Space-Time Coding phần 5 potx

Decoding of STBC 107 The decision statistics for X c 4 can be expressed as ˜x 1 = n R  j=1 (r j 1 h ∗ j,1 +r j 2 h ∗ j,2 +r j 3 h ∗ j,3 +r j 4 h ∗ j,4 +(r j 5 ) ∗ h j,1 +(r j 6 ) ∗ h j,2 +(r j 7 ) ∗ h j,3 +(r j 8 ) ∗ h j,4 ) = ρ 4 x 1 + n R  j=1 (n j 1 h ∗ j,1 + n j 2 h ∗ j,2 + n j 3 h ∗ j,3 + n j 4 h ∗ j,4 + (n j 5 ) ∗ h j,1 + (n j 6 ) ∗ h j,2 + (n j 7 ) ∗ h j,3 + (n j 8 ) ∗ h j,4 ) ˜x 2 = n R  j=1 (r j 1 h ∗ j,2 −r j 2 h ∗ j,1 −r j 3 h ∗ j,4 +r j 4 h ∗ j,3 +(r j 5 ) ∗ h j,2 −(r j 6 ) ∗ h j,1 −(r j 7 ) ∗ h j,4 +(r j 8 ) ∗ h j,3 ) = ρ 4 x 2 + n R  j=1 (n j 1 h ∗ j,2 − n j 2 h ∗ j,1 − n j 3 h ∗ j,4 + n j 4 h ∗ j,3 + (n j 5 ) ∗ h j,2 − (n j 6 ) ∗ h j,1 − (n j 7 ) ∗ h j,4 + (n j 8 ) ∗ h j,3 ) ˜x 3 = n R  j=1 (r j 1 h ∗ j,3 +r j 2 h ∗ j,4 −r j 3 h ∗ j,1 −r j 4 h ∗ j,2 +(r j 5 ) ∗ h j,3 +(r j 6 ) ∗ h j,4 −(r j 7 ) ∗ h j,1 −(r j 8 ) ∗ h j,2 ) = ρ 4 x 3 + n R  j=1 (n j 1 h ∗ j,3 + n j 2 h ∗ j,4 − n j 3 h ∗ j,1 − n j 4 h ∗ j,2 + (n j 5 ) ∗ h j,3 + (n j 6 ) ∗ h j,4 − (n j 7 ) ∗ h j,1 − (n j 8 ) ∗ h j,2 ) ˜x 4 = n R  j=1 (−r j 1 h ∗ j,4 −r j 2 h ∗ j,3 +r j 3 h ∗ j,2 −r j 4 h ∗ j,1 −(r j 5 ) ∗ h j,4 −(r j 6 ) ∗ h j,3 +(r j 7 ) ∗ h j,2 −(r j 8 ) ∗ h j,1 ) = ρ 4 x 4 + n R  j=1 (−n j 1 h ∗ j,4 − n j 2 h ∗ j,3 + n j 3 h ∗ j,2 − n j 4 h ∗ j,1 − (n j 5 ) ∗ h j,4 − (n j 6 ) ∗ h j,3 + (n j 7 ) ∗ h j,2 − (n j 8 ) ∗ h j,1 ) where ρ 4 = 2 4  i=1 n R  j=1 |h j,i | 2 . (3.63) To decode the rate 3/4 code X h 3 , the receiver constructs the decision statistics as follows ˜x 1 = n R  j=1   r j 1 h ∗ j,1 + (r j 2 ) ∗ h j,2 + (r j 4 − r j 3 )h ∗ j,3 2 − (r j 4 − r j 3 ) ∗ h j,3 2   ˜x 2 = n R  j=1   r j 1 h ∗ j,2 − (r j 2 ) ∗ h j,1 + (r j 4 + r j 3 )h ∗ j,3 2 + (−r j 3 + r j 4 ) ∗ h j,3 2   108 Space-Time Block Codes ˜x 3 = n R  j=1   (r j 1 + r j 2 )h ∗ j,3 √ 2 + (r j 3 ) ∗ (h j,1 + h j,2 ) √ 2 + (r j 4 ) ∗ (h j,1 − h j,2 ) √ 2   Similarly, to decode the rate 3/4 code X h 4 , the receiver constructs the decision statistics as follows ˜x 1 = n R  j=1 (r j 1 h ∗ j,1 + (r j 2 ) ∗ h j,2 + (r j 4 − r j 3 )(h ∗ j,3 − h ∗ j,4 ) 2 − (r j 3 + r j 4 ) ∗ (h j,3 + h j,4 ) 2 ) ˜x 2 = n R  j=1 (r j 1 h ∗ j,2 − (r j 2 ) ∗ h j,1 + (r j 4 + r j 3 )(h ∗ j,3 − h ∗ j,4 ) 2 + (−r j 3 + r j 4 ) ∗ (h j,3 + h j,4 ) 2 ) ˜x 3 = n R  j=1   (r j 1 + r j 2 )h ∗ j,3 √ 2 (r j 1 − r j 2 )h ∗ j,4 √ 2 + (r j 3 ) ∗ (h j,1 + h j,2 ) √ 2 + (r j 4 ) ∗ (h j,1 − h j,2 ) √ 2   3.7 Performance of STBC In this section, we show simulation results for the performance of STBC on Rayleigh fading channels. In the simulations, it is assumed that the receiver knows the perfect channel state information. The bit error rate (BER) and the symbol error rate (SER) for STBC with 3 bits/s/Hz and a variable number of transmit antennas are shown in Figs. 3.7 and 3.8, respectively. The performance of an uncoded 8-PSK is plotted in the figures for comparison. 