本論文主要探討在時變合作式網路中有多個半雙工（half-duplex）之中繼點，並使用放大後前送（amplify-and-forward）的分散式波束成形（distributed beamforming）。藉由完整的通道狀態資訊（channel state information, CSI），傳輸的波束成形可以有效的達到多樣性及編碼增益。然而，要維持完整通道狀態資訊已知的狀態，需耗費非常大的資源來交換通道的整體資訊。首先，我們採取廣義Lloyd演算法（Generalized Lloyd Algorithm, GLA）去設計碼書（codebook）用以減少平均訊雜比（signal to noise ratio, SNR），並且透過量化通道狀態資訊來減少回授率。除此之外，利用通道時變的相關性去壓縮所需回傳訊息的大小，而達到分散式波束成形。將時變通道以一階有限狀態的馬可夫鏈表示，稱作馬可夫鏈通道狀態（channel state Markov chain）。然後根據不同通道狀態之間轉移機率的特性，我們提出兩種方法來壓縮回傳的位元數。一種方法是藉由摒棄部分不易於被目前狀態所轉移的通道狀態來壓縮回傳；另一種方法是保留所有的通道狀態，並利用轉移機率的特性使用霍夫曼編碼（Huffman Coding）壓縮回傳位元數。模擬也指出使用壓縮回傳的分散式波束成形效果近於使用無窮多個位元回傳。

This thesis proposes a distributed beamforming technique in wireless networks with half-duplex amplify-and-forward relays. With full channel state information, it has been shown that transmit beamforming is able to achieve significant diversity and coding gain. However, it takes large amount of overhead. First, we adopt the Generalized Lloyd Algorithm to design codebooks which minimize average SNR, and reduce the feedback rate by quantizing the channel state information. Furthermore, we utilize the correlation property of time-varying channels to compress the size of feedback message required to accomplish distributed beamforming. We model time-varying channels as a first-order finite-state Markov chain, namely the emph{channel state Markov chain}. Then, we propose two methods to compress the feedback bits according to the property of the transition probabilities among different channel states. One method is to compress the feedback by discarding some channel states which is less likely to be transited given current state. In the other method, we reserve all channel states and adopt Huffman coding to compress the feedback bits based on the transition probabilities. Simulations also show that distributed beamforming with compressed feedback performs closely to the beamforming with infinite feedback.

Abstract ii
List of Tables v
List of Figures vi
1 Introduction 1
2 Background and Related Work 5
2.1 Transmit Beamforming in Multi-Input Single-Output (MISO) System 5
2.2 Transmit Beamforming in Multi-Input Multi-Output (MIMO) System
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Distributed Beamforming in Cooperative Networks . . . . . . . . . . 9
2.3.1 Optimal Beamforming in Amplify-and-Forward System with
Unlimited Feedback . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.2 Distributed Beamforming in Amplify-and-Forward System with Finite Feedback 12
3 System Model 17
4 Codebook Design 22
4.1 The Generalized Lloyd Algorithm . . . . . . . . . . . . . . . . . . . . 23
4.2 Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 Compression of Feedback Messages in Time-Varying Channels 31
5.1 Channel State Markov Chain . . . . . . . . . . . . . . . . . . . . . . . 31
5.2 Compression of Feedback Message . . . . . . . . . . . . . . . . . . . . 37
6 Simulation Results 41
6.1 Comparison between GLP and GLA Codebooks . . . . . . . . . . . . 42
6.2 Performance Results of GLA Codebooks . . . . . . . . . . . . . . . . 43
6.3 Feedback Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.3.1 Transition Probability . . . . . . . . . . . . . . . . . . . . . . . 44
6.3.2 Bit Error Rate of Comparison . . . . . . . . . . . . . . . . . . . 46
6.3.3 Different autocorrelation coefficient ρ . . . . . . . . . . . . . . 47
7 Conclusion 50
Appendix 51
Bibliography 55

