5.3 Two-steps Design of Precoder System
6.2.2 Proposed Two-step System
In this example, we show the BER performance of two step system in section 5.3. The two step system has separate selection criteria for FV and FQ. We first select FV. The criteria we considered for FV are sum of squared error covariance criterion in (5.8) (SS), trace function of Re criterion in [5] (TrMSE) and BER criterion in [7] (BER). After FV is chosen, FQ is selected from CFQ for the chosen FV. The selection criteria for choosing FQinclude the AM-GM selection criterion (AMGM), BER criterion, and sum of squared error covariance criterion (SS).
Figure 6.18 shows the BER performance using different selection criterion for FQ
when FV is chosen using TrMSE criterion. In this example Mr = 4, Mt = 5, M = 4, Rb = 8 and BV = 4 and BQ = 4. Both CFV and CFQ are generated by RVQ method [6]. It can be observed that the difference between AM-GM criterion, BER criterion and SS criterion is almost negligible. Hence, we can use the low complexity AM-GM selection criterion for FQ.
4 6 8 10 12 14 16 18 20 22 24
10−4 10−3 10−2 10−1
P0/N 0
BER
BER AMGM SS
Figure 6.18: BER of different selection criterion of FQ for Mr = 4, Mt = 5, M = 4 and Rb = 8
The BER performance for different FV selection criteria is shown in Fig.
6.19. The system has Mr = 4, Mt = 5, M = 4 and Rb = 8. The matrix FQ
is chosen using the AM-GM criterion. The feedback bits B = 8 are equally divided for CFV and CFQ, BV = 3 and BQ = 3. We compare sum of squared error covariance criterion, trace function of Recriterion and BER-based selection criterion for choosing Mt× M precoder matrix FV. The BER curves are close but trace function of Re (total MSE) criterion is slightly better than the other two selection criteria. TrMSE criterion selects the total MSE minimizing FV
but the other two criteria consider total MSE minimizing and error covariance equalizing together. Therefore, the AM-GM selection criterion for CFQ combined with the trace function of Re for CFV provides a low-complexity design for two step system.
4 6 8 10 12 14 16 18 20 22 24
10−4 10−3 10−2 10−1
P0/N0
BER
BER SS TrMSE
Figure 6.19: BER of different selection criterion of FV for Mr4 =, Mt= 5, M = 4 and Rb = 8
Comparison between one-step and two-step system. In figure 6.20 the BER per-formance of one-step system and two-step system are compared. In this example Mr = 4, Mt = 5, M = 4 and Rb = 8. The feedback bit B = 8. For one-steps system, a precoder codeook of size 2B from [7] is prepared. The BER criterion in (3.1) is employed. For two-step system, BV = 4 and BQ = 4. The code-books CFV and CFQ are generated using RVQ method [6]. TrMSE criterion is used for FV selection and AMGM criterion is used for FQ selection. A similar hierarchical system (hierarchical) with two precoder codebooks [32] is also pro-vided. From [32], the feedback bits allocation of hierarchical system is 6 bits for one Grassmanian codebook and 2 bits for one rotational based codebook. The performance of BER optimal precoder with infinite feedback bits is also plotted.
Observing from figure 6.20, the BER optimal precoder has the best BER, and the curve of the two-step system is close to that of one-step system. In addition, The performance of two-steps system is slightly better than hierarchical system. Note that the number of searches for two-step system is 32, the numbers of searches for one-step system and hierarchical system are respectively 256 and 68.
4 6 8 10 12 14 16 18 20 22 24 10−4
10−3 10−2 10−1
P0/N 0
BER
Opt. precoder (∞ bits) 1 step (# of searches=256) hierarchical (# of searches=68) 2 step (# of searches=32)
Figure 6.20: BER comparison of one-step and two-steps systems for Mr = 4, Mt= 5, M = 4 and Rb = 8
Chapter 7 Conclusions
In this paper we first proposed to feedback only bit allocation for MIMO systems with limited feedback and the system is called a BA systems. Secondly, for precoder system with limited feedback, we describe two insightful properties of the BER optimal precoder. Motivated by these two properties, we develop two selection criteria for conventional one-step system and propose a two-steps design In proposed BA system, the augmented precoder is assumed to be known to both transmitter and receiver. With the BA scheme, the bits can be nonuniformly loaded. By allowing general bit allocation, bits can be allocated according to the channel. We have also shown that the proposed BA system can achieve diversity order of MrMt using log2(Mt) bits. The optimal augmented precoder can be any square unitary matrix. Furthermore, the unconstrained bit allocation is derived. Using the unconstrained bit allocation, we develop an efficient method for selecting the BER-minimizing bit allocation vectors from the codebook.
