Exact Bit Error Probability of Orthogonal Space-Time Block Codes with Quadrature Amplitude Modulation

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1 JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 0, NO. 3, SEPTEMBER Exact Bit Error Probability of Orthogonal Space-Time Block Code with Quadrature Amplitude Modulation Sang-Hyo Kim, Jae-Dong Yang, and Jong-Seon No Abtract: In thi paper, the performance of generic orthogonal pace-time block code OSTBC) introduced by Alamouti [], Tarokh [3], and Su and Xia [] i analyzed. We firt define onedimenional component ymbol error function ODSEF) from the exact expreion of the pairwie error probability of an OSTBC. Utilizing the ODSEF and the bit error probability BEP) expreion for quadrature amplitude modulation QAM) introduced by Cho and Yoon [9], the exact cloed-form expreion for the BEP of linear OSTBC with QAM in quai-tatic Rayleigh fading channel are derived. We alo derive the exact cloed-form of the BEP for ome OSTBC which have at leat one meage ymbol tranmitted with unequal power via all tranmit antenna. Index Term: Bit error probability BEP), orthogonal pace-time block code OSTBC), pairwie error probability PEP), quadrature amplitude modulation QAM), pace-time block code, pacetime code. approximating log likelihood ratio LLR). Cho and Yoon [9] derived the general BEP expreion of rectangular QAM in AWGN channel. But, the exact cloed-form expreion for BEP of OSTBC with QAM ha not been reported o far. In thi paper, we define the one-dimenional ymbol error function ODSEF) for each ymbol in an OSTBC a given in [0], which can be obtained from the exact PEP expreion [4], [5]. We alo define term homogeneou and nonhomogeneou which ditinguih the power allocation uniformity of a certain meage ymbol over given tranmit antenna. It i hown that, for all homogeneou and ome nonhomogeneou OSTBC, the ODSEF can be expreed in cloed-form. Then, uing the general BEP expreion for QAM [9] and the ODSEF, we derive the exact cloed-form expreion for the BEP of all homogeneou [], [] and ome known nonhomogeneou OST- BC []. I. INTRODUCTION Multiple input multiple output MIMO) ytem with pacetime code which were introduced by Tarokh, Sehadri, and Calderbank [], ignificantly outperform ingle tranmit antenna ytem due to their tranmit diverity. Alamouti propoed a imple tranmit diverity cheme with two tranmit antenna which employ the complex orthogonal deign []. Tarokh, Jafarkhani, and Calderbank [3] generalized the Alamouti cheme to pace-time block code from orthogonal deign which we call orthogonal pace time block code OST- BC). The OSTBC have the advantage that they guarantee full diverity and low decoding complexity linear with repect to the number of the meage ymbol in a codeword matrix. A part of ome effort to analyze the nature of pace-time code, Simon [4] and Taricco and Biglieri [5] independently worked on the exact expreion for the pairwie error probability PEP). Lu, Wang, Kumar, and Chugg [6] alo derived the exact PEP and the bit error probability BEP) of BPSK and QPSK for ome OSTBC. Uing the reult, the exact ymbol error probability SEP) of OSTBC with quare quadrature amplitude modulation QAM) wa derived [7]. Recently, Raju, Annavajjala, and Chockalingam [8] analyzed the BEP of OSTBC by Manucript received November 4, 007 ; approved for publication by Xiang- Gen Xia, Diviion I Editor, February 04, 008. S.-H. Kim i with the School of Information and Communication Engineering, Sungkyunkwan Univerity. J.-D. Yang and J.