10 15 20 25 30 35 10 −4 10 −3 10 −2 10 −1 Bit Error Probability 3 bits/sec/Hz SNR (dB) uncoded 2Tx 1Rx 3Tx 1Rx 4Tx 1Rx Figure 3.7 Bit error rate performance for STBC of 3 bits/s/Hz on Rayleigh fading channels with one receive antenna Performance of STBC 109 10 15 20 25 30 35 10 −4 10 −3 10 −2 10 −1 10 0 Symbol Error Probability 3 bits/sec/Hz SNR (dB) uncoded 2Tx 1Rx 3Tx 1Rx 4Tx 1Rx Figure 3.8 Symbol error rate performance for STBC of 3 bits/s/Hz on Rayleigh fading channels with one receive antenna For transmission with two transmit antennas, the rate one code X c 2 and 8-PSK modulation are employed. For three and four transmit antennas, 16-QAM and the rate 3/4 codes X h 3 and X h 4 are used, respectively. Therefore, the transmission rate is 3 bits/s/Hz in all cases. For Fig. 3.7, we can see that at the BER of 10 −5 , the code X h 4 is better by about 7dB and 2.5 dB than the code X c 2 and the code X h 3 , respectively. Figures 3.9 and 3.10 show BER and SER performance, respectively, for STBC of 2 bits/s/Hz with two, three, and four transmit antennas and one receive antenna on Rayleigh fading channels. The STBC with two transmit antennas is the rate one code X c 2 with QPSK modulation. The STBC with three and four transmit antennas are the rate 1/2 codes X c 3 and X c 4 , respectively, with 16-QAM modulation. It can be observed that at the BER of 10 −5 , the code with four transmit antennas gains about 5 dB and 3 dB relative to the codes with two and three transmit antennas, respectively. The BER and SER performance for the codes with 1 bit/s/Hz, a variable number of the transmit antennas and a single receive antenna are illustrated in Figs. 3.11 and 3.12, respectively. The STBC with two transmit antennas is the rate one code X c 2 with BPSK modulation. The STBC with three and four transmit antennas are the rate 1/2 codes X c 3 and X c 4 , respectively, with QPSK modulation. It can be observed that at the BER of 10 −5 ,the code with four transmit antennas is superior by about 8 dB and 2.5 dB to the codes with two and three transmit antennas, respectively. The simulation results show that increasing the number of transmit antennas can provide a significant performance gain. The increase in decoding complexity for STBC with a large number of transmit antennas is very little due to the fact that only linear processing is required for decoding. In order to further improve the code performance, it is possible to concatenate an outer code, such as trellis or turbo code, with an STBC as an inner code. 110 Space-Time Block Codes 5 10 15 20 25 30 10 −4 10 −3 10 −2 10 −1 Bit Error Probability 2 bits/sec/Hz SNR (dB) uncoded 2Tx 1Rx 3Tx 1Rx 4Tx 1Rx Figure 3.9 Bit error rate performance for STBC of 2 bits/s/Hz on Rayleigh fading channels with one receive antenna 5 10 15 20 25 30 10 −4 10 −3 10 −2 10 −1 10 0 Symbol Error Probability 2 bits/sec/Hz SNR (dB) uncoded 2Tx 1Rx 3Tx 1Rx 4Tx 1Rx Figure 3.10 Symbol error rate performance for STBC of 2 bits/s/Hz on Rayleigh fading channels with one receive antenna Performance of STBC 111 6 8 10 12 14 16 18 20 22 24 26 28 10 −4 10 −3 10 −2 10 −1 Bit Error Probability 1 bits/sec/Hz SNR (dB) uncoded 2Tx 1Rx 3Tx 1Rx 4Tx 1Rx Figure 3.11 Bit error rate performance for STBC of 1 bits/s/Hz on Rayleigh fading channels with one recei ve antenna 6 8 10 12 14 16 18 20 22 24 26 10 −4 10 −3 10 −2 10 −1 Symbol Error Probability 1 bits/sec/Hz SNR (dB) uncoded 2Tx 1Rx 3Tx 1Rx 4Tx 1Rx Figure 3.12 Symbol error rate performance for STBC of 1 bits/s/Hz on Rayleigh fading channels with one receive antenna 112 Space-Time Block Codes 3.8 Effect of Imperfect Channel Estimation on Performance In this section, the effect of imperfect channel state information on the code performance is discussed. We start with the description of the channel estimation method used in the simulations [6]. The channel fading coefficients are estimated by inserting pilot sequences in the transmitted signals. It is assumed that the channel is constant over the duration of a frame and independent between the frames. In general, with n T transmit antennas we need to have n T different pilot sequences P 1 ,P 2 , ,P n T . At the beginning of each frame transmitted from antennas i, a pilot sequence P i consisting of k symbols P i = (P i,1 ,P i,2 , ,P i,k ) (3.64) is appended. Since the signals at the receive antennas are linear superpositions of all trans- mitted signals, the pilot sequences P 1 ,P 2 , ,P n T are designed to be orthogonal to each other. During the channel estimation, the received signal at antenna j and time t can be repre- sented by r j t = n T  i=1 h j,i P i,t + n j t (3.65) where h j,i is the fading coefficient for the path from transmit antenna i to the receive antenna j and n j t is the noise sample at receive antenna j and time t. The received signal and noise sequence at antenna j can be represented as r j = (r j 1 ,r j 2 , ,r j k ) n j = (n j 1 ,n j 2 , ,n j k ) (3.66) The receiver estimates the channel fading coefficients h j,i by using the observed sequences r j . Since the pilot s equences P 1 ,P 2 , ,P n T are orthogonal, the minimum mean square error (MMSE) estimate of h j,i is given by [6] ˜ h j,i = r j · P i ||P i || 2 = h j,i + n j · P i ||P i || 2 = h j,i + e j,i (3.67) where e j,i is the estimation error due to the noise, given by e j,i = n j · P i P i · P i (3.68) Since n j t is a zero-mean complex Gaussian random variable with single-sided power spectral density N 0 , the estimation error e j,i has a zero mean and single-sided power spectral density N 0 /k [6]. Effect of Antenna Correlation on Performance 113 0 2 4 6 8 10 12 14 16 18 10 −3 10 −2 10 −1 10 0 SNR (dB) Frame Error Rate Ideal CSI non−ideal CSI Figure 3.13 Performance of the STBC with 2 bits/s/Hz on correlated slow Rayleigh fading channels with two transmit and two receive antennas The performance of the STBC with imperfect channel state information at the receiver is shown in Fig. 3.13. In the simulation, QPSK modulation and the rate one code X c 2 with two transmit and two receive antennas are employed. It