For precoder feedback system, two simple selection criterion are developed for square and rectangular precoders respectively, and a two-step system is designed for reducing the complexity. These selection criteria for one-step system are easy to compute and provide BER performance close to the BER-based selection criterion. The two-step system contains two precoder matrices in transmitter and lower the number of searches. Many interesting problem remain to be solved, such as the design of bit allocation codebook and the feedback bits allocation for two-step system.
Bibliography
[1] D. Love, R. Heath, V. Lau, D. Gesbert, B. Rao, and M. Andrews, “An overview of limited feedback in wireless communication systems,” IEEE Journal on Selected Areas in Communications, vol. 26, no. 8, pp. 1341–1365, 2008.
[2] A. Ekpenyong, Y. Huang, T. Instrum, and C. San Diego, “Feedback constraints for adaptive transmission,” IEEE Signal Processing Magazine, vol. 24, no. 3, pp. 69–78, 2007.
[3] T. Eriksson and T. Ottosson, “Compression of feedback for adaptive trans-mission and scheduling,” Proceedings of the IEEE, vol. 95, no. 12, pp. 2314–
2321, 2007.
[4] S. Sanayei and A. Nosratinia, “Antenna selection in MIMO systems,” IEEE Communications Magazine, vol. 42, no. 10, pp. 68–73, 2004.
[5] D. Love and R. Heath, “Limited feedback unitary precoding for spatial multi-plexing systems,” IEEE Transactions on Information Theory, vol. 51, no. 8, pp. 2967–2976, 2005.
[6] W. Santipach and M. Honig, “Asymptotic performance of MIMO wireless channels with limited feedback,” in IEEE Military Communications Confer-ence, 2003. MILCOM 2003, vol. 1, 2003.
[7] S. Zhou and B. Li, “BER criterion and codebook construction for finite-rate precoded spatial multiplexing with linear receivers,” IEEE Transactions on Signal Processing, vol. 54, no. 5, pp. 1653–1665, 2006.
[8] R. Blum, “MIMO with limited feedback of channel state information,” in Proc. IEEE Int. Conf. Acoust., Speech and Sig. Proc, vol. 4, 2003, pp. 89–92.
[9] J. Roh and B. Rao, “Design and analysis of MIMO spatial multiplexing systems with quantized feedback,” IEEE transactions on signal processing, vol. 54, no. 8, pp. 2874–2886, 2006.
[10] Y. Kim, H. Lee, S. Park, and I. Lee, “Optimal precoding for orthogonal-ized spatial multiplexing in closed-loop MIMO systems,” IEEE Journal on Selected Areas in Communications, vol. 26, no. 8, pp. 1556–1566, 2008.
[11] S. Thoen, L. Van der Perre, B. Gyselinckx, M. Engels, and H. IMEC, “Perfor-mance analysis of combined transmit-SC/receive-MRC,” IEEE Transactions on Communications, vol. 49, no. 1, pp. 5–8, 2001.
[12] R. Heath Jr, S. Sandhu, and A. Paulraj, “Antenna selection for spatial multiplexing systems with linearreceivers,” IEEE Communications Letters, vol. 5, no. 4, pp. 142–144, 2001.
[13] R. Yellapantula, Y. Yao, and R. Ansari, “Unitary precoding and power con-trol in MIMO systems with limited feedback,” in IEEE Wireless Communi-cations and Networking Conference, 2006. WCNC 2006, vol. 3, 2006.