-S. No are with the Department of EECS, INMC, Seoul National Univerity. Thi work in part wa preented in IEEE International Sympoium on Information Theory 003,Yokohama, July 003. Thi reearch wa upported by the MEST, the MKE, and the MOLAB, Korea, through the fotering project of the Laboratory of Excellency and by the Korea Reearch Foundation Grant funded by the Korean GovernmentMOEHRD). KRF D0004) /08/$0.00 c 008 KICS II. SYSTEM MODEL Let L t be the number of tranmit antenna and L r the number of receive antenna of MIMO ytem. Let X =[x n,i ] be the N L t codeword matrix of an OSTBC, where x n,i i the ymbol tranmitted from the ith tranmit antenna at the nth time. We conider the quai-tatic Rayleigh fading channel, where fading i aumed to be contant over the duration of each codeword matrix and independent. Let α i,j be the channel coefficient from the ith tranmit antenna to the jth receive antenna and A = [α i,j ] be the channel matrix. α i,j are independent complex Gauian random variable with zero mean and unit variance. Then, the N L r received matrix Y =[y n,j ] i given by Y = ρ E m XA + W where y n,j i the received ymbol of the jth receive antenna at the nth time, ρ the average ignal to noie ratio SNR), E m the average energy tranmitted from the L t tranmit antenna during a ymbol period, and W =[w n,j ] the noie matrix which conit of independent, identically ditributed i.i.d.) complex Gauian random variable with zero mean and unit variance. The perfect channel etimation i aumed. In thi paper, the quare and rectangular linear OSTBC are conidered, where the codeword matrice have columnwie orthogonality. Let be the meage vector of length L given by =,,, L ), which i encoded into the codeword matrix X. Letb be the number of bit per meage ymbol k.let C) be a mapping from an L -tuple complex meage vector to an N L t codeword matrix. For linear OSTBC, each element in a codeword matrix i a linear combination of the meage ymbol k and their complex conjugate.

2 54 JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 0, NO. 3, SEPTEMBER 008 Variou linear complex OSTBC have been introduced in [], [3], and [] which are lited a ) C =, ) L =,N = L t =, code rate =, 3 C = ) ) L =3,N =4,L t =3, code rate = 3/4,, ) 3 0 C 3 = , 3) 0 3 L =3,N =4,L t =4, code rate = 3/4, C 4 = , 4) L =7,N =,L t =5, code rate = 7/. It i obviou that they are columnwie orthogonal. All column vector in an OSTBC do not alway have the ame magnitude. The quared magnitude of the ith column in a linear OSTBC C) correpond to the total energy tranmitted from the ith tranmit antenna during N ymbol time, that i, g k,i k 5) where g k,i denote the multiplicity of k in the quared magnitude of the ith column of C). Uing 5) and the columnwie orthogonality of the linear OSTBC, we have C) H C) = diagg k,,g k,,,g k,lt } k 6) where ) H denote the Hermitian operator and diag} the diagonal matrix. The linear OSTBC can be claified according to the regularity of the value of g k,i. A ymbol k in an OSTBC i called homogeneou if nonzero value of g k,i are contant and otherwie, nonhomogeneou. We call an OSTBC homogeneou if all ymbol in the code are homogeneou and otherwie, nonhomogeneou. Mot known OSTBC [], [3], [] are homogeneou code, while Su and Xia [] introduced two nonhomogeneou code, and one of them i C 4 in 4). Let E be the average ymbol energy of k. Thu the average energy E m tranmitted from all the L t tranmit antenna during a ymbol period i given a E m = N L t L g k,i E. i= In thi paper, we conider only QAM with Gray map [9]. A rectangular M-ary QAM, M = I J, i compoed of I-ASK for the real part of k and J-ASK for the imaginary part. Let b be the number of bit which are modulated to a ymbol k.then we have M = b.leti = b, and J = b,. The minimum ditance among contellation i aumed to be d. Foragiven average ymbol energy E,wehaved = 3E /I + J ). III. ONE-DIMENSIONAL SYMBOL ERROR FUNCTION A. Symbol Decoding of OSTBC Uing the columnwie orthogonality, the linear OSTBC can be decoded ymbol by ymbol [3], [3]. In thi ection the ymbol by ymbol