International Journal of Software Engineering and It Application Vol. 9, No. 1 (015), pp. 381-388 http://dx.doi.org/10.1457/ijeia.015.9.1.34 Aymptotic Diverity Analyi of Alamouti Tranmit Diverity with Quai-ML Decoding Algorithm in Time-Selective Fading Channel Hoojin Lee Dept. of Information and Communication Engr., Hanung Univerity, Korea hjlee@hanung.ac.kr Abtract Thi paper addree the aymptotic diverity performance of Alamouti tranmit diverity technique epecially in time-elective fading channel. In particular, a linear quai-maximum-likelihood (QML) decoding method i employed for the Alamouti tranmit diverity ytem to olve the error floor problem induced by the conventional linear ML decoding method. By judiciouly utilizing the derived aymptotic cloed-form formula of ymbol pairwie error rate (SPER), it i theoretically verified that the aymptotic diverity order achieved by the QML decoding algorithm become and 1 in quai-tatic and time-elective fading channel, repectively. Keyword: Alamouti tranmit diverity, quai-maximum-likelihood (QML) decoding, time-elective fading channel, ymbol pairwie error rate (SPER) 1. Introduction Alamouti tranmit diverity technique i well-known a an effective method to exploit patial diverity and thu to mitigate the detrimental effect of fading channel which alo ha the attractive property in that full-diverity tranmiion can be achieved along with a low-complexity linear maximum-likelihood (LML) decoding algorithm for a wirele communication ytem incorporating two tranmit antenna and complex ignal contellation [1-6]. Accordingly, the Alamouti cheme ha been widely adopted in everal wirele communication and networking tandard uch a, IEEE 80.16-009, IEEE 80.11n, 3GPP LTE, etc. The aforementioned advantage of Alamouti cheme however, can be eaily lot epecially in time-elective fading channel (e.g., mobile radio environment), ince the orthogonality inherent in the Alamouti cheme i detroyed by the time-elective fading, o that the full-diverity gain obtained from the conventional low-complexity LML decoding cannot be achieved, even howing an error floor of the error rate performance (e.g., bit-error rate (BER), ymbol-error rate (SER), etc.,) in the high ignal-to-noie ratio (SNR) regime, which ultimately lead to the evere performance degradation. In order to reolve thi problem, there have been everal effort to deign efficient decoding algorithm [4-6]. In particular, a linear QML decoding method for Alamouti tranmit diverity technique wa propoed in [4] over time-elective fading channel, which ha the ame decoding complexity a the conventional LML decoding and how the ame error rate performance a the LML decoding in quai-tatic fading channel (i.e., timenonelective fading channel). In [7], the error rate performance aement wa carried out, where a cloed-form formula for the ymbol pairwie error rate (SPER) and a correponding union upper bound on the SER are derived for the Alamouti cheme employing the low-complexity QML decoding algorithm. From the derived expreion given in [7], however, it i difficult to obtain analytical inight into the achievable aymptotic diverity order a the everity of time-electivity of fading channel. Therefore, in thi paper, the reearcher preent an accurate aymptotic cloed-form ISSN: 1738-9984 IJSEIA Copyright c 015 SERSC
International Journal of Software Engineering and It Application Vol. 9, No. 1 (015) approximating formula for the SPER with the aid of the formula derived in [7]. Furthermore, by judiciouly exploiting the formula, the reearcher alo evaluated the aymptotic diverity order achieved by the QML decoding over both quai-tatic and time-elective fading channel. Throughout thi paper, the reearcher ued the following notation. The upercript * () and () H ~ (, ) x denote complex conjugate and Hermitian operation, repectively. x C N m tand for a circular ymmetric complex Gauian variable x with mean m and variance. P r( ) x and E {} denote the probability and expectation, repectively.. Sytem Model and Decoding Algorithm Throughout thi paper, the reearcher conidered the Alamouti tranmit diverity ytem employing two tranmit antenna and one receive antenna (i.e., N, N 1 ) for T R brevity. From [1]-[7], the input-output