡ − + ⎤ ⎪ ⎣⎦ ⎩ (17) where M = L /2 is an integer, and θ is a real number betwe en 0 and 2 π / M . Note that in (17), the param eter M is related to the spectral efficiency Eff by Eff = 2(log 2 2 M )/ N T (since L = 2 M ) for a full-ra te QO-STBC. Hence a unitary DSTM scheme with a wide range of spectral efficiency can be syste matically designed fro m (17) by adjustin g M . The parameter θ provides an extra degree of freedom to maximize t he diversity and coding gain of the resultant unitary DSTM sche me. Theorem 1 : For the unitary DSTM constellation set defined in (17), the optimum value of θ (in the sense of the Performance Optimi zation Criterion (15)(iii)) is π / M when M is even, and is π /2 M or 3 π /2 M when M is odd. Proof of Theorem 1 : The proof is given in four cases, considering different values of k and l . Case 1: 1 ≤ k, l ≤ L/2, an d k ≠ l. Since b k and b l are always zero in this case, ∆ b kl is always zero, but not ∆ a kl (since k ≠ l ). Hence the determinant value in (15)(iii) can never be zer o, and its value is independent of θ . This implies that, in this case, full diversity is always achieved, and the coding gain does not depend on θ . Case 2 : L/2 < k, l ≤ L and k ≠ l. The proof is similar to Case 1 with the ro les of ∆ a kl and ∆ b kl interchanged. Accepted by IEEE Trans . Vehicular Technology 10 Case 3: 1 ≤ k ≤ L/2 a nd L/ 2 < l ≤ L. For this case, the determinant value in (15)(iii) can be simplified to: () ( ) () ( ) 2 (e x p 2 / e x p 2 ( /2 ) / ) 1 det 16 ( exp 2 / exp 2( / 2) / ) 2 2 jk M j l L M jk M j l L M ππ θ ππ θ ⎡⎤ ⎡⎤ + ⎡ − + ⎤ × ⎣⎦ ⎣ ⎦ ⎢⎥ = ⎢⎥ ⎡⎤ − ⎡ − + ⎤ ⎢⎥ ⎣⎦ ⎣ ⎦ ⎣⎦ (18) () 2 2 4 22 = sin 1 , where 2 2 sin 1 km L L km l - M M MM 2 np 0nk - mM - M ππ θ π ⎛⎞ ⎛⎞ −− ≤ ≤ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎛− ⎞ =≤ ≤ ⎜⎟ ⎝⎠   where k , l , m , n are inte gers, an d / p M2 θ π  is a real numb er betwe en 0 and 1 incl usively . (a) M even (b) M odd Figure 1 Optimization of constellation rotation angle / 2p M θ π = To maximize the determinant va lue in (18), we first c onsider even va lues of M . As shown by the triangular markers in Figure 1(a), the function si n[2 n π / M ] (solid line) is zero if n = 0 or M /2. This results in a zero determinant value in (18) and the resultant unitary DSTM will not deliver full diversity. In order to achieve full diversity a nd maximum coding gain, a right shift p can be introduced to obtain the function sin[2( n-p ) π / M ] (dashed line) such that its minimum absolute values at integer values of n are non-zero and maximized, as shown by the circular markers i n Figure 1(a). Clearly, this optimum point is reached when p = 0.5, which corresponds to θ = π / M . Accepted by IEEE Trans . Vehicular Technology 11 Similarly , as indicated by the circular marker s in Figure 1(b) , the optimu m p value for odd values of M is 0.25 or 0.75, which corresponds to θ = π /2 M or 3 π /2 M . Case 4: 1 ≤ l ≤ L/2 and L/2 < k ≤ L. The p roof is sim ilar t o Case 3. In summary, as the determinant value in (15)(iii) does not depend on θ for Cases 1 and 2, and the opti mum θ value has been derived for Cases 3 and 4, Theor em 1 is proved. ■ There ar e a few im portant points to take note about th e pairwise constellation set proposed in (14) and its optimum design stated in Theo rem 1: 1. The constellation set sp ecified in (14) is not th e same as the conventio nal PSK with constellation rotation described in [9-13]. It is a special “p airwise/join t” constellation set that assigns 2 constellation points to 2 code symbols at a ti me, i.e. c 1 and c 4 , or c 2 and c 3 in C Q4 . In contrast, coherent QO-STBC ty pically assigns constellation points to individual codeword symbols independently (i.e. symbol by symbol), not 2 symbols at a time. Therefore our results in Theorem 1 is fundamentally different from the constellation rotation results reported in [9- 13], as Theorem 1 per tains s pecif ically to the pr opos ed pairwise constellation set (14). 