is assumed that the channel is described as a slow Rayleigh fading model with constant coefficients over a frame of 130 symbols. The pilot sequence inserted in each frame has a length of 10 symbols. The simulation results show that due to imperfect channel estimation, the code performance is degraded by about 0.3 dB compared to the case of ideal channel state information. Note that the degradation in code performance also accounts for the loss of the signal energy by appending the pilot sequences. If the number of transmit antennas is small, the performance degradation due to the channel estimation error is small. However, as the number of transmit antennas increases, the sensitivity of the system to channel estimation error increases [6]. 3.9 Effect of Antenna Correlation on Performance Figure 3.14 shows the performance of the STBC with 2 bits/s/Hz on correlated slow Rayleigh fading channels with two transmit and two receive antennas. We assume that the transmit antennas are not correlated but the receive antennas are correlated. The receive antenna correlation matrix is given by  R =  1 θ θ 1  (3.69) where θ is the correlation factor between the receive antennas. In the simulation, the correla- tion factor is chosen to be 0.25, 0.5, 0.75 and 1. It can be observed that the code performance 114 Space-Time Block Codes 0 5 10 15 20 25 10 −3 10 −2 10 −1 10 0 SNR (dB) Frame Error Rate cor=0 cor=0.25 cor=0.5 cor=0.75 cor=1 Figure 3.14 Performance of the STBC with 2 bits/s/Hz on correlated slow Rayleigh fading channels with two transmit and two receive antennas is slightly degraded when the correlation factor is 0.25. However, relative to the case with uncorrelated antennas, the code is getting worse by 0.7 dB and 1.6 dB at a FER of 10 −2 for the correlation factors of 0.5 and 0.75, respectively. When the channels are fully correlated, the penalty on the code performance is about 4.2 dB at the same FER. Bibliography [1] S. M. Alamouti, “A simple transmit diversity technique for wireless communications”, IEEE Journal Select. Areas Commun., vol. 16, no. 8, pp. 1451–1458, Oct. 1998. [2] A. Wittneben, “A new bandwidth efficient transmit antenna modulation diversity scheme for linear digital modulation”, in Proc. IEEE ICC93, pp. 1630–1634, 1993. [3] V. Tarokh, H. Jafarkhani and A. R. Calderbank, “Space-time block codes from orthog- onal designs”, IEEE Trans. Inform. Theory, vol. 45, no. 5, pp. 1456–1467, July 1999. [4] V. Tarokh, H. Jafarkhani and A. R. Calderbank, “Space-time block coding for wireless communications: performance results”, IEEE J. Select. Areas Commun., vol. 17, no. 3, pp. 451–460, Mar. 1999. [5] V. Tarokh, A. Naguib, N. Seshadri and A. R. Calderbank, “Combined array processing and space-time coding”, IEEE Trans. Inform. Theory, vol. 45, no. 4, pp. 1121–1128, May 1999. [6] V. Tarokh, A. Naguib, N. Seshadri and A. R. Calderbank, “Space-time codes for high data rate wireless communication: Performance criteria in the presence