考量互耦合效應下稀疏陣列之幾何形狀設計及效能分析

Sparse Array Design and Performance Analysis with Mutual Coupling

王廷崧(Ting-Sung Wang)

指導教授 ： 劉俊麟

國立臺灣大學/電機資訊學院/電信工程學研究所/碩士(2021年)

https://doi.org/10.6342/NTU202102312

摘要

在陣列信號處理領域，有多個擺放於不同位置的傳感器接收環境中訊號源發出的信號，我們可以藉由適當的演算法從接收到的信號中估計感興趣的訊號源資訊，例如到達方向。陣列信號處理領域中，到達方向估計的應用十分廣泛，在雷達、天文學、聲納、通信等諸多領域受到普遍關注。傳統的等間距線性陣列在給定 N 個傳感器下最多只能解析 N-1 個不相關源的到達方向，而稀疏陣列可以使用 O(N) 量級個傳感器分辨 O(N^2) 量級個不相關源的到達方向。然而傳感器之間的非理想效應，互耦合，會導致到達方向估計的表現變差。雖然我們可以透過去耦合來減輕耦合效應，但該方法需要很高的計算複雜度，且對耦合模型不匹配很敏感。反之，與等間距線性陣列相比，使用稀疏陣列可以有效地減緩耦合效應，並於到達方向估計提供滿意的表現。
本論文的主要貢獻可分為三個部分。第一部分中，由於大多數的已知一維稀疏陣列在考慮互耦合的情況下仍有自己的缺點，因此我們提出了可擴展陣列，它滿足良好的特性，使得其擴展後的陣列，稱 M 次擴展陣列，成為一個能有效對抗互耦合的稀疏陣列。數值模擬說明，從均勻自由度、權重函數及考慮互耦合效應的到達方向估計實驗來看，M 次擴展陣列均擁有優秀的表現。
第二部分中，因平面陣列可以同時估計訊號源的方位與仰角，在實際陣列信號處理中比線性陣列更為通用。因此，半開盒陣列、雙層半開盒陣列及沙漏陣列這些基於直角網格的平面稀疏陣列在2017年被其他學者研發出來，以有效減緩互耦合效應。然而，由於六邊形取樣對於在圓形區域中的有限頻帶信號為最佳取樣方案，因此我們提出了基於三角形網格的新型二維稀疏陣列，稱縮減六邊形陣列。數值模擬說明，以考量互耦合效應的到達角度估計來看，縮減六邊形陣列很大可能優於那些基於直角網格的平面稀疏陣列。
在第三部分中，由於到達方向估計的模擬結果可能會受限於所使用的估計器其準確性與兼容性，我們研究了利用二維稀疏陣列估計到達方向的克拉馬-羅限。我們在現有論文的基礎上提出了基於二維陣列的克拉馬-羅限表達式、二維秩條件及其充分條件，並進一步研究考量互耦合的數據模型其克拉馬-羅限的存在性與表達式。

關鍵字

稀疏陣列 ； M次擴展陣列 ； 縮減六邊形陣列 ； 互耦合 ； 到達方向估計 ； 克拉馬-羅限

並列摘要

In array signal processing, multiple sensors located at different positions receive signals emitted from sources in the environment. We can estimate interested source information such as directions of arrival (DOA) from received signals via appropriate algorithms. DOA estimation in array signal processing is widely applied and has attracted considerable attention in many fields such as radar, astronomy, sonar, and communications. Traditional uniform linear arrays (ULA) resolve at most N-1 uncorrelated source directions given N sensors, while sparse arrays can distinguish O(N^2) uncorrelated sources with O(N) sensors. Nevertheless, mutual coupling, the nonideal effect on sensors, results in the degradation on the performance of DOA estimation. Although we can mitigate the coupling effect by decoupling, such approach requires high computational complexity and is sensitive to model mismatch. On the contrary, sparse arrays can potentially alleviate the coupling effect and provide satisfactory performance for DOA estimation compared to ULA.
The main contribution of this thesis can be divided into three sections. In the first part, since most of known one-dimensional (1D) sparse arrays have their own disadvantages in the presence of mutual coupling, we propose Extensible ARrays (EAR), which satisfy nice properties so that their extension, called the EXtended Array with M-extension operation (EXAM), becomes a sparse array potentially against mutual coupling. Numerical simulations demonstrate the exceptional performance of EXAM from the view of uniform degree of freedoms (uDOF), weight functions, and practical DOA estimation with mutual coupling.
In the second part, planar arrays are more general than linear arrays for array signal processing in practice, since they can estimate the azimuth and elevation of the sources simultaneously. Accordingly, two-dimensional (2D) sparse arrays based on the square lattice such as half open box arrays (HOBA), HOBA with two layers, and hourglass arrays were proposed by C.-L. Liu and P. P. Vaidyanathan in 2017 to alleviate the coupling effect. However, due to the fact that hexagonal sampling is the optimal sampling scheme for signals that are bandlimited over a circular region, we propose novel 2D sparse arrays located on the triangular lattice called the REduced HExagonal ARrays (REHEAR). Numerical simulations show that REHEAR potentially prevail over those planar arrays based on the square lattice in the presence of mutual coupling.
In the third part, since the results of the simulation for practical DOA estimation may be affected by the accuracy and compatibility of estimators we used, we study the Cramér-Rao bound (CRB) for 2D sparse arrays. We propose CRB expressions, 2D rank condition, and sufficient conditions for 2D rank condition based on the existing paper, and we further investigate the existence and expression of CRB for the case that the data model is in the presence of mutual coupling.

並列關鍵字

sparse array ； EXAM ； REHEAR ； mutual coupling ； DOA estimation ； Cramér-Rao bound

參考文獻

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