2. The “zero” sy mbols in a k or b k in (17) do not reduce the code rate of the proposed DSTM scheme by half. This is be cause every constellation pair { a k , b k } in (17 ) represen ts 2 code symbols in the DSTM codeword, hence the “zero” symbols in a k or b k actually carry information – they are not null symbol s. Since there are 4 code symb ols in C Q4 and they will be represented by 2 pairs of complex co nstellation points drawn from (17), the proposed DSTM scheme effectively transmit 4 complex symbols in 4 sym bol times, hence its code rate remains as 1, which is the same as the original C Q4 . V. P ERFORMANCE R ESULTS We now compare our two proposed double-sy mbol-decodable unitary DSTM schemes (one based on O-STBC with spherical code, the other based on QO-STBC with optimized joint constellation set specified in (17) and Theore m 1 ) against each other, as well as against existing unitary DSTM schemes. In Table 2, we compare the coding gain and de coding complexity of our proposed DSTM schemes Accepted by IEEE Trans . Vehicular Technology 12 against those based on square O-STBC [7] and group codes [1, 2] for four trans mit antennas. We can see that both proposed unitary DSTM schemes provide higher coding gain than t he O-STBC and group- code DSTM schemes, at both spectral efficiency values of 1.5bps/Hz and 2bps/Hz. Our proposed DSTMs also have lower decoding co mplexity than the group-cod e DSTM, because they can be decoded with two paralle l decoders, each with a decoding sear ch space dimensi on of 8 at 1.5bps/Hz and 16 at 2 bps/Hz. Although the unitary DSTM based on rate-3/4 square O-STBC with 64PSK [7] h as an even lower decoding complexity at a spectral efficiency 1.5bps/Hz, it suffers from a much lower coding gai n. At a spectral efficiency of 2 bps/Hz , our propos ed DSTMs have higher coding gai ns than, and equal decoding se arch space d imension as, the D STM ba sed on rate-1/2 square O-STBC with 16PSK [7]. It should also be noted that the use of joi nt modulation enables our DSTM schemes to fle xibly support various spectral efficie ncy values. This is not possible fo r the O-STBC and QO-STBC DSTM sch emes proposed in [15,16]. Table 2 Comparison of coding gain and decoding search s pace per decoder of unitary DSTM sch emes fo r four transmit an tennas Spectral efficiency Unitary DSTM scheme Constellation Coding gain Number of parallel decoders Decoding search space p er decoder 1.5 [1, 2] 64PSK 1.85 1 64 1.5 Propos ed scheme with rate -3/4 O-ST BC (Section III) Spherical code (3d, 8 points) 2.95 2 8 1.5 Propos ed scheme with rate-1 QO -STBC (Section IV) (17) with M = 4, θ = π / 4 2.83 2 8 1.5 Rate-3/4 O-STBC [7] QPSK 2.70 3 4 2 [1, 2] 256PSK 0.78 1 256 2 Propos ed scheme with rate-3/4 O-ST BC (Section III) Spherical code (3d, 16 points) 1.55 2 16 2 Propos ed scheme with rate-1 QO -STBC (Section IV) (17) with M = 8, θ = π / 8 1.17 2 16 Accepted by IEEE Trans . Vehicular Technology 13 2 Rate-1/2 O-STBC [7] 16-PSK 0.31 2 16 In Figure 2, we compare the block error rate (BLER ) performance of our proposed DSTM schemes with those report ed in [6, 7] for four transmit and one rece ive antennas. Co mpared with the 2 bps/Hz DSTM based on rate-1/2 square O-STBC [7], both our proposed DSTM schemes have much better BLER performance, which agrees with the superior coding gain observation already made in Table 2. Compared with the Sp(2) DSTM schem e of spectral efficiency 1.94 bps/Hz and decoding search space dimension of 225 (obtained from [6 ] with M =5 , N =3), our proposed DSTM schemes perform no more than 1dB worse, but both have much smaller decoding search space dimension o f 16 and a slightly higher spectral efficiency . 10 12 14 16 18 20 22 24 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 BLER SN R R ate - 1/2 O- ST BC 2b ps/H z [7 ] SP( 2 ) 1.9 4b ps/H z [6 ] R ate - 1 Q O - ST BC 2b ps/H z [pr o p o se d ] R a te- 3/4 O - ST BC SP 2 b p s/H z [p r op ose d ] Figure 2 Block error ra tes of different DSTM schemes for four tx and one rx antennas Table 3 compares the coding gain and decoding search space per decoder of three unitary DSTM schemes for eight transmit antennas. The first DSTM is our proposed DSTM bas ed on a rate-1/2 O- STBC with spherical code. In t his setting, we jointly modulate two complex symbols. In order t o achieve a s pectral effi ciency of 1.5bps/Hz , we em ploy a spherical code of four dimensions (two complex symbols is equivalent to four real dimensions) and 6 4 points. It can be decoded by two parallel decoders, each with a decoding search space of 64. The second DSTM under comparison is our Accepted by IEEE Trans . Vehicular Technology 14 proposed DSTM based o n rate-3/4 QO-STBC with the joint constellation set specified in ( 17) and Theorem 1 . In this setting, we jointly modulate two co mplex sy mbols . This DS TM can b e deco ded by three parallel decoders, each with a search space of 16. The thi rd DSTM under