LLR based BER Analysis of Orthogonal STBCs using QAM on ...

"!"+=Z m´Ð [—-1g @ " ( d 5+£G"G5+ %"¤¤¤¤¤¤¤¥G 5+g @ (ÛÊÑ¦Û ÑÊ££ £ ££ ££ £££££ ©§§§§§§§5""Û"A"%¤£¤¤¤¤¤¤¤¤¤¤¤¤¤¤¤¥£ £ Û £ Ê ££ £ Ê£££££££ÊÛ££ Ñ Ñ . Ê Û¹¢P¤P¥" %£5¹¢Q*¤Q¥"£%££ £ § ¦ §§©+§§§§§§§§§§§§"§"""10 −1Average Probability <strong>of</strong> Bit Error10 −210 −310 0 Average SNR per bit (dB)C ¦ÛÑ£¦Û . ÑÑ ÑC Û ÑÊ ÑÑ. ÑSimulated <strong>BER</strong> <strong>using</strong> True <strong>LLR</strong>’s, 2Tx, 2Rx antennasAnalytical <strong>BER</strong> <strong>using</strong> apprx. <strong>LLR</strong>s, 2Tx, 2 Rx antennasSimulated <strong>BER</strong> <strong>using</strong> True <strong>LLR</strong>s, 2Tx, 1 Rx antennasAnalytical <strong>BER</strong> <strong>using</strong> apprx. <strong>LLR</strong>s, 2Tx, 1 Rx antennasÑ. Ñ . 10 −40 5 10 15ÑÑFig. 2. Comparison <strong>of</strong> the analytical <strong>BER</strong> evaluated <strong>using</strong> approximate<strong>LLR</strong>s vs the simulated <strong>BER</strong> <strong>using</strong> the <strong>LLR</strong>s without approximation. 16-<strong>QAM</strong> with rate-1 STBC (Alamouti code). 2Tx/2Rx and 2Tx/1Rx antennas., andis the well known Alamouti code with parameters," + >> is a - UJ- diagonal matrixwith ¹¢© diagonal element, ¥, <strong>of</strong> the - @¥

10 −110 −1Average Probability <strong>of</strong> Bit Error10 −210 −310 −410 0 γ b(dB)No Diversity2Tx, 1Rx2Tx, 2Rx2Tx, 4Rx2Tx, 10RxAverage Probability <strong>of</strong> Bit Error10 −210 −310 −410 0 γ b(dB)No Diversity3Tx, 1Rx3Tx, 2Rx3Tx, 4Rx3Tx, 10Rx10 −50 2 4 6 8 10 12 14Fig. 3. <strong>BER</strong> performance <strong>of</strong> 16-<strong>QAM</strong> with 2 transmit antennas V e¡ _£¢_¥¤_ ¡§¦andreceive antennas <strong>using</strong> rate-1 STBC (Alamouti code).Fig. 4.10 −50 2 4 6 8 10 12 14<strong>BER</strong> performance <strong>of</strong> 16-<strong>QAM</strong> with 3 transmit antennas V e andreceive antennas <strong>using</strong> rate-1/2 STBC.¡ _¥¢_£¤_ ¡§¦<strong>of</strong> the individual bits forming the <strong>QAM</strong> symbol, and hencethe average <strong>BER</strong> <strong>of</strong> the system. Although the analysis wasgiven only for 16-<strong>QAM</strong> in this paper, the approach appliesto the <strong>BER</strong> analysis <strong>of</strong> -<strong>QAM</strong> systems for any value <strong>of</strong>(any arbitrary mapping <strong>of</strong> bits to <strong>QAM</strong> symbols) for anySTBC from COD with linear processing. We presented theanalytical <strong>BER</strong> results for 16-<strong>QAM</strong> with ) (2-Tx, -Rx) antennas<strong>using</strong> Alamouti code (rate-1 STBC), ) (3-Tx, -Rx)antennas <strong>using</strong> a rate-1/2 STBC and ) (5-Tx, -Rx) antennas<strong>using</strong> a rate-7/11 STBC. The <strong>LLR</strong>s derived can alsobe used as s<strong>of</strong>t inputs to decoders for various coded <strong>QAM</strong>schemes, including turbo coded <strong>QAM</strong> with space-time coding.Average Probability <strong>of</strong> Bit Error10 −110 −210 −310 −410 −5No Diversity5Tx, 1 Rx5Tx, 2Rx5Tx, 4Rx5Tx, 10Rx10 0 γ b(dB)REFERENCES[1] G. J. Foschini and M. J. Gans, “On limits <strong>of</strong> wireless communicationsin a fading environment when <strong>using</strong> multiple antennas,” Wireless Pers.Commun., vol. 6, pp. 311–335, 1998.