交替方向乘子法 (Alternating Direction Method of Multipliers 簡稱 ADMM) 是一種簡單且有用的演算法，它適用於分配凸集最佳化，主要是在把大型的全域問題，轉換成小型區域性的問題。
現今，我們所需要處理的資料愈來愈龐大，電腦再處理的時間上，也會越來越長， 所以如果能有效地把資料分割成幾個小區塊，個別來處理，勢必能縮短電腦運算的時間。
ADMM 主要結合了雙重拆解法 (Dual Decomposition) 和增強的拉格朗日法 (Augmented Lagrangian Methods) 的優點，同時也避免了乘子法 (Method of Multipliers) 缺點，因為利用乘子法有可能求出的解是不收斂的。
目前有許多的分散式系統都很適合使用 ADMM 來做處理，例如雲端系統及分散式的天線系統等，他們主要的特性，都是將很大量的資料，分散到多個電腦或基地台來做處理，降低處理的時間和成本。

本篇前半部主要是探討上行鏈路的問題，並配合不同的疊代次數，來觀察其錯誤率的變化。本篇後半部，則是探討下行鏈路的 ADMM 應用，及其訊號對干擾與雜訊比 (Signal to Interference plus Noise Ratio 簡稱 SINR) 的變化藉由參數的調整。
本篇論文的貢獻，主要是利用 ADMM 來當作分散式的解調工具，並且和其他的非分散式解調方法來做比較，以探討其優缺點。

Currently, data size that we have to deal with is growing bigger and bigger. This fact implies that the computing time and computing power for dealing with the data is demanded. A way to circumvent the difficulty is as follows: Divide the data into several small blocks and then process these small blocks by several computers. Therefore, we need a tool for the decomposition-coordinated procedure. Alternating direction method of multipliers (ADMM) is a powerful algorithm for the mentioned purpose and has widely used in distributed optimizations. With ADMM algorithm, a big global optimization problem can be decomposed into several small local optimization problems.



ADMM algorithm has been used in several recent distributed systems such as cloud systems and distributed antenna systems. In this thesis, we aim to apply the ADMM in a distributed antenna system. For the uplink setting, we develop a distributed demodulation algorithm, where multiple base stations collaborate with each other for data detection. On the other hand, for the downlink setting, we develop a distributed beamforming design algorithm, where multiple base stations collaborate to form a beamforming for mitigating the inter-cell interference. Finally, simulations are conducted to verify the efficiency of our designs.

摘要 i
Abstract ii
目錄 iii
圖次 iv
表次 v
1 緒論 1
1.1 研究背景與動機 1
1.2 論文架構 2
2 相關技術的介紹 3
2.1 ADMM技術概念 3
2.1.1 雙生子 3
2.1.2 乘子法 4
2.1.3 ADMM 5
2.1.4 一維範問題 7
2.1.5 一致性問題 8
2.1.6 分享式問題 9
2.2 ADMM 演算法使用時機 10
2.3 多重輸入與輸出技術簡介 11
2.4 多重輸入與輸出優點 12
2.5 多重輸入與輸出通道模型 13
2.6 等化器介紹 13
2.7 openMP介紹 14
3 ADMM在上行鏈路下的模型分析18
4 ADMM在下行鏈路下的模型分析22
5 模擬結果27
5.1 上行鏈路模擬 27
5.2 下行鏈路模擬 32
5.3 openMP模擬 39
6 結論41
參考文獻42
附錄43
.1 附錄I Soft Thresholding 43

[1] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, ``Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers,' Foudations and Trends in Machine Learning., vol. 3, no. 1, pp. 1-122, 2011.


[2] V. K. Nguyen, and J. S. Evans, ``Multiuser Transmit Beamforming via Regularized Channel Inversion: A Large System Analysis,' in Proc. IEEE Global Telecomm. Conf., Nov.-Dec. 2008.


[3] R. Muharar, and J. S. Evans, ``Downlink Beamforming with Transmit-side Channel Correlation: A Large System Analysis,' in IEEE International Conf. on Communications (ICC 2011), Kyoto, Japan, Jun. 2011.



[4] W. Stallings, Computer Organization and Architecture. New Jersey: Prentice Hall, 2010.


[5] B. Chapman, G. Jost, and R. V. D. Pas, Using OpenMP: Portable Shared Memory Parallel Programming. The MIT Press, 2008.