comparison is the DSTM ba sed on ra te-1/2 O-ST BC from [7] that employs 8-PSK . Decodi ng it requir es four pa rallel decoders, each with a search space of 8. T able 3 shows that both our proposed DSTM sche mes have higher codin g gain than the third s cheme. Interestingly , among our two proposed sche mes, the one based on O-STBC with s pherical code has a higher coding gain, but als o larger decoding search space, than that based on QO-STBC (un like the case of four transmit an tennas). In T able 3 we also demonstrate that our propose d QO-STBC DSTM scheme ca n be extended to eight tran smit antennas and still maintains pair-wise decoding complexity. Su ch ex tension to eight transmit antennas is not possible for the non-linear DSTM reported in [16]. Table 3 Compa rison of coding gains a nd decodi ng search space per decoder of unitary DSTM for eight transmit antennas Spectral efficiency Unitary DSTM scheme Constellation Coding gain Number of parallel decoder Decoding search space per decoder 1.5 Proposed scheme with rate-1/2 O-S TBC (Section III) Spherical code (4d, 64 points) 2.08 2 64 1.5 Proposed scheme with rat e-3 /4 QO-S TB C (Section IV) (17) with M = 8, θ = π / 8 1.56 3 16 1.5 Rate- 1/2 O-STBC [7] 8-PSK 1.17 4 8 Accepted by IEEE Trans . Vehicular Technology 15 6 8 10 12 14 16 18 20 10 -4 10 -3 10 -2 10 -1 10 0 BLER SN R R a te- 1 / 2 O - ST BC (8 PSK) [7 ] Rate-3/ 4 QO-S T B C [ propos ed] Rate-1/ 2 O-S T BC S C [ propos ed] Figure 3 Block error rates of for eight tx an d one rx antennas at 1.5 bits/sec/Hz The decoding performan ce of the three D STM sche mes from Ta ble 3 are co mpared in Figur e 3. We can see that the best- performing sc heme is our propos ed DST M based on rate -1/2 O- STBC wit h spherical code, followed by our proposed DSTM based on rate-3/4 QO-STBC, and lastly the DSTM based on rate-1/2 square O-STBC from [7]. This gene rally agrees with the coding gain ranking in Table 3, but we can observe that the BLER performan ce difference between our proposed O-STBC and QO- STBC schemes is not as large as what their coding gain difference may suggest. VI. C ONCLUSIONS Two new unitary differential sp ace-time modula tion (DSTM) sche mes with low decoding complexity are proposed. The first d esign is based on O-STBC with joint constellation constructed fr om spherical code. The main idea is to jointly modulate multiple symbols using a set of joint constellation points constructed based on spherical codes The second design is based on unitary m atrices constructed from double-symbol-decodable QO-STBC. The main id ea is to force the QO -STBC codeword to be a unitary matrix by using pair-wise s ymbol modulation with a specially design ed constellation set. Our proposed unitary DSTM sche mes have much smaller decoding search spac e per decoder than the DSTM schem es repo rted in [1 – 6] , with comparable or better coding g ain or decoding performance. Compared wi th the O-STBC DSTM r eported in [ 7], at a bloc k error rate of 10 -3 or lower, both our Accepted by IEEE Trans . Vehicular Technology 16 proposed DSTM schemes give more than 3dB and 1dB decoding performance gain for 4 and 8 transmit antennas respectively, with only a slight increase in decoding c omplexity. ACKNOWLEDGMEN T The authors would like to than k the anony mous reviewers and the associate editor for their comments that greatly improve this paper. REFERENCES [1] B. L. Hughes, “Differe n tial space-time modulation”, IEEE Trans. on Info. Theory , vol: 46, Nov. 2000, pp. 2567–2578. [2] B. L. 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S mith and other s, “Tab les of Spherical Code s”, published el ectronica lly at www.research.att.c om/~nj as/packing s/ Appendix A Optimal 16-Point 3-Dimension Spherical Code a k b k c k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0.089527456 0.68133324 8 -0.166642852; -0.418469 889 0.54508083 1 0. 16664285 2; 0.057360 813 0.436534568 0.553305 8; -0.268116 333 0.34923677 3 - 0.553305 8; -0.6813332 48 0.089527456 -0.166642852; -0.5450808 31 -0.4184698 89 0.166642852; -0.436534 568 0.05736081 3 0.5533 058; -0.349236 773 -0 .2681163 33 -0.55 33058; -0.089527 456 -0 .6813332 48 -0.1 66642852 ; 0.41846988 9 -0.545 080831 0.1666428 52; -0.057360 813 -0.43653 4568 0.553305 8; 0.26811633 3 -0.349 236773 -0.5533 058; 0.68133324 8 -0.089 527456 -0.1666 42852; 0.54508083 1 0.41846 9889 0.16 6642852; 0.43653456 8 -0.05736 0813 0.553305 8; 0.349236773 0.268116333 -0.5533058.