在大規模多天線系統中，最大的問題在於如何從組合的訊號終將個別的訊號解調出來，但是最佳的解調器極其複雜。而被設計用在壓縮感知的近似信息傳遞演算法，因為AMP演算法可以減少運算複雜度因此，吸引了許多研究者用來解決大規模多天線系統的問題。本篇論文中我們把重點放在如何以固定點數實現AMP演算法。為了實現此架構，我們在AMP演算法中提出近似平均值與變異數估測方程式 這些估測方程式是利用近似對數和得到，且取出結果的指數。近似對數和是由雅各比演算法且利用地回修正的方法所得到。為了更符合我們的問題，我們也提供了近似修正方程式的修改。接著利用近似方程式來取代平均值和變異數方程式。這些近似的應用影響了演算法的收斂性，為了對抗這個問題我們修改算法的初始阻尼和提供新的阻尼演算法的位元錯誤率效能 接著我們將這個演算法轉為固定點數，目標是得到變數的字元長度 最後我們利用硬體來實現這個演算法。

In massive multiple input and multiple output (MIMO) systems the challenge is the detection of the individual signals from the composite signal in the large system limit. The optimal detector becomes prohibitively complex. The approximate message passing (AMP) algorithm, designed for compressed sensing, has attracted researchers to counter this problem due to its reduced complexity in the large system limit. For this reason the AMP algorithm has been used for detection in massive MIMO systems. In this thesis we focus on implementing this algorithm in fixed-point format. To obtain an implementation friendly architecture, we propose approximations for the mean and variance estimation functions within the algorithm. These estimation functions are obtained using the log-sum approximation then taking the exponent of the result. The log sum approximation is obtained by the Jacobean logarithm with a correction function recursively. We also provide a modification of the correction function for the approximations that best suits our case. We then substitute the mean and variance estimation functions with the approximations. The application of these approximations affect the convergence of the algorithm, to counter that we modify the initial damping strategy of the algorithm and plot a BER performance for the algorithm of the newly damped version. We then transform this algorithm to fixed point with an aim of obtaining word lengths for the variables. We finally provide a hardware architecture for the algorithm.

Thesis Validation letter …………………………………………………………………………. i
Acknowledgements ……………………………………………………...….…………………… ii
Abstract (Chinese) ...…………………………………………………..………………………....iii
Abstract (English) ...…………………………………………………………………...………... iv
Chapter 1: Introduction …………………………………………………..……………………… 1
1.1 Introduction…………………………………………...…………………………...1
1.2 Background ………………………………………………………...……………. 3
1.2.1 MIMO Detection …………….….……………………………………….. 3
1.2.2 Fixed point theory …………...………..…………………………………. 5
1.2.3 Hardware architecture ……………………….……………………………7
1.3 Problem statement ……………………………...…………………………………8
1.4 Purpose of the study………………………………………...……………………..9
1.5 Significance of the study ……………………………...…………………………..9
1.6 Basic assumptions …………………………………...…………………………..10
1.7 Limitations …………………………………………...………………………….10
1.8 Summary …………………………………………………...……………………11
Chapter 2: MIMO-AMP algorithm derivation …………………….……………..……………..12
2.1 Introduction of the AMP algorithm...………………………...……….……………..12
2.2 MIMO - AMP algorithm ……………………………………….……………………13
2.2.1 Bayes theorem ……………………………………...……………………...13
2.2.2 Belief Propagation ……………………………………..………………….15
2.2.3 Relaxed Belief Propagation ………………...……………………………..16
2.2.4 Approximate Message Passing………………...…………………………..17
2.2.5 Damping ……………………………………………………...……………19
2.3 Approximations of mean and variance estimation functions ………………….…….21
2.3.1 Approximation of mean estimation function …………………………...…21
2.3.2 Approximation of variance estimation function …..…………………...…...25
2.4 Correction function modification …………………………………………….……...26
2.5 Convergence of AMP algorithm with approximated estimation function …………..27
2.6 Summary ……………………………………………………………….……………29
Chapter 3: Fixed point transformation …………………………………………………..………30
3.1 Introduction ……………………………………………………….…………………30
3.2 The fi object ……………………………………………………….………………...30
3.3 Effects of arithmetic operations on the word-length …….…………………………..32
3.3.1 Addition and subtraction …………………………………………………...32
3.3.2 Multiplication ……………………………………………………...………33
3.3.3 Division ………………………………………………………...………….33
3.3.4 Complex numbers arithmetic ……………………………...………………34
3.4 Word length analysis of the AMP algorithm ………………………………….…….38
3.5 Fixed point transformation procedure ……………………………………….……....41
3.6 Summary …………………………………………………………….………………44
Chapter 4: Hardware architecture ………………………………………..……………………..45
4.1 Introduction …………………………………………………………….……………45
4.2 Functions’ pseudo codes ………………………………………………….…………45
4.2.1 AMP algorithm function pseudo-code ...................................................…...45
4.2.2 Approximated estimate function pseudo-code ……………………………..46
4.2.3 Mean approximation function pseudo-code ………………………....…….48
4.2.4 Log sum-approximation function of complex numbers …………………...49
4.2.5 Log-sum approximation function for real numbers ………………………...50
4.3 Hardware architecture ………………………………………………….……………51
4.4 Summary …………………………………………………….………………………58
Chapter 5: Results and Analysis ……………………………………………..………………….59
5.1 Introduction …………………………………………………….……………………59
5.2 Performance plots of AMP algorithm variants …………………………….………..60
5.3 Fixed point AMP algorithm performance analysis …………………………….……63
5.3.1 64-QAM performance analysis ………………………………...………….63
5.3.2 16-QAM performance analysis ………………………...………………….67
5.4 Summary …………………………………………………………………….………68
Chapter 6: Conclusion and Future work ……………………………………………..………….70
6.1 Conclusion …………………………………………………………………………..70
6.2 Future work ………………………………………………………….………………70
References …………………………………………………….…………………………….…...72
Appendix A ……………………………………………………………………………..……….75
Appendix B ………………………………………………………………………………...……77

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