maximum likelihood decoding i reviewed. Let Y =[y, y,, y Lr ] be a received ignal matrix correponding to the tranmiion of X through the known channel A, where y j =y,j,y,j,,y N,j ) T. The maximum-likelihood decoder chooe the codeword matrix ˆX =[ˆx n,i ] which minimize the deciion metric given by D = N L r y n,j L t γ α i,j ˆx n,i n= j= where γ = ρ/e m.leta =[a, a,, a Lr ]= γa, where a j = γα,j, γα,j,, γα Lt,j) T. Uing the columnwie orthogonality of OSTBC, the deciion metric in 7) can be rewritten a L r L D = y j ˆXa j b k = H k ŝ k H k + C 8) j= i= where the poitive real number H k i given by 7) L r L t H k = γ g k,i α i,j 9) j= i= and b k i a function of the received ignal y n,j and the known channel coefficient α i,j. It wa rigorouly derived in [3]. Let b k = b k/h k, which i dependent on y n,j and α i,j. The deciion metric D then reduce to D = H k b k ŝ k + C = H k D k + C 0)

3 S.-H. KIM et al.: EXACT BIT ERROR PROBABILITY OF ORTHOGONAL SPACE-TIME where the deciion metric D k for each ymbol i defined a D k = b k ŝ k. ) Since the H k and C are not dependent on the meage ymbol ŝ k, the maximum-likelihood deciion metric D can be minimized by minimizing each ymbol deciion metric D k, independently, which mean that it i poible to decode each ymbol, independently. The deciion metric D k can eaily be proved to be written a D k = b k ŝ k = k + e k ˆ k ) H k where e k i a function of the noie term w n,j and channel coefficient α i,j. Each b k i only dependent on the correponding ymbol k o that the code can be decoded in ymbol by ymbol manner. B. One-Dimenional Symbol Error Function Equation ) how that the deciion boundary i fixed for any k and H k, where maximum likelihood decoding i performed ymbolwiely. Thu, we have an analogy between the detection of QAM in AWGN channel [9] and that in thi tranmit diverity channel except for the different noie characteritic. In the decoding of C), the error probability for k i determined by the tatitic of b k = k + n k,wheren k = e k /H k in ). Intead of the complementary error function, we ue ODSEF which i ymbol error function regarding the tatitic of n k. The ODSEF i defined for each k in OSTBC C) a Q k l γ)= Pr b k k >l). The ODSEF can be repreented a the PEP of two codeword uch that two correponding meage vector are different in only one element. Let x and y be two meage vector of length L, in which all correponding element are equal except x k y k and uppoe x k y k =l. From the exact expreion for the PEP of pace-time code [4], the ODSEF i given by Q k l γ)=prcx) Cy)) = π/ L t [ + g ] Lr k,i π in θ γl dθ 0 i= where PrCx) Cy)) i the probability of erroneou decoding to Cy) when Cx) i tranmitted. Uing the Simon reult [4], [4], for a homogeneou ymbol in the code, the ODSEF can be derived in the cloed-form a π/ Q k l γ)= π 0 = [ + γ ] LDL r in θ g kl dθ g k γl +g k γl 4 + g k γl ) L DL r m=0 ) m } ) m m 3) where g k,i = g k or 0 and L D L t i the number of nonzero g k,i, i L t, i.e., the diverity order for k in the code. For a nonhomogeneou ymbol which ha two ditinct nonzero g k,i, the ODSEF can alo be derived in the cloedform uing Simon reult ee 5A.58) of [4] ). Two nonhomogeneou OSTBC introduced by Su and Xia [] belong to thi cae. A an example, the 7/ OSTBC C 4 in 4) i nonhomogeneou becaue the ymbol,, and 3 are nonhomogeneou. Applying Simon reult5a.58) of [4]), we have the cloed-form ODSEF for a Q l [ Lr γ)= 4Lr where B k = C k = A k 5Lr k L r n=0 k=0 ), k ) n An 5Lr ), n B k I k γ l ) Lr ) A k = ) Lr +k k L r )! [ c I k c) = + c + k n= L r n= n k+ 4L r k=0 ] C k I k γl ) k 5L r n), n )!! n! n + c) n where!! denote the double factorial. For the other nonhomogeneou ymbol and 3 of C 4 in 4), their ODSEF have the ame form. IV. BEP OF ORTHOGONAL SPACE TIME BLOCK CODES WITH QAM M-ary QAM can be conidered a a two-dimenional ASK. Conider an M-ary QAM which employ I-ASK and J-ASK in each dimenion, where M = b, I = b,,andj = b,. Every ymbol k in an OSTBC can be tranmitted a arbitrary M-QAM. Cho and Yoon derived the general expreion for the BEP of two-dimenional ASK in AWGN channel [9]. Uing their method [9] and the ODSEF, the exact cloed-form expreion for the BEP of OSTBC i derived in thi ection. Let P k ) be the BEP for the ymbol k. Then, the BEP for the code C i given by P C) = L ] P k ). 