[2] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time blockcoding for wireless communications: Performance results,” IEEE Jl.Sel. Areas in Commun., vol. 17, no. 3, pp. 451–460, 1999.[3] W. Su and X.-G. Xia, “On space-time block codes from complex orthogonaldesigns,” Wireless Pers. Commun., vol. 25, no. 1, pp.1-26,April 2003.[4] H. Shin and J. H. Lee, “Exact symbol error probability<strong>of</strong> orthogonal space-time block codes,”http://cctlab01.snu.ac.kr/nrl/conference/no1547.pdf[5] M. K. Simon, “Evaluation <strong>of</strong> average bit error probability for spacetime coding <strong>based</strong> on a simpler exact evaluation <strong>of</strong> pairwise error probability,”Jl. Commun. Networks, vol. 3, no. 3, September 2001.[6] G. Taricco and E. Biglieri, “Exact pairwise error probability <strong>of</strong> spacetimecodes,” IEEE Trans. Inform. Theory, vol. 48, pp. 510–513, February2002.[7] S. M. Alamouti, “A simple transmit diversity technique for wirelesscommunications,” IEEE Jl. Sel. Areas in Commun., vol. 16, no. 8, pp.1451–1458, October 1998.[8] R. Pyndiah, A. Picard and A. Glavieux, “Performance <strong>of</strong> block turbocoded 16-<strong>QAM</strong> and 64-<strong>QAM</strong> modulations,” Proc. IEEE GLOBE-COM’95, pp. 1039–1043, Singapore, November 1995.[9] A. J. Viterbi, “An intuitive justification and a simplified implementation<strong>of</strong> the MAP decoder for convolutional codes,” IEEE Jl. Sel. Areas inCommun., vol. 16, no. 2, pp. 260–264, 1998.[10] J. W. Craig, “A new, simple and exact result for calculating the probability<strong>of</strong> error for two-dimensional signal constellations,” Proc. IEEEMILCOM’91, pp. 571–575, 1991.10 −60 2 4 6 8 10 12 14Fig. 5. <strong>BER</strong> performance <strong>of</strong> 16-<strong>QAM</strong> with 5 transmit antennas V e¡ _¥¢_£¤_ ¡§¦andFig. 6.Average Probability <strong>of</strong> Bit Errorreceive antennas <strong>using</strong> rate-7/11 STBC.10 010 −110 −210 −310 −4No diversity (1Tx, 1Rx)2Tx, 2Rx, Alamouti code3Tx, 2Rx Scheme (rate−1/2)5Tx, 2Rx Scheme (rate−7/11)10 −5AWGN10 −60 2 4 6 8 10 12 14γ b(dB)<strong>BER</strong> performance <strong>of</strong> 16-<strong>QAM</strong> with different <strong>STBCs</strong> –)2Tx antennas<strong>using</strong> rate-1 STBC (Alamouti code),) 3 Tx antennas <strong>using</strong> rate-1/2STBC,)5Tx antennas <strong>using</strong> rate-7/11 STBC. Number <strong>of</strong> receive antennas,` e ¢ .