[14] J. Roh, B. Rao, T. Inc, and C. Sunnyvale, “Efficient feedback methods for MIMO channels based on parameterization,” IEEE Transactions on Wireless Communications, vol. 6, no. 1, pp. 282–292, 2007.
[15] M. Sadrabadi, A. Khandani, and F. Lahouti, “Channel feedback quantiza-tion for high data rate MIMO systems,” IEEE Transacquantiza-tions on Wireless Communications, vol. 5, no. 12, pp. 3335–3338, 2006.
[16] J. Liu, D. Xu, R. Zhao, and L. Yang, “Adaptive transmission for finite-rate feedback mimo systems,” in Wireless Communications and Networking Conference, 2008. WCNC 2008. IEEE, march 2008, pp. 666 –671.
[17] R. Heath Jr and D. Love, “Multimode antenna selection for spatial multi-plexing systems with linear receivers,” IEEE Transactions on Signal Pro-cessing, vol. 53, no. 8 Part 2, pp. 3042–3056, 2005.
[18] D. Love and R. Heath Jr, “Multimode precoding for MIMO wireless sys-tems,” IEEE Transactions on Signal Processing, vol. 53, no. 10 Part 1, pp.
3674–3687, 2005.
[19] V. Raghavan, V. Veeravalli, and A. Sayeed, “Quantized Multimode Precod-ing in Spatially Correlated Multiantenna Channels,” IEEE Transactions on Signal Processing, vol. 56, no. 12, pp. 6017–6030, 2008.
[20] X. Song and H. Lee, “Multimode precoding for MIMO systems: Perfor-mance bounds and limited feedback codebook design,” IEEE Transactions on Signal Processing, vol. 56, no. 10 Part 2, pp. 5296–5301, 2008.
[21] C. Mun, “Quantized Principal Component Selection Precoding for Spatial Multiplexing with Limited Feedback,” IEEE Transactions on Communica-tions, vol. 56, no. 5, 2008.
[22] P. P. Vaidyanathan, S.-M. Phoong, and Y.-P. Lin, Signal processing and optimization for tranceiver systems. Cambridge University Press, 2010.
[23] J. G. Proakis, Digital communications. McGraw-Hill, 2001.
[24] D. Love and R. Heath Jr, “Necessary and sufficient conditions for full di-versity order in correlated Rayleigh fading beamforming and combining sys-tems,” IEEE Transactions on Wireless Communications, vol. 4, no. 1, pp.
20–23, 2005.
[25] K. H. Rosen, Discrete mathematics and its applications. McGraw-Hill, 2003.
[26] A. James, “Distributions of matrix variates and latent roots derived from normal samples,” The Annals of Mathematical Statistics, vol. 35, no. 2, pp.
475–501, 1964.
[27] Z. Chen, J. Yuan, and B. Vucetic, “Analysis of transmit antenna selection/maximal-ratio combining in Rayleigh fading channels,” IEEE Transactions on Vehicular Technology, vol. 54, no. 4, pp. 1312–1321, 2005.
[28] T. LO, “Maximum Ratio Transmission,” IEEE transactions on communica-tions, vol. 47, no. 10, pp. 1458–1461, 1999.
[29] G. Dudevoir, J. Chow, J. Cioffi, and S. Kasturia, “Vector coding for T1 transmission in the ISDN digital subscriber loop,” in IEEE Int. Conf. Com-munications, 1989, pp. 536–540.
[30] L. Ord´o˜nez, D. Palomar, A. Pag`es-Zamora, and J. Fonollosa, “Minimum BER linear MIMO transceivers with adaptive number of substreams,” IEEE Transactions on Signal Processing, vol. 57, no. 6, 2009.
[31] ——, “High-SNR analytical performance of spatial multiplexing MIMO sys-tems with CSI,” IEEE Transactions on Signal Processing, vol. 55, no. 11, p.
5447, 2007.
[32] Y. Hunag, D. Xu, L. Yang, and W. Zhu, “A limited feedback precoding system with hierarchical codebook and linear receiver,” IEEE transactions on wireless communications, vol. 7, no. 12 Part 1, pp. 4843–4848, 2008.