of channel estimation errors, mobility, and multiple paths”, IEEE Trans. Commun., vol. 47, no. 2, pp. 199–207, Feb. 1999. Bibliography 115 [7] V. Tarokh and H. Jafarkhani, “A differential detection scheme for transmit diversity”, IEEE J. Select. Areas Commun., vol. 18, pp. 1169–1174, July 2000. [8] H. Jafarkhani and V. Tarokh, “Multiple transmit antenna differential detection from generalized orthogonal designs”, IEEE Trans. Inform. Theory, vol. 47, no. 6, pp. 2626– 2631, Sep. 2001. [9] B. L. Hughes, “Differential space-time modulation”, IEEE Trans. Inform. Theory, vol. 46, no. 7, pp. 2567–2578, Nov. 2000. [10] B. M. Hochwald and T. L. Marzetta, “Unitary space-time modulation for multiple- antenna communications in Rayleigh flat fading”, IEEE Trans. Inform. Theory, vol. 46, no. 2, pp. 543–564, Mar. 2000. [11] B. M. Hochwald and W. Sweldens, “Differential unitary space-time modulation”, IEEE Trans. Communi., vol. 48, no. 12, Dec. 2000. [12] B. Hochwald, T. L. Marzetta and C. B. Papadias, “A transmitter diversity scheme for wideband CDMA systems based on space-time spreading”, IEEE Journal on Selected Areas in Commun., vol. 19, no. 1, Jan. 2001, pp. 48–60. [13] T. S. Rappaport, Wireless Communications: Principles and Practice, Prentice Hall, 1996. This page intentionally left blank [...]... 4 TSC 5 optimum 5 generator sequences g1 g2 g3 g1 g2 g3 g1 g2 g3 g1 g2 g3 g1 g2 g3 g1 g2 g3 = [(0, 4), (4, 0)] = [(0, 2), (2, 0)] = [(0, 1), (5, 0)] = [(0, 2), (2, 0)] = [(0, 4), (4, 0)] = [(4, 5) , (1, 4)] = [(0, 4), (4, 4)] = [(0, 2), (2, 2)] = [(0, 1), (5, 1), (1, 5) ] = [(0, 4), (4, 0)] = [(0, 2), (2, 0)] = [(2, 0), (6, 5) , (1, 4)] = [(0, 4), (4, 4)] = [(0, 2), (2, 2), (2, 2)] = [(0, 1), (5, 1),... antennas are chosen to be 0. 25, 0 .5, 0. 75 and 1 It can be observed that the code performance is slightly degraded, relative to the system with independent sub-channels, when the correlation factor is 0. 25 However, the code performance is getting worse by 0 .5 dB and 1.3 dB compared to the case with uncorrelated antennas at a FER of 10−2 for the correlation factors of 0 .5 and 0. 75, respectively When the... [(0, 4), (4, 4)] = [(0, 2), (2, 2), (2, 0)] = [(3, 5) , (0, 0), (4, 0)] det r tr 2 2.0 4.0 2 4.0 4.0 2 3 .51 5 6.0 2 4.0 4.0 2 3 .51 5 8.0 2 7.029 7.172 as references It is clear from the table that for a given memory order, the proposed optimum codes have the same minimum rank as the TSC codes, but a larger minimum determinant which can result in a larger coding gain 4.3.2 Optimal STTC Based on the Trace... =4 R n =2 T n =3 T nT=4 −1 Frame Error Probability 10 −2 10 −3 10 0 5 10 15 SNR (dB) 20 25 30 Figure 4.19 Performance comparison of the 8-state 8-PSK codes based on the trace criterion with two, three and four transmit antennas on slow fading channels 0 10 nR=1 nR=2 nR=4 nT=2 nT=3 nT=4 −1 Frame Error Rate 10 −2 10 −3 10 0 5 10 15 20 25 30 SNR (dB) Figure 4.20 Performance comparison of the 16-state 8-PSK...4 Space-Time Trellis Codes 