relationhip of Alamouti cheme i expreed a r(1) h (1) h (1) v(1) 1 1 * * * * r ( ) h ( ) h ( ) v ( ) 1 r H v, (1) where r,, and v are the received ignal vector, the tranmitted ymbol vector, and the noie vector with v ( ) ~ C N (0, ), repectively. H i the effective channel matrix, where v h () repreent the channel coefficient from the i th tranmit antenna to the receive antenna, i and the index inide () tand for the time index. From [4]-[7], the channel coefficient in time-elective fading channel can be approximated a ( 1) ( ) 1 ( 1), i i i h t h t n t * where E { h ( t ) h ( t 1)} with 0 1 denote the fading correlation parameter to i i characterize the degree of channel time-variation, and n ( ) ~ C N (0,1). In addition, i according to Jake model in [8], the reearcher alo have J ( f ), where J () i 0 D 0 the zero-order Beel function of the firt kind and f i the relative Doppler frequency. D When the channel i quai-tatic, H i orthogonal, which mean that by multiplying H H H in (1), the Gramian matrix G H H become diagonal. Then, by uing the conventional LML decoding algorithm, the two tranmitted ymbol can be eaily decoupled and then decoded. However, when the channel i time-elective, the effective channel matrix i no longer orthogonal and the off-diagonal term of G are not zero. Thu, the LML decoding uffer from an error floor in the high SNR region, which lead to the need for more effective decoding method. To overcome the aforementioned problem, till maintaining the low-complexity of decoding procedure, the linear QML decoding algorithm wa propoed in [4] with the aid of the following orthogonal combining matrix tranformation a z z 1 Z for z z 1 1 1 z z z 1 1 1 Z. () z 1 Then, the reearcher can eaily find that the orthogonality at H H i achieved through combining r with H in (1). Thu, after ome traightforward manipulation, the reearcher obtained the following linear QML deciion metric a ˆ a rg m in r ( i ) i for i 1, (3) A where A i the contellation of the tranmitted ymbol, r () i an element of H r, and * * h (1) h ( ) h (1) h ( ). The reearcher note that the orthogonal combining matrix 1 1 38 Copyright c 015 SERSC
International Journal of Software Engineering and It Application Vol. 9, No. 1 (015) given in () enable low-complexity linear coding, a hown in (3), however, the element of H v are not white-noie (i.e., correlated), o that the correponding noie enhancement can lead to ome performance degradation. 3. Performance Evaluation 3.1. SPER Derivation for QML Decoding Algorithm Auming that the channel tate information i known at the receiver, the SPER of Alamouti tranmit diverity with QML decoding, conditioned on the fading coefficient, i expreed a [7] P r H P r rˆ rˆ h ( ) h (1) h ( ) h (1) 1 1 P r ˆ ˆ r v h ( ) h (1) 1 (4) Q h ( ) h (1) 1 where tand for a pairwie error event, Q () i the Q-function [9], / E repreent the normalized quared Euclidean ditance, E / v i the average SNR at the receiver, and E denote the total tranmit power on the two tranmit antenna per ymbol duration. The reearcher noted that the ubcript in are dropped due to the ymmetric tructure of Alamouti tranmit diverity. Then, by averaging P r H over H in (4), the average SPER can be traightforwardly formulated a [7] P r ;, 0 A Q f d (5) 1 3 1 3 5 1 F 1, ; ;, ; 3; 1 F 4 1 where f A i the probability denity function (PDF) of / ( h ( ) h (1) ) and 1 F 1 (, ; ; ) i the Gau hypergeometric function [10, (07.3.0.0001.01)], which i implemented in mot of the well-known mathematical oftware package, uch a MATLAB, MATHEMATICA, MAPLE, etc. It i obviou that the formula in (5) doe not provide inightful information for Alamouti tranmit diverity with QML decoding, uch a achievable diverity order. Thu, it i needed to evaluate the aymptotic behavior of the SPER in the high SNR regime. To thi end, with the aid of F 1 (, ; ; 0 ) 1 in [10, (07.3.03.0001.01)] for, the reearcher can aymptotically approximate (5) a 1 3 P r ;, 4 (6) where the fading correlation parameter i bounded 0 1 in accordance with the everity of time-elective fading. Copyright c 015 SERSC 383
