對擴充辛格陣列的差協同陣列和能量圖形的分析

Abstract

在陣列信號處理中，來向角（DoA）估計和波束成形是兩個重要主題。在 DoA 估計中，線性稀疏陣列因其在差協同陣列中具有較大的中央均勻線性陣列 （ULA）段而具有優勢。在波束成形中，陣列的性能由其輻射方向圖決定。輻射方 向圖可以使用能量圖形來表示。然而，像嵌套陣列（NA）和互質陣列（CA）這 樣的稀疏陣列的能量圖形通常會表現出較高的旁瓣高度。這些較高的旁瓣高度會 導致波束成形性能下降。為了抑制稀疏陣列的高旁瓣高度，一些研究人員使用了 數位波束成形或最佳化方法去處理。然而，這些方法需要額外的實施成本。另一 方面，我們可以尋找具有低旁瓣高度的稀疏陣列。 在本論文中，我們分析了具有 N 個物理感測器的擴充辛格陣列（ESA）的差 協同陣列和能量圖形。ESA 的差協同陣列也具有一個大的中央 ULA 段，類似於 其他稀疏陣列。我們證明了ESA 的中央 ULA 段的大小是 O(N2 )。相較於 ULA， ULA 的中央 ULA 段的大小則是 O(N)。ESA 在 DoA 估計的優勢會在實驗模擬當 中說明。 在能量圖形方面，我們在特定角度時有一些保證值。然後我們證明了ESA 的零到零波束寬度（NNBW）是 O( 1 N2 )、平均旁瓣高度（MSLL）是 O( 1 N )。我 們還將 ESA 與 NA、CA 和 ULA 在能量圖形方面進行了比較。ULA 的 NNBW 是 O( 1 N )，而 NA、CA 的 NNBW 在特定情況下亦是 O( 1 N )。因此，ESA 擁有較窄的波束寬度和空間解析度上的優勢。在實驗模擬當中，ESA 的峰值旁瓣高 度（PSLL）介於-9 到-15 分貝之間。另一方面，ULA 的 PSLL 是-13 分貝、NA 的 PSLL 大約是-6 分貝，而 CA 的大約是-5.5 分貝。 在陣列信號處理中，平面（2D）陣列是另一個值得注意的技術。這些陣列可 以看作是線性（1D）陣列的變化。一些 ESA 可以調整成為平面稀疏陣列，稱為 2D ESA。2D ESA 在二維差協同陣列和二維能量圖形方面保留了與 ESA 相似的 可證明性質。2D ESA 的二維差協同陣列也包含了一個較大的中央連續部分。2D ESA 的二維能量圖形在特定角度下也有保證值。

In array signal processing, Direction of Arrival (DoA) estimation and beamforming are two prominent topics. In DoA estimation, linear sparse arrays offer an advantage due to their large central Uniform Linear Array (ULA) segment in the difference coarray. In beamforming, the performance of arrays is determined by their radiation patterns. The radiation pattern can be described by the power pattern. However, the power patterns of sparse arrays like Nested Arrays (NAs) and Coprime Arrays (CAs) often exhibit high sidelobe levels. These higher sidelobe levels can result in degraded performance in beamforming. To suppress the high sidelobe levels of sparse arrays, some researchers have applied digital beamforming or min processing techniques. Nevertheless, these methods require additional implementation costs. On the other hand, we might explore a sparse array with a low sidelobe level. In this thesis, we analyze the difference coarray and power pattern of the Extended Singer Array (ESA) with N physical sensors. The difference coarray of the ESA also has a large central ULA segment, similar to other sparse arrays. We prove that the size of the central ULA segment of the ESA is of O(N2 ). Compared to the ULA, the size of the central ULA segment of the ULA is of O(N). The advantage of the ESA on DoA estimation is demonstrated in simulation. On the power pattern, we have some guaranteed values at specific angles. Then we show that the Null-to-Null BeamWidth (NNBW) of the ESA is of O( 1 N2 ) and the Mean SideLobe Level (MSLL) of the ESA is of O( 1 N ). We also compare the ESA with the NA, CA, and ULA in terms of their power patterns. The NNBW of the ULA is of O( 1 N ), and those of the NA and CA are also of O( 1 N ) in certain situation. Hence, the ESA features a narrower beam width and has an advantage in spatial resolution. In simulation, the Peak Sidelobe Level (PSLL) of the ESA fluctuates between -9 dB and -15 dB. On the other hands, the PSLL of the ULA is -13 dB, the PSLL of the NA is about -6 dB and that of the CA is about -5.5 dB. In array signal processing, planar (2D) arrays represent another noteworthy technique. These arrays can be viewed as a variation of linear (1D) arrays. Some ESAs can be adapted into planar sparse arrays, referred to as 2D ESAs. 2D ESAs retain similar provable properties to ESAs in terms of the 2D difference coarray and the 2D power pattern. The 2D difference coarray of the 2D ESA also contains a larger central contiguous part. The 2D power pattern of the 2D ESA also has guaranteed values at specific angles.