[1] A. Nosratinia, T. Hunter, and A. Hedayat, “Cooperative communication in wireless networks,” Communications Magazine, IEEE, vol. 42, pp. 74 – 80, oct. 2004.
[2] J. Laneman, D. Tse, and G. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Transactions on Information
Theory, vol. 50, pp. 3062 – 3080, dec. 2004.
[3] Y. Jing and H. Jafarkhani, “Network beamforming using relays with perfect channel information,” IEEE Transactions on Information Theory, vol. 55, pp. 2499 –2517, june 2009.
[4] Y. Jing and H. Jafarkhani, “Single and multiple relay selection schemes and their achievable diversity orders,” IEEE Transactions on Wireless Communications, vol. 8, pp. 1414 –1423, march 2009.
[5] Y. Jing and B. Hassibi, “Distributed space-time coding in wireless relay networks,” IEEE Transactions on Wireless Communications, vol. 5, pp. 3524 –3536, december 2006.
[6] E. Koyuncu, Y. Jing, and H. Jafarkhani, “Distributed beamforming in wireless relay networks with quantized feedback,” IEEE Journal on Selected Areas in Communications, vol. 26, pp. 1429 –1439, october 2008.
[7] A. Gersho and R. Gray, Vector quantization and signal compression. Kluwer Academic Press, 1992.
[8] K. Huang, R. Heath, and J. Andrews, “Limited feedback beamforming over temporally-correlated channels,” IEEE Transactions on Signal Processing,
vol. 57, pp. 1959 –1975, may 2009.
[9] S. Zhou, Z. Wang, and G. Giannakis, “Quantifying the power loss when transmit beamforming relies on finite-rate feedback,” IEEE Transactions on Wireless Communications, vol. 4, pp. 1948 – 1957, july 2005.
[10] K. Mukkavilli, A. Sabharwal, E. Erkip, and B. Aazhang, “On beamforming with finite rate feedback in multiple-antenna systems,” IEEE Transactions on
Information Theory, vol. 49, pp. 2562 – 2579, oct. 2003.
[11] N. J. A. Sloane, “How to pack lines, planes, 3-spaces, etc..” [Online]: http://www2.research.att.com/~njas/grass/index.html.
[12] P. Xia and G. Giannakis, “Design and analysis of transmit-beamforming based on limited-rate feedback,” IEEE Transactions on Signal Processing,
vol. 54, pp. 1853 – 1863, may 2006.
[13] D. Love, J. Heath, R.W., and T. Strohmer, “Grassmannian beamforming for
multiple-input multiple-output wireless systems,” IEEE Transactions on Information Theory, vol. 49, pp. 2735 – 2747, oct. 2003.
[14] K. Amiri, D. Shamsi, B. Aazhang, and J. Cavallaro, “Adaptive codebook for beamforming in limited feedback mimo systems,” in 42nd Annual Conference on Information Sciences and Systems, 2008. CISS 2008., pp. 994 –998, march 2008.
[15] B. Mondal and J. Heath, R.W., “An upper bound on snr for limited feedback mimo beamforming systems,” in Information Theory Workshop, 2004. IEEE, pp. 408 – 412, oct. 2004.
[16] A. Barg and D. Nogin, “Bounds on packings of spheres in the grassmann manifold,” IEEE Transactions on Information Theory, vol. 48, pp. 2450 – 2454, sep 2002.
[17] Y. Zhao, R. Adve, and T. Lim, “Beamforming with limited feedback in amplify-and-forward cooperative networks - [transactions letters],” IEEE Transactions on Wireless Communications, vol. 7, pp. 5145–5149, december 2008.
[18] V. Havary-Nassab, S. Shahbazpanahi, A. Grami, and Z.-Q. Luo, “Distributed beamforming for relay networks based on second-order statistics of the channel state information,” IEEE Transactions on Signal Processing, vol. 56, pp. 4306–4316, sept. 2008.
[19] P. Fertl, A. Hottinen, and G. Matz, “A multiplicative weight perturbation scheme for distributed beamforming in wireless relay networks with 1-bit feedback,” in IEEE International Conference on Acoustics, Speech and Signal Processing, 2009. ICASSP 2009., pp. 2625 –2628, april 2009.
[20] P. Fertl, A. Hottinen, and G. Matz, “Perturbation-based distributed beamforming for wireless relay networks,” in Global Telecommunications Conference,
2008. IEEE, pp. 1 –5, 30 2008-dec. 4 2008.
[21] Y. Jing and H. Jafarkhani, “Beamforming in wireless relay networks,” in Information Theory and Applications Workshop, 2008, pp. 142 –150, 27 2008-feb. 1 2008.
[22] J. H. Conway, R. H. Hardin, and N. J. A. Sloane, “Packing lines, planes, etc.: packings in Grassmannian spaces,” Experiment. Math, vol. 5, no. 2, pp. 139–159, 1996.
[23] P. Xia, S. Zhou, and G. Giannakis, “Achieving the welch bound with difference sets [optimal complex codebook design applications],” in Acoustics, Speech, and Signal Processing, 2005. Proceedings. ICASSP ’05). IEEE International Conference on, vol. 3, pp. iii/1057 – iii/1060 Vol. 3, march 2005.
[24] Z. Yi and I.-M. Kim, “Joint optimization of relay-precoders and decoders with partial channel side information in cooperative networks,” IEEE Journal
on Selected Areas in Communications, vol. 25, pp. 447 –458, february 2007.
[25] R. S. Gilks, W. R. and D. J. Spiegelhalter, Markov Chain Monte Carlo in Practice. Chapman and Hall, 1995.
[26] D. Huffman, “A method for the construction of minimum-redundancy codes,” Proceedings of the IRE, vol. 40, pp. 1098 –1101, sept. 1952.