International Journal of Software Engineering and It Application Vol. 9, No. 1 (015) For the cae of quai-tatic fading channel (i.e., 1 ), the formula and correponding aymptotic approximate expreion of SPER are implified, repectively, into 3 5 1 P r ;, 1 F, ; 3; 1 P r ;, 1 4 3, (7). (8) 4 It i intereting that the formula in (7) i aymptotically equivalent to the formula given in [11, (9)], which i derived by the ue of Taylor erie expanion technique. In addition, for the cae of fat fading channel (i.e., fading channel are uncorrelated in time with 0 ), the reearcher have 1 3 1 P r ;, 0 F 1, ; ; 1 P r ;, 0 1, (9). (10) Furthermore, by exploiting the derived SPER formula given in (5)-(10), the union upper and lower bound on the SER of Alamouti cheme with QML decoding can be obtained a [1] 1 P P P r ;, U B A P m a x P r ;, LB A, A (11) where A denote the cardinality of a contellation A. 4.. Aymptotic Diverity Order for QML Decoding Algorithm The diverity order i known to be generally defined a the magnitude of the lope of the average error rate veru SNR on a log-log cale in the high SNR region. Then, the aymptotic and intantaneou diverity order can be expreed, repectively, a d lo g P r ;, lim (1) lo g lo g P r ;, dˆ lo g P r ;, P r ;, (13) where it i clear that d lim dˆ. Then, ubtituting the formula in (5)-(10) into (1) yield the correponding aymptotic diverity order for Alamouti tranmit diverity ytem with QML decoding over variou fading channel, which can be ummarized a 384 Copyright c 015 SERSC
SER International Journal of Software Engineering and It Application Vol. 9, No. 1 (015) d A la m o u ti Q M L 1 lo g lim 1, 0 lo g 1 3 lo g 4 lim 1, 0 1 lo g 3 lo g 4 lim, 1 lo g which indicate that the QML decoding can reolve the error floor problem revealed in A la m o u ti L M L the conventional LML decoding (i.e., d 0 for 0 1 ) over time-elective fading channel. It i, however, noteworthy that the QML decoding algorithm till experience ome performance degradation in that full-diverity (i.e., d A la m o u ti N N ) T R i not achieved in time-elective fading channel, which i due to the inherent noie enhancement problem, a mentioned in Section. (14) 10 0 Alamouti Tranmit Diverity with N T = and N R =1 10-1 10-10 -3 10-4 10-5 10-6 LML, =0.0000 LML, =0.95 LML, =0.9755 LML, =0.9911 LML, =1.0000 QML, =0.0000 QML, =0.95 QML, =0.9755 QML, =0.9911 QML, =1.0000 0 5 10 15 0 5 30 35 40 SNR [db] Figure 1. Average SER veru SNR for Alamouti tranmit diverity with N, N 1 and QPSK over Quai-Static and Time-Selective Fading T R Channel (i.e., 0 1) Copyright c 015 SERSC 385
Intantaneou Diverity Order International Journal of Software Engineering and It Application Vol. 9, No. 1 (015) 4. Numerical Reult In thi ection, ome numerical reult are preented to verify the accuracy of the analytical reult given in the previou ection. Thu, Figure 1 how the SER v. SNR curve of the Alamouti tranmit diverity ytem with two tranmit antenna and one receive antenna, Gray-coded QPSK modulation, and both LML and QML decoding method, over variou Rayleigh fading channel (i.e., 0 1). For the variou channel configuration, we conider the time-elective fading channel with 0.9 9 1 1, 0.9 7 5 5, 0.9 5, which, for example, correpond to the relative Doppler frequency f 0.0 3 D 0.0 5, 0.0 7, repectively, from Jake model. The reearcher alo conider two further cae uch a quai-tatic fading channel (i.e., 1 ) and fat fading channel (i.e., 0 ). Specifically, a hown in Figure 1, it i obviou that the QML decoding i more uperior to the LML decoding in time-elective fading channel (including the fat fading channel) without howing error floor at high SNR, which the LML decoding i unable to avoid. Furthermore, from the examination of the lope in the high SNR region, we can oberve that the SER curve from QML decoding how the diverity order of and 1 over quai-tatic and time-elective fading channel, repectively, which i apparently equivalent to the analytical reult given in (14).,.5 Alamouti Scheme with N T =, N R =1, QML decoding 1.5 1 0.5 =0.0000 =0.95 =0.9755 =0.9911 =1.0000 0 0 10 0 30 40 50 SNR [db] Figure. SPER-baed Intantaneou Diverity Order Derived in (13) veru SNR for Alamouti Tranmit Diverity with N, N 1 and QPSK To further demontrate the aymptotic diver order achieved by the QML decoding againt the time-electivity of fading channel, the reearcher depict the intantaneou diveritie obtained from (13) veru SNR in Fig.. The reultant intantaneou diverity T R 386 Copyright c 015 SERSC
