臺灣博碩士論文加值系統

互質陣列(coprime array) 是由兩個均勻線性陣列(uniform linear array) 組成，互質陣列可以分辨出比感測器數量更多的來源數量。然而，互質陣列的差陣列(difference coarray) 是非連續的，也就是說，差陣列中存在空洞(hole)。近年來，有許多基於陣列內插的入射角度估測方法(direction-of-arrival estimation) 被提出，包括最大熵(entropy)Toeplitz 矩陣補全，最小核範數(nuclear norm) Toeplitz 矩陣補全和重建Toeplitze 共變異矩陣。這些方法透過陣列內插技術，將不連續的差陣列內插為連續的均勻線性陣列，因此，基於陣列內插的入射角度估測方法可以利用互質陣列接收到的所有資訊進行入射角度估測。然而，陣列內插技術涉及求解凸函數最優化問題(convex optimization problem)，因此計算所需要的時間顯著地增加。在本論文中，我們提出了一種高效率的陣列內插方法透過核範數最小化和么正轉換(unitary transformation)，模擬結果說明，該方法可以大幅減少計算所需要的時間且和其他陣列內插方法擁有非常相近的入射角度估測精準度。

A coprime array is composed of two uniform linear arrays (ULAs). A coprime array can resolve more the number of sources than the number of sensors. However, the difference coarray of coprime arrays is non-consecutive. Namely, there are holes in the difference coarray. Recently, coarray interpolation-based direction-of-arrival (DOA) estimation methods, such as the maximum entropy Toeplitz matrix completion method, the minimum nuclear norm Toeplitz completion method, and the reconstructed Toeplitz covariance matrix method, have been proposed. These methods generate a consecutive uniform linear array from the non-consecutive difference coarray through array interpolation. Thus, these methods can utilize all the information received by the coprime array for DOA estimation. However, the coarray interpolation technique involves solving a convex optimization problem. It significantly increases the computational time. In this thesis, we propose an efficient coarray interpolation method via nuclear norm minimization and unitary transformation. Simulation results demonstrate that the proposed method can dramatically reduce computational time. Moreover, the proposed method and other coarray interpolation-based DOA estimation methods achieve a quite similar DOA estimation accuracy.

誌謝iii
摘要v
Abstract vii
1 Introduction 1
1.1 Comparison between Uniform Linear Arrays and Sparse Arrays . . . . . 1
1.1.1 Uniform Linear Arrays (ULA) . . . . . . . . . . . . . . . . . . . 2
1.1.2 Sparse Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Main Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 DOA Estimation for Coprime Array 7
2.1 The Coprime Array Structure . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Signal Model of the Coprime Array . . . . . . . . . . . . . . . . . . . . 9
2.3 The Spatial Smoothing MUSIC Method . . . . . . . . . . . . . . . . . . 12
2.4 Coarray MUSIC without Spatial Smoothing . . . . . . . . . . . . . . . . 16
2.5 Coarray Interpolation via Nuclear Norm Minimization . . . . . . . . . . 19
3 Proposed Methods 21
3.1 Method 1: Efficient Coarray Interpolation Method via Unitary Transformation
(Unitary-NNM) . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.1 The Unitary Transformation . . . . . . . . . . . . . . . . . . . . 22
3.1.2 Coarray Interpolation for Even-Dimensional Matrix . . . . . . . 25
3.1.3 Coarray Interpolation for Odd-Dimensional Matrix . . . . . . . . 34
3.2 Method 2: Real-Valued Coarray Interpolation Method (Real-NNM) . . . 38
4 Simulation Results 41
5 Conclusion 55
A Proof of Lemma 3.1.9 57
B Low-rank Terms of the matrixeBV 59
Bibliography 61

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