基於感測器錯誤與陣列對稱性的稀疏陣列穩健性分析

Abstract

在一維稀疏陣列中，多個感測器以不同的間距被擺放在一直線上。稀疏陣列能夠用O(N)個感測器分辨出O(N^2)個不相關訊號源，原因是差異協列中間之連續片段有O(N^2)的長度，而差異協列定義為陣列中任兩個感測器位置的差所形成的集合。基於差異協列的角度估計器如協同陣列多信號分類，可以使用在差異協列上的資料來估計訊號源角度。經驗上來說，差異協列容易被感測器錯誤所影響，感測器錯誤會隨機發生並造成感測器無法準確地接收信號，因此錯誤的感測器會從陣列中被移除。一旦有感測器被移除，稀疏陣列便不保證能擁有辨別O(N^2)個不相關訊號源的優點。 傳統均勻線性陣列比稀疏陣列更穩健，但其最多只能辨別N-1個不相關訊號源。一個強化稀疏陣列穩健性的方法是將陣列做對稱。本篇論文提出一些關於對稱陣列的性質，例如，被研究過的「廣義1-脆弱性」之上下限。此外，我們證明將互質陣列做對稱後，它可以達到廣義1-脆弱性的下限。 如果每個感測器失敗的機率為p，且互相獨立，那差異協列中間連續片段的期望值可以用機率質量函數的觀點被推導出來。此期望值是變數p的一元多項式。除了能量化一個陣列的穩健性之外，此期望值還適合去比較不同陣列的效能。最後透過模擬的結果，可以展現出在感測器錯誤影響下，對稱陣列的優勢。

In one-dimensional sparse arrays, multiple sensors are placed on a line with different intervals. Sparse arrays are able to distinguish O(N^2) uncorrelated sources with O(N) sensors. The reason is that the difference coarray, defined as the differences between any two sensors of an array, has a central uniform linear array (ULA) segment of length O(N^2). The coarray-based angle estimators such as coarray MUltiple SIgnal Classification (MUSIC) can use the data on the difference coarray to estimate the source directions. Empirically, difference coarrays are easily influenced by sensor failures. They will occur randomly and cause the sensors not to receive the signals accurately. Therefore, the faulty sensors will be removed from the array. Once the sensors are removed, sparse arrays are not guaranteed to have the advantage of identifying the O(N^2) uncorrelated sources. Traditional ULA are known to be more robust than sparse arrays, but they can only resolve at most N-1 uncorrelated sources. A method that can enhance the robustness of sparse arrays is to symmetrize the array. This thesis advances some properties related to the symmetrical arrays. For instance, the upper bound and the lower bound of the generalized 1-fragility are studied. Additionally, we prove that coprime arrays can achieve the lower bound of the generalized 1-fragility after symmetrizing them. If each sensor fails independently with probability p, then the expected value of the size of the central ULA segment in the difference coarray can be derived from the view of the probability mass function (PMF). The expected value is an unary polynomial with the variable p. Besides quantifying the robustness of an array, the expected value is suitable to compare the performance of different arrays. Finally, the benefits of symmetrical arrays under the influence of sensor failures will be shown through the simulation results.