我們會在這篇論文中探討如何在一個毫米波(Millimeter Wave) 頻段的多重輸入多重輸出系統(MIMO) 中做波束搜尋，來找到天線陣列的出設角度(AoD) 和入射角度(AoA)。由於毫米波頻段有非常大的路徑損失，因此我們需要更多的天線並實做波束成型技術來克服它，為了實作波束成型我們需要完整的通道資訊, 但在毫米波頻段下通道資訊是非常難取得的。然而在此頻段下真正能到達接收端的路徑是非常少的，因此有些文獻提出了新的通道估測方法，去找出這幾條路徑，也就是所謂的波束搜尋技術。最常見的方法叫做窮盡搜尋法，此方法是把所有的出設角度與入射角度都做一次配對，並找出有最大訊號強度的角度組合，但此方法的缺點就是搜尋的時間非常的長，複雜度也相當的高。

因此，之後也有文獻提出適應性的演算法並搭配二分搜尋法來縮短搜尋的時間，而在此篇論文中，我們會利用邏輯拼圖的概念，來提出一種名為平衡拼圖波束搜尋法的技術，並且針對單路徑以及多路徑的通道設計了不同的演算法。我們也設計了最佳的臨界值，使得此方法的頻譜使用效率更好，最後也利用了做波束搜尋所需的階段數來當作判斷複雜度的標準，並做了複雜度的分析，在我們的模擬結果中顯示出此方法能將複雜度更降低，並且有和二分搜尋法差不多的頻譜使用效率，而所付出的代價就是在每次波束搜尋的階段裡必須多回傳一個位元。

We focus on the angle of departure and angle of arrival searching problem for millimeter wave MIMO channel in the thesis. The path loss is high in the mmWave band. Hence, we need to use the massive MIMO and do the beamforming to combat these drawbacks. To implement the beamforming, we have to get the complete channel information. But the channel information is difficult to achieve in mmWave channel. However, the path which can arrive the receiver is few. Hence, there are literature proposed the new channel estimation method to find these paths, and it is called beam search. The most common method is exhaustive search, it trains over all possible beam directions for both the transmitter and
the receiver, and find the best beam pair which have maximum signal magnitude. However, the method takes much time to do the beam search, and it also has high complexity.

Hence, there are literature also proposed the adaptive algorithm with bisection method to shorten the searching time. In this thesis, we propose a new beam search method which is called balance puzzle beam search method by using the concept of logical puzzle. We design the different algorithm for the single-path and multi-path case and design the optimal threshold to make the spectral efficiency better. Finally, we consider the number of stages as a criterion for judging complexity, and do the complexity analysis. Simulation result show that the balance puzzle beam search method can reduce the complexity and have the
comparable spectral efficiency compared to the bisection method, and the cost is to feedback one more bit in each stage.

1 Introduction 1
1.1 Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Motivation and Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 The Proposed Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Background 4
2.1 MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Spatial Multiplexing [1] . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Beamforming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 System Model [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Multi-Resolution Codebook Design for Different Beam Search Scheme . . . . 11
2.3.1 The Bisection Beam Search Method 1 [2] . . . . . . . . . . . . . . . . 11
2.3.2 The Bisection Beam Search Method 2 [3] [4] [5] . . . . . . . . . . . . 15
2.3.3 The Exhaustive Beam Search Method [5] . . . . . . . . . . . . . . . . 18
2.3.4 Beam Search for Multi-Path Channel [4] . . . . . . . . . . . . . . . . 19
3 Proposed Beam Search Scheme 21
3.1 The Concept of the Balance Puzzle Method . . . . . . . . . . . . . . . . . . 21
3.2 Two Types of the Balance Puzzle Methods . . . . . . . . . . . . . . . . . . 23
3.2.1 Balance Puzzle Method with Binary Grouping . . . . . . . . . . . . . 23
3.2.2 Balance Puzzle Method with Ternary Grouping . . . . . . . . . . . . 27
3.3 The Balance Puzzle Beam Search Method . . . . . . . . . . . . . . . . . . . 28
3.3.1 Balance Puzzle Beam Search Method for Single-Path Channel . . . . 28
3.3.2 Balance Puzzle Beam Search Method for Multi-Path Channel . . . . 31
3.4 The Criterion for Judging Balance or Not . . . . . . . . . . . . . . . . . . . 35
3.4.1 Threshold Optimization for the Single-Path Channel . . . . . . . . . 35
3.4.2 Threshold Optimization for the Multi-Path Channel . . . . . . . . . 37
3.5 Complexity of the Balance Puzzle Method . . . . . . . . . . . . . . . . . . . 39
4 Simulation Results 44
4.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2 Spectral Efficiency among Different Beam Search Scheme . . . . . . . . . . 45
4.3 Complexity Analysis among Different Beam Search Scheme . . . . . . . . . . 48
5 Conclusions 52

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