4.1 Introduction Space-time block codes can achieve a maximum possible diversity advantage with a simple decoding algorithm It is very attractive because of its simplicity However, no coding gain can be provided by space-time block codes, while non-full rate space-time block codes can introduce bandwidth expansion In... channel estimation is about 0.2 dB at a FER of 10−3 4 .5 Design of Space-Time Trellis Codes on Fast Fading Channels Code Design Criteria Sets III and IV are applied to construct good STTC for fast Rayleigh fading channels For an STTC, the symbol-wise Hamming distance δH between two paths 140 Space-Time Trellis Codes 0 10 cor=0 cor=0. 25 cor=0 .5 cor=0. 75 cor=1 −1 Frame Error Probability 10 −2 10 −3 10 0... 3 4 4 4 5 generator sequences g1 g2 g3 g1 g2 g3 g1 g2 g3 = [(2, 1, 3, 7), (3, 4, 0, 5) ] = [(4, 6, 2, 2), (2, 0, 4, 4)] = [(0, 4, 4, 4), (4, 0, 2, 0)] = [(2, 4, 2, 2), (3, 7, 2, 4)] = [(4, 0, 4, 4), (6, 6, 4, 0)] = [(7, 2, 2, 0), (0, 7, 6, 3), (4, 4, 0, 2)] = [(0, 4, 0, 3), (4, 4, 4, 3)] = [(0, 2, 4, 2), (2, 3, 7, 1), (2, 2, 7, 5) ] = [(4, 2, 6, 5) , (4, 2, 0, 7), (3, 7, 2, 6)] r det tr 2 – 16 .58 6 2 –... of Space-Time Trellis Codes on Slow Fading Channels Table 4.1 Upper bound of the rank values for STTC nT =2 nT = 3 nT = 4 nT = 5 nT ≥ 6 2 2 2 2 2 2 2 3 3 3 2 2 3 3 4 2 2 3 3 4 2 2 3 3 4 =2 =3 =4 =5 =6 ν ν ν ν ν nR nR 4 4 3 3 2 2 1 1 2 3 4 (a) 2 ≤ ν ≤ 5 Figure 4.4 5 nT 3 2 4 5 nT (b) ν ≥ 6 The boundary for applicability of the TSC and the trace criteria achievable rank is min(nT , l) Consider the constraint... 2Rx −1 Frame Error Rate 10 −2 10 −3 10 0 5 10 15 SNR (dB) Figure 4.11 Performance comparison of the QPSK codes based on the trace criterion on slow fading channels with three transmit and two receive antennas 133 Performance Evaluation on Slow Fading Channels 0 10 4−state 8−state 32−state 64−state outage 2b/s/Hz 4Tx 2Rx −1 Frame Error Rate 10 −2 10 −3 10 0 5 10 15 SNR (dB) Figure 4.12 Performance comparison... with two transmit antennas and memory orders 3 to 5 based on the rank & determinant criteria are shown in Table 4.4 The TSC codes [4] are also considered 1 25 Design of Space-Time Trellis Codes on Slow Fading Channels Table 4.3 Optimal QPSK STTC with three and four transmit antennas for slow fading channels based on rank & determinant criteria nT ν 3 4 3 5 3 6 4 6 generator sequences g1 g2 g1 g2 g1 g2 . 0. 25, 0 .5, 0. 75 and 1. It can be observed that the code performance 114 Space-Time Block Codes 0 5 10 15 20 25 10 −3 10 −2 10 −1 10 0 SNR (dB) Frame Error Rate cor=0 cor=0. 25 cor=0 .5 cor=0. 75 cor=1 Figure. Space-time block codes from orthog- onal designs”, IEEE Trans. Inform. Theory, vol. 45, no. 5, pp. 1 456 –1467, July 1999. [4] V. Tarokh, H. Jafarkhani and A. R. Calderbank, Space-time block coding. 17, no. 3, pp. 451 –460, Mar. 1999. [5] V. Tarokh, A. Naguib, N. Seshadri and A. R. Calderbank, “Combined array processing and space-time coding , IEEE Trans. Inform. Theory, vol. 45, no. 4, pp.