International Journal of Software Engineering and It Application Vol. 9, No. 1 (015) order via variou, epecially in the high SNR regime are identical to the analytical reult derived in (14), where the reearcher can alo oberve that the convergence peed of the intantaneou diverity order i inverely proportional to the value of. 5. Concluion The reearcher analytically evaluated the aymptotic diverity performance of Alamouti tranmit diverity ytem employing QML decoding algorithm, particularly in time-elective fading channel. By judiciouly deriving and utilizing the aymptotic cloed-form approximate formula of SPER, the reearcher apparently demontrated that the achievable aymptotic diverity order become and 1 over quai-tatic and timeelective fading channel, repectively, which reveal that the QML decoding i capable of overcoming the error floor effect induced by the conventional LML decoding in the high SNR regime. Finally, the reearcher noted that the derived analytical reult enable u to efficiently predict the diverity performance of the Alamouti tranmit diverity with QML decoding method, epecially in time-elective fading environment. Acknowledgment Thi work wa financially upported by Hanung Univerity. Reference [1] S. M. Alamouti, A Simple Tranmit Diverity Technique for Wirele Communication, IEEE J. Select. Area Commun., vol. 16, no. 8, (1998), pp. 1451-1458. [] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Space-Time Block Coding for Wirele Communication: Performance Reult, IEEE J. Select. Area Commun., vol. 17, no. 3, (1999), pp. 451-460. [3] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Space-Time Block Code from Orthogonal Deign, IEEE Tran. Inf. Theory, vol. 45, no. 5, (1999), pp. 1456-1467. [4] T. A. Tran and A. B. Seay, A Generalized Linear Quai-ML Decoder of OSTBC for Wirele Communication over Time-Selective Fading Channel, IEEE Tran. Wirele Commun., vol. 3, no. 3, (004), pp. 855-864. [5] A. Vielmon, Y. Li, and J. R. Barry, Performance of Alamouti Tranmit Diverity over Time-Varying Rayleigh-Fading Channel, IEEE Tran. Commun., vol. 3, no. 5, (004), pp. 1369-1373. [6] D.-B. Lin, P.-H. Chiang, and H.-J. Li, Performance Analyi of Two-Branch Tranmit Diverity Block- Coded OFDM Sytem in Time-Varying Multipath Rayleigh-Fading Channel, IEEE Tran. Veh. Technol., vol. 54, no. 1, (005), pp. 136-148. [7] V.-B. Pham and W.-X. Sheng, Performance Analyi of Alamouti Space-Time Block Code over Time- Selective Fading Channel, Wirele Per. Commun., vol. 73, no. 3, (013), pp. 401-413. [8] W. C. Jake, Microwave mobile communication, Wiley, New York, (1974). [9] I. S. Gradhteyn and I. M. Ryzhik, Table and Integral, Serie, and Product, 5 th ed. Academic, (1994). [10] Wolfram Reearch: The Wolfram function ite. Available: http://function.wolfram.com. [11] Y. R. Wei and M. Wang, Analyi of Single-Symbol Detectable Space-Time Block Code with COD and GCIOD, Wirele Per. Commun., vol. 56, no., (011), pp. 315-331. [1] M. K. Simon and M. S. Alouini, Digital Communication over Fading Channel, 1 t ed. Wiley, (000). Copyright c 015 SERSC 387
International Journal of Software Engineering and It Application Vol. 9, No. 1 (015) Author Hoojin Lee, he received hi B.S. degree from the School of Electrical Engineering, Seoul National Univerity, Seoul, Korea, in 1997, and hi M.S. and Ph.D. degree in electrical and computer engineering from the Univerity of Texa at Autin, Autin, TX, USA, in 00 and 007, repectively. From 008 to 009, he worked a a ytem and architecture engineer in the Algorithm and Standard Team, Cellular Product Group of Freecale Semiconductor, Inc., Autin, TX, USA. Since 009, he ha been working at the Department of Information and Communication Engineering in Hanung Univerity, Seoul, Korea, where he i currently an Aociate Profeor. Hi current reearch interet are in the area of wirele communication, including multiple-input multiple-output (MIMO) communication incorporating pace-time coding and decoding, orthogonal frequency diviion multiplexing (OFDM) communication, and advanced ignal proceing for communication. 388 Copyright c 015 SERSC