關於差分共陣列在使用最小孔數陣列的波束成形中的角色

On the Role of the Difference Coarray in Beamforming using Minimum Hole Arrays

黃柏誠(Po-Cheng Huang)

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(2020年)

https://doi.org/10.6342/NTU202003149

摘要

在陣列訊號處理中，我們可以由多個擺放在不同位置的天線接收訊號，分別做訊號的濾波處理，並將濾波後的結果組合在一起，此動作為空間濾波，若陣列系統接收的是單頻信號，訊號經過空間濾波器所得到的輸出相當於訊號被一個具有特定波束圖的天線接收，因此空間濾波器也被稱為波束成形器。我們能夠根據某些特定結構的波束圖來設計權重函數，也能根據某些特定的波束成形器輸出來設計權重函數。因為權重函數和波束圖之間的數學結構，相似於權重函數和頻譜圖的數學結構，所以我們可以將特定結構的波束圖設計問題，連結到離散時間訊號處理中的濾波器設計問題做分析。重要的是，根據數學關係，由權重函數得到波束圖的過程，也等效於具有特定結構的Difference Co-array(DCA)計算陣列響應的過程，雖然DCA僅是一個數學上存在的虛擬陣列，但相比原始的陣列，DCA具有較多的權重點數，基於此特性，我們可以將波束圖的設計問題，轉換到DCA權重函數所對應的陣列響應設計問題，並從中得到對波束成形系統新的分析方式與設計。根據特定波束成形器輸出來建構權重函數設計問題的研究有許多，Minimum Variance Distortionless Response(MVDR) beamformer是其中一種，以得到最小的干擾源平均輸出功率為目標做設計，並由等效的最佳化問題得到了最佳的權重函數。
本篇論文主要分成兩個部份，第一個部份我們分析了MVDR beamformer所產生的陣列增益。經過分析，MVDR beamformer隨著不同INR(power ratio of single interference signal and local noise)而變化的陣列增益效果受到兩個方向向量做內積的絕對值平方的影響，產生了四種不同的變化情形。在了解兩個方向向量做內積的絕對值平方造成陣列增益函數的重要影響之後，我們也分析不同陣列擺放方式，對兩個方向向量做內積的絕對值平方所造成的影響，並進一步說明MVDR beamformer的陣列增益效果和陣列擺放方式的重要關聯性。
第二個部份我們根據Minimum Hole Array(MHA)的特殊DCA結構進行分析，進而提供一個設計DCA權重函數的演算法。數值模擬的部份說明了，比起在原始陣列上設計權重函數，我們有機會藉由設計DCA權重函數得到更理想的波束圖，相當於波束圖具有更小的main beamwidth、更低的sidelobe level、更接近0的null效果以及改善其他在實務應用上重要的參數。

關鍵字

雜訊消除 ； 最小變異數無失真響應波束成形器 ； 最小孔數陣列 ； 陣列響應圖合成 ； 窗形函數

並列摘要

In array signal processing, we can receive signals from multiple antennas placed in different locations, filter signals separately, and combine the filtering signals. Such signal processing operation is spatial filtering. If we receive single-frequency signals as the input of the spatial filter, the output of spatial filter is equivalent to the output of a single-antenna system with a specific beampattern, so the spatial filter is also called a beamformer. We can design the weight function based on a beampattern with specific structures, or design the weight function based on the specified output of a beamformer. Since the mathematical structure between the beampattern and the frequency spectrum is similar, we can convert the beampattern design problem into a filter design problem in discrete-time signal processing. Most importantly, based on the mathematical structure, the beampattern of a weight function is equivalent to the array response of a difference co-array (DCA) with specific structure. Although DCA is just a mathematical virtual array, DCA has more weight taps than the original array. Based on this advantage, we can convert the beampattern design problem on original array into the array response design problem on DCA. The new mathematical formulation of the beampattern design problem provides us a new approach to design and analyze a beamformer. Tremendous researches have been devoted to the weight function design problem based on specified output of a beamformer, Minimum Variance Distortionless Response (MVDR) beamformer is the most significant one of them. It is designed with the goal of forming unit gain on DOA of the desired signal while minimizing the average power of the output of interference signals and noise.
This thesis is mainly divided into two parts, the first part is analysis of array gain generated by MVDR beamformer. We found that the array gain of a MVDR beamformer can be classified into four different cases based on the absolute square value of inner product operation of two steering vectors. With the understanding of importance about the absolute square value of inner product operation of two steering vectors to array gain of a MVDR beamformer, we also discuss the importance of array location setting to the absolute square value of inner product operation of two steering vectors. In the overall discussion on MVDR beamformer, we found that array location setting is a significant factor for array gain.
In the second part, we analyze the mathematical structure of DCA and then present an algorithm for DCA weight function design by taking advantage of the special properties on MHA. Numerical simulation shows that, compared to designing weight function on original array, we have more opportunities to obtain desired beam pattern, which has narrower main beamwidth, lower sidelobe level, deeper null, and other important parameters in practical applications, by designing DCA weight function.

並列關鍵字

noise reduction ； MVDR beamforming ； Minimum Hole Array ； array pattern synthesis ； window function

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