4) Thu, we have to derive P k ).AnM-QAM ymbol k hould be divided into I-ASK and J-ASK. For b, bit in I-ASK, the error probability of the mth bit i derived a m )I Pm I k)= I i=0 m i + I ) m i I m ) Q k i +) ) } γd 5)

4 56 JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 0, NO. 3, SEPTEMBER 008 where x denote the larget integer not greater than x. Equation 5) i directly derived from 9) in [9] by replacing erfc ) with Q k ). Thu, the BEP of I-ASK for k i given by P I k )= b, and the BEP for k i derived a b, m= P I m k ) P k )= b P I k )b, + P J k )b, ). For the homogeneou ymbol k in 5), Q k ) can be given in the cloed-form expreion a 3) and then P k ) can alo be given in the cloed-form becaue it i the ummation of ODSEF. For quare M-QAM, we have P k )=P I k ),wherem = I I. Here are the example of the BEP of quare M-QAM uch a QPSK, 6QAM, 64QAM, and 56QAM: Fig.. Exact bit error probability of Alamouti cheme for QPSK, 6QAM, 64QAM, and 56QAM. P QP SK = L L S Q k γd), 3 4 Q k γd)+ Q k γ3d) P 6QAM = L 4 Q k } γ5d), 7Q k γd)+6q k γ3d) P 64QAM = L Q k γ5d)+q k γ9d) Q k } γ3d), P 56QAM = 5Q k γd)+4q k γ3d) L 3 Q k γ5d)+5q k γ9d)+4q k γd) 5Q k γ3d) 4Q k γ5d)+5q k γ7d) +4Q k γ9d) 3Q k γd) Q k γ3d) + Q k γ5d) Q k } γ9d). In Fig., we plot the exact BEP and imulated reult of Alamouti cheme with everal quare QAM in quai-tatic Rayleigh fading channel where L r =, andtheyexactly match. For every homogeneou OSTBC, the cloed-form BEP i available. Moreover, the exact cloed-form expreion for the BEP of all known nonhomogeneou code [] can be derived uing their ODSEF becaue they have only two nonzero ditinct g k,i. Fig. how the BEP performance curve of nonhomogeneou C 4 obtained by imulation and cloed-form expreion. V. CONCLUSION In thi paper, uing the ODSEF, we have derived the exact cloed-form expreion for the BEP of linear OSTBC with quare and rectangular QAM. For all the homogeneou OS- TBC, the ODSEF and the BEP for QAM can be repreented Fig.. Exact bit error probability of C 4 for QPSK, 6QAM, 64QAM, and 56QAM. in the cloed-form. For ome known nonhomogeneou OST- BC introduced by Su and Xia [], the ODSEF and the BEP for QAM can alo be derived in cloed-form. Therefore, the exact BEP of all known OSTBC [], [], [], [] with QAM can be given in the cloed-form expreion. REFERENCES [] V. Tarokh, N. Sehadri, and A. R. Calderbank, Space-time code for high data rate wirele communication: Performace analyi and code contruction, IEEE Tran. Inf. Theory, vol. 44, no., pp , Mar [] S. M. Alamouti, A imple tranmitter diverity cheme for wirele communicatin," IEEE J. Sel. Area Commun., vol. 6, no. 8, pp , Nov [3] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Space-time block code from orthogonal deign, IEEE Tran. Inf. Theory, vol. 45, no. 5, pp , July 999. [4] M. K. Simon, Evaluation of average bit error probability for pace-time coding baed on a impler exact evaluation of pairwie error probability, J. Commun. Netw., vol. 3, pp , Sept. 00. [5] G. Taricco and E. Biglieri, Exact pairwie error probability of pace-time code, IEEE Tran. Inf. Theory, vol. 48, no., pp , Feb. 00. [6] H. Lu, Y. Wang, P. V. Kumar, and K. M. Chugg, Remark on pace-time code including a new lower bound and an improved code, IEEE Tran. Inf. Theory, vol. 49, no. 0, pp , Oct. 003.

