在第五代無線通訊系統當中，傳送端將會具有大規模的天線數量，更進一步的說，5G系統將會使用數百根，甚至是數千根天線，也就是所謂的多輸入多輸出技術(MIMO)，這樣的設計為系統帶來許多好處，比如最大化頻譜效率與更大的通道吞吐量，然而，一些演算法的高複雜度會導致系統難以被實作，這就是在實現這樣的高維度系統中最需要克服的難題。
當我們考慮在MIMO接收端的符元判斷問題，常見的訊號檢測器演算法如Maximum a Posteriori (MAP) 將因為複雜度問題而不能再被使用，這些常見的訊號檢測器算法的複雜度將會隨著系統的維度而有指數層級遞增，起因於這些演算法的每次疊代都需要計算遞迴迴圈，所以，一個低複雜度的檢測器演算法將會成為實現大規模MIMO系統的主要需求。
在這篇論文當中，我們有兩個主要貢獻，第一，我們提出一個名為期望傳播演算法的低複雜度檢測器算法來處理Sparse Code Multiple Access (SCMA) 檢測器問題。期望傳播演算法可以利用指數族函數來近似事後機率的邊際分佈，而指數族函數的機率比起近似前的函數更加容易計算，所以期望傳播演算法非常適合用來處理高維度與高等級調變的複雜系統。我們提出了理論分析來估測期望傳播演算法在SCMA系統的效能表現，可以看到其效能在傳送端與接收端天線數增加時可以貼近最佳檢測效能，有了這樣的理論根據，我們開始研究以星座點旋轉來增加自由度方法的必要性，我們發現在上行鏈路當中，我們也可以用不同的通道響應來增加不同使用者的可辨識度。因此，在SCMA編碼器上附加一個旋轉量並非必要，而當我們刪除這一個步驟，就可以在編碼與解碼兩端省下許多不必要的計算量。
第二，我們提出了一個名為分散式期望傳播的演算法來輔助大規模MIMO系統，這個演算法可以達到勝過Approximate Message Passing(AMP) 的效能，我們也研究了在不同反矩陣維度時的期望傳播演算法複雜度，原始演算法中，期望傳播 演算法的反矩陣維度等於傳送端天線的維度個數，藉由實現分散式系統，我們可以將反矩陣的維度減少至原始演算法的1/C倍，C代表的是分散式系統中的一個參數。我們也提供了分散式系統的理論分析。

ABSTRACT
In near future fifth generation of wireless system (5G), a huge number of transceivers antennas will be employed. Speci cally, the 5G system will employ hundreds even thousands antennas that known as massive Multiple-Input Multiple-Output (MIMO) technology. It brings a lot of advantages such as maximization of spectral e ciency (SE) and larger channel capacity [1], [2]. However, the implementation of high dimensional antennas results a technical issue that needs to be solved. This mainly issue regarding unaffordable complexity.
Considering the symbols detection at the massive MIMO receiver side, the conventional symbols detector algorithm such as Maximum a Posteriori (MAP) can no longer be used due to the unaffordable complexity. The conventional detector algorithm complexity increases exponentially with the dimension of the system because it needs to calculate the feedback loop operation in every iteration. Therefore, a low complexity detector algorithm becomes the main requirement in order to implement the massive MIMO systems.
In this thesis, we propose two major contributions. First, we propose a low complexity detection method for the SCMA detector named the expectation propagation algorithm (EPA). The EPA approximates the marginal distribution of the posterior probability by using an exponential family [3]. Given that the probability in exponential family is easy to compute, the EPA is suitable to deal with high order and
dimensional system. We also provide theoretical analysis to evaluate the performance
of EPA SCMA. We show that the EPA for SCMA can achieve near optimal detection
performance as the numbers of transmit and receive antennas grow. With the theoretical promise, we investigate the necessity of constellation rotation, which is used to increase the degree of freedom [4, 5]. We show that for the uplink scheme, channel responses from di erent users vary and thus increase the identi ability of each user.Therefore, appending a rotation value in SCMA encoder is unnecessary. The removal of the rotation, value can omit many unnecessary calculations not only in decoding but also in SCMA encoding.
Second, we propose a novel algorithm i.e. decentralized expectation propagation algorithm (EPA) to support massive MU-MIMO system which outperforms decentralized AMP [6]. We also investigate the EPA complexity which lies on the dimension of the EP inverse matrix. Originally, the dimension of the EP inverse matrix is equal with the dimension of transmitter antennas. By implementing the partially decentralized system, we signi cantly reduce the dimension of the inverse matrix to become C times smaller than the original one, where C denotes the number of the decentralized system
we have. In addition, we provide the theoretical analysis for each decentralized EP systems.

TABLE OF CONTENTS
Page
論文審定書 ........................................................................................................................................ i
ACKNOWLEDGMENTS ..................................................................................................... ii
ABSTRACT .......................................................................................................................... iii
TABLE OF CONTENT ........................................................................................................ vi
LIST OF TABLES ............................................................................................................ viii
LIST OF FIGURES ......................................................................................................... ix
NOTATIONS ................................................................................................................... x
ABBREVIATIONS ......................................................................................................... xi
1 Introduction and Motivation .......................................................................................... 1
1.1 Multiple Input Multiple Output System ................................................................. 1
1.2 Symbol Detection in Massive MIMO Systems ...................................................... 3
1.3 Thesis Organization............................................................................................... 3
2 Expectation Propagation Algorithm and MU-MIMO Systems .................................... 5
2.1 Single Loop Expectation Propagation ................................................................... 5
2.1.1 EP Message Passing ............................................................................... 7
2.1.2 Detail of EP Algorithm .......................................................................... 9
2.1.3 EP Computational Complexity............................................................. 10
2.2 Future Work For Expectation Propagation ......................................................... 13
2.2.1 Double Loop EP ................................................................................... 13
2.2.2 Approximation of Inverse matrix ......................................................... 14
2.3 EP State Evolution .............................................................................................. 15
2.4 Massive MU-MIMO Systems ............................................................................. 16
3 Implementing Expectation Propagation on Next Generation Wireless Systems ........ 18
3.1 MPA SCMA and EPA SCMA ............................................................................ 18
3.1.1 System Model ....................................................................................... 19
3.1.2 MPA SCMA ......................................................................................... 21
3.1.3 EPA SCMA .......................................................................................... 23
3.1.4 MPA and EPA Complexity Analysis ................................................... 23
vii
3.2 Decentralized Expectation Propagation in massive MU-MIMO System ........... 24
3.2.1 System Model ....................................................................................... 25
3.2.2 Fully Decentralized Expectation Propagation (FD-EP) Algorithm ..... 26
3.2.3 Partial Decentralized Expectation Propagation (PD-EP) Algorithm ... 30
3.2.4 Decentralized EP Performance Analysis.............................................. 31
4 Experimental Result and Discussion .......................................................................... 33
4.1 EPA SCMA ......................................................................................................... 33
4.2 Decentralized EP ................................................................................................. 36
5 Summary ..................................................................................................................... 43
LIST OF REFERENCES ............................................................................................... 45

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