在文獻中，已經有許多線性限制 (Linearly constraints) 的可適性陣列波束成型演算法 (Adaptive array beamforming algorithm ) 被提出，用以討論有關干擾源抑制 (Jammer cancellation) 的問題。線性限制最小變異 (Linearly constrained minimum-variance; LCMV ) 波束成型器 (Beam-former) 是其中被認為能有效抑制干擾訊號的重要方法之一。另外，在有關傳統適應性演算法方面，最小平方和(Recursive least square; RLS) 演算法與最小均方值 (Least mean square; LMS) 演算法是兩種最常被使用的適應性技術。相較於最小均方值(LMS)適應性演算法，最小平方和(RLS)演算法有較佳的收斂速度、均方誤差 (Means-square error; MSE ) 和參數追蹤能力 (Tracking capability)。然而最小平方和演算法，在有限的位元數 (Finite precision) 情況下，會有數值不穩定 (Numerical instability) 的現象，此問題可以藉由 QR分解-RLS（QRD-RLS）演算法來解決。 基本上，它可以採用由吉文斯旋轉（Givens rotation） 方法以計算出 QR 分解的輸入數據矩陣（Input data matrix），再利用反向代回 (Backward substitution) 方法來求解 LS 權重向量 (Weights vector) 。如此，可以減少計算量及改善數值不穩定等問題。 因為在實際應用上， 在求解最佳權重向量過程中需要在每次疊代後，對應的執行反向代回。 然而，反向代回在 VLSI 設計及利用DSP的技術來實現硬體時，其所採用陣列結構 (Systolic array structure) 中是一種昂貴的操作 (Costly)，為了改善這個問題我們可以採用所謂反向QRD-RLS (Inverse QRD-RLS)演算法以求取最佳權重向量。 再則，採用傳統的線性限制最小變異 (LCMV) 法則的演算法時經常會因為不匹配效應 (Mismatch effect) ，而使得系統效能降低 (Performance degradation)。文獻中已證實如果能採用所謂的常模 (Constant modulus; CM) 方法，可以有效改善上述問題。 常模方法可有效抑制干擾源。 然而，在無線通訊系統中，在資訊傳輸過程中如果干擾源 (Jammer user) 的功率強度較理想用戶 (Desired user) 訊號強的情況下。 單獨利用常模 ( CM) 法可能會發生所謂的捕獲問題 (Capture Problem)， 即接收端檢測器會鎖住功率強的干擾訊號而無法將功率弱的理想用戶訊號檢測出來。一般會結合加入與理想用戶相關的限制性條件，已改善捕獲問題(Capture Problem)，稱之為限制性常模 (LCCM) 法則。 本論文希望探討如何利用限制性常模 (LCCM) 法之最小平方值法則 (Least Square Criterion)，推導出改良式的可適性線性常模反向QR分解遞回式最小平方和 (Inverse QRD-RLS) 演算法，以解決上述所探論的不匹配效應及有效抑制干擾源的問題。再則， 因為反向QR分解法可以利用平行計算 (Systolic array) 來提高運算速度，也可同時有效消除多重使用著的相互干擾 (Multiple access interference; MAI )。

Various adaptive array beamforming algorithms with linearly constraints have been proposed to deal with the nulling processes for jammer cancellation. Among them the linearly constrained minimum variance (LCMV) Beam-former has been extensively used to effectively suppressing the interference signals. While the so-called recursive least squares (RLS) adaptive algorithm is considered to be one of the significant approaches to offer better convergence rate, steady-state means-square error (MSE), and parameter tracking capability. It is normally performed better than the adaptive least mean square (LMS) based algorithms. But, the widespread acceptance of the RLS filter has been impeded by a sometime unacceptable numerical performance in limited precision environments. To overcome this problem, an alternative numerical stable RLS algorithm, referred to as the QR-decomposition RLS (QRD-RLS) algorithm, is adopted. Basically, it computes the QR decomposition of the input data matrix using Givens rotation and solving the LS weight vector by the back substitution to reduce the computations, numerical instability, and other issues. Since in practical applications, the solution to the optimal weight vector during updating iteration, the corresponding implementation of the backward substitution must be performed. The backward substitution is a costly operation in the Systolic array structure used in VLSI design and DSP technology for implementing hardware. To avoid this problem, the inverse QRD-RLS algorithm is one of the good candidates to approaching the minimum weight vector solution. On the other hand, the use of LCMV will result in system performance degradation when a mismatch effect occurred. It has been proved in the literature that the above problems can be effectively improved if the constant modulus (CM) criterion is employed. For instances, in wireless communication systems, if the power of the jammer user is relatively higher than the desired user during the transmission of the information, the use of the constant modulus approach could effectively suppress the interference sources. To avoid the capture problem when the constant modulus approach is apply, a linear constrained CM approach is considered. In this thesis, we discuss the use of the LCCM approach to derive the adaptive LCCM inverse QR-RLS algorithm (IQRD-RLS algorithm) for solving the mismatch effect and interference sources suppression problems. Furthermore, the inverse QR decomposition method can make use of the Systolic array to improve the speed of operation, and at the same time to effectively eliminate the multiple user interference (MAI).

CHAPTER 1 INTRODUCTION	1
CHAPTER 2 Recursive Least Square Based Adaptation Algorithms	4
2.1 Introduction	4
2.2 Least Square Solutions	4
2.2.1 Recursive Least Squares (RLS) algorithm	4
2.2.2 QR Decomposition for Recursive Least Squares Adaptive Filters	6
2.2.3 The Inverse QRD-RLS Algorithm	7
2.3 Linear Constrained RLS Algorithm	9
2.3.1 Linear Constrained RLS Algorithm	9
2.3.2 Optimal Linearly Constrained QRD-LS Filter 10
2.3.3 The Adaptive LC-IQRD-RLS Filtering Algorithm 11
2.4 The GSC Implementation of the LC-IQRD-RLS Algorithm	15
CHAPTER 3 Adaptive Linear Constrained Inverse QRD-RLS Beamforming Algorithm	20
3.1 Introduction	20
3.2 Linearly Constrained Minimum Variance IQRD-RLS Beamforming Algorithm	20
3.3 Linearly Constrained Constant Modulus with IQRD-RLS Algorithm	25
3.4 Computer simulation	29
CHAPTER 4 CONCLUSIONS	34
Appendix A	35
References	38
 


LIST OF FIGURES
Fig.3.1 SINR performance comparison of the beam-former using different approaches without pointing error	30
Fig.3.2 Beam pattern in terms of nulling capability with different approaches	31
Fig 3.3 The learning curves in terms of MSE for different approaches	31
Fig 3.4 SINR performance comparison of the beam-former using different approaches with pointing error	32
Fig 3.5 Beam pattern in terms of nulling capability using different approaches, with pointing error	33
Fig 3.6 The learning curves in terms of MSE for different approaches, with pointing error	33

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