5 S.-H. KIM et al.: EXACT BIT ERROR PROBABILITY OF ORTHOGONAL SPACE-TIME [7] S.-H. Kim, I.-S. Kang, and J.-S. No, Symbol error probability of orthogonal pace-time block code with QAM in low Rayleigh fading channel, IEICE Tran. Commun., vol. 87-B, no., Jan [8] M. S. Raju, R. Annavajjala, and A. Chockalingam, BER analyi of QAM on fading channel with tranmit diverity, IEEE Tran. Wirele Commun. vol. 5, no. 3, Mar [9] K. Cho and D. Yoon, On the general BER expreion of one- and twodimenional amplitude modulation, IEEE Tran. Commun., vol. 50, no. 7, pp , July 00. [0] S.-H. Kim, I.-S. Kang, and J.-S. No, Exact bit error probability of orthogonal pace-time block code with quadrature amplitude modulation," in Proc. IEEE ISIT 003, June 9 July 4, 003, p. 63. [] W. Su and X.-G. Xia, Two generalized complex orthogonal pace-time block code of rate 7/ and 3/5 for 5 and 6 tranmit antenna, IEEE Tran. Inf. Theory, vol. 49, no., pp , Jan [] X.-B. Liang, A high-rate orthogonal pace-time block code, IEEE Commun. Lett., vol. 7, no. 5, pp. 3, May 003. [3] X. Li et al., A quaring method to implify the decoding of orthogonal pace-time block code, IEEE Tran. Commun., vol. 49, no. 0, pp , Oct. 00. [4] M. K. Simon and M.-S. Alouini, Digital Communication over Fading Channel: A Unified Approach to Performance Analyi, NewYork,NY: John Wiley & Son, Inc., Aug Sang-Hyo Kim received hi B.S., M.S., and Ph.D. in electrical engineering from Seoul National Univerity, Seoul, Korea in 998, 000, and 004, repectively. From 004 to 006, he wa a Senior Engineer at Samung Electronic. He viited Univerity of Southern California a a viiting cholar from 006 to 007. In 007, he joined the School of Information and Communication Engineering, Sungkyunkwan Univerity, Suwon, Korea where he i currently an Aitant Profeor. Hi reearch interet include error correcting code, peudo random equence, cooperative communication, cognitive radio ytem. Jae-Dong Yang received the B.S. in electronic engineering from Seoul National Univerity, Seoul, Korea, in 00. Since 003, he i a Ph.D. candidate in the Department of Electrical Engineering and Computer Science in Seoul National Univerity through the M.S./Ph.D. joint coure. Hi reearch area include pace-time code, MIMO, coding theory, information theory, and wirele communication ytem. Jong-Seon No received the B.S. and M.S.E.E. degree in electronic engineering from Seoul National Univerity, Seoul, Korea, in 98 and 984, repectively, and the Ph.D. degree in electrical engineering from the Univerity of Southern California, Lo Angele, in 988. He wa a Senior MTS at Hughe Network Sytem from February 988 to July 990. He wa alo an Aociate Profeor in the Department of Electronic Engineering, Konkuk Univerity, Seoul, Korea, from September 990 to July 999. He joined the faculty of the Department of Electrical Engineering and Computer Science, Seoul National Univerity, in Augut 999, where he i currently a Profeor. Hi area of reearch interet include error-correcting code, equence, cryptography, pace-time code, LDPC code, and wirele communication ytem.