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

國立臺灣大學電機資訊學院電信工程學研究所 碩士論文

Graduate Institute of Communication Engineering College of Electrical Engineering and Computer Science

National Taiwan University Master Thesis

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

Po­Cheng Huang

指導教授：劉俊麟博士 Advisor: Chun­Lin Liu, Ph.D.

中華民國 109 年 7 月 July, 2020

誌謝

回首這兩年的碩士生涯，第一個要感謝的人，是我的指導教授，劉 俊麟教授。經過這兩年來，每一週和教授的研究討論，我了解到了做 研究的方式、解決問題的方法、資料整理的重要性以及時間管控的方 式，並從教授的帶領中逐漸成長。感謝教授在研究討論時不厭其煩地 指導以及在我感到失落時給予鼓勵，使我的碩士生涯能夠有系統的學 習成長，也使我能夠抵抗壓力不斷的努力前進，而這些都是我往未來 邁進時的重要養分。希望未來，我能將教授在做研究或做事情的嚴謹 態度，以及待人處事的謙卑態度，放在職場上，甚至是生活上，也希 望有昭一日，能夠在小有成就之時，報答教授這兩年的教導之恩。

接下來我想感謝台灣大學電信所的老師。在這兩年修習的所有課程 中，每當上完課時，我總會從課堂老師用心、清楚以及耐心的教學中 得到新的專業知識和研究啟發，並且，我也常常在課堂中獲得許多學 習的樂趣。感謝老師專業且充滿熱情的教學，促使我在學習專業科目 上能夠順利地前進。在研究所生涯中，同儕也是推動我成長的一大助 力。感謝同儕不吝嗇的分享學習方式，使得我有機會精進自己。

最後，我想感謝在求學路上，一路支持著我的家人。當初決心報考 研究所時，家人就給予我全力的支持，順利考上台灣大學之後，家人 在生活上的幫忙，也使我能全心全意地做研究。感謝爸爸、媽媽以及 姊姊一路以來，充當我的避風港，當我的堅強後盾，希望未來學有所 成之時，能夠和家人一起分享豐收的喜悅。

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 Co­array(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

Abstract

In array signal processing, we can receive signals from multiple anten­

nas placed in different locations, filter signals separately, and combine the filtering signals. Such signal processing operation is spatial filtering. If we receive single­frequency signals as the input of the spatial filter, the output of spatial filter is equivalent to the output of a single­antenna 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 discrete­time signal processing. Most importantly, based on the mathematical structure, the beampattern of a weight function is equivalent to the array response of a difference co­array (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 con­

vert the beampattern design problem on original array into the array response design problem on DCA. The new mathematical formulation of the beam­

pattern design problem provides us a new approach to design and analyze a beamformer. Tremendous researches have been devoted to the weight func­

tion design problem based on specified output of a beamformer, Minimum Variance Distortionless Response (MVDR) beamformer is the most signifi­

cant 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 in­

ner product operation of two steering vectors to array gain of a MVDR beam­

former, 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 advan­

tage of the special properties on MHA. Numerical simulation shows that, compared to designing weight function on original array, we have more op­

portunities 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.

Keywords: noise reduction, MVDR beamforming, Minimum Hole Array, array pattern synthesis, window function

Contents

口試委員會審定書 iii

誌謝 v

摘要 vii

Abstract ix

1 Introduction 1

2 Review of beamforming 5

2.1 Array Model . . . 5

2.2 Beamforming . . . 7

2.2.1 Array Response and Beampattern . . . 8

2.3 Minimum Variance Distortionless Response Beamforming . . . 10

2.4 Difference Co­array location set . . . 13

2.5 Minimum Hole Array . . . 14

3 Analysis of Array Gain on MVDR Beamformer and the Importance of Array Configuration 19 3.1 Introduction . . . 19

3.2 Data Model . . . 20

3.2.1 Auto­Correlation Matrix of Interferer Signal and Noise . . . 21

3.2.2 Optimal Weight Vector and Optimal Value of MVDR Beamformer 23 3.2.3 Array Gain . . . 23

3.3 Main Result . . . 24

3.3.1 Array Gain of MVDR Beamformer . . . 25

3.3.2 Pratical Case of Array Gain of MVDR Beamformer . . . 29

3.4 Simulation . . . 39

3.5 Conclusion . . . 42

4 Weight Function Design on Difference Coarray using Minimum Hole Arrays 45 4.1 Introduction . . . 45

4.2 Data Model . . . 46

4.2.1 Recieved Signal Model and Beampattern of Array System . . . . 46

4.2.2 Definition of Difference Co­array Weight Function . . . 47

4.3 Main Result . . . 48

4.3.1 DCA Weight Function on MHA . . . 48

4.3.2 Analysis and Design of Magnitude Part of DCA Weight Function on MHA . . . 49

4.3.3 Analysis and Design of Phase Part of DCA Weight Function on MHA . . . 54

4.3.4 MHA Weight Function Obtained from DCA Weight Function . . 55

4.4 Simulation . . . 57

4.4.1 Kaiser Window Function with N_K = 23 . . . 57

4.4.2 Design of Original MHA Weight Function With N = 5 . . . . 58

4.4.3 Design of Magnitude of DCA weight Function With N = 5 . . . 60

4.4.4 Design of Phase of DCA Weight Function With N = 5 . . . . 67

4.4.5 Beampattern . . . 70

4.4.6 MHA Weight Function Obtained from Weight Function Designed on DCA . . . 74

4.4.7 Kaiser Window Function with N_K = 99 . . . 78

4.4.8 Comparison of Designation on MHA and DCA With N = 10 for real function design . . . 80

4.5 Conclusion . . . 84

5 Conclusion 85

6 Future Work 87

Bibliography 89

List of Figures

2.1 The operation of beamforming can be illustracted above. . . 8 2.2 The difference lag function with N = 5 on ULA is shown above. . . . 15 2.3 The difference lag function with N = 5 on MRA is shown above. . . . . 15 2.4 The difference lag function with N = 5 on arbitrary array is shown above. 16 2.5 The difference lag function with N = 5 on MHA is shown above. . . . . 17

3.1 The array gain A_opt(IN R) of case 1 is shown above. . . . 39 3.2 The array gain A_opt(IN R) of case 2, which B = 0.05 is ranged by 0 <

B ≤ _N¹, is shown above. . . 39 3.3 The array gain Aopt(IN R) of case 3, which B = 0.55 is ranged by _N¹ <

B < 1, is shown above. . . . 40 3.4 The array gain A_opt(IN R) of case 4 is shown above. . . . 40 3.5 The array gain of four cases are shown together. . . 41 3.6 The blue line and the orange line correspond to the lower bound and upper

bound, respectively. . . 41 3.7 The blue line and the orange line correspond to the lower bound and upper

bound, respectively. . . 42

4.1 The array weight function w^K_S(n) is produced from Kaiser window func­

tion w^K(n) for 0≤ n ≤ (NK − 1) with NK = 23 and β = 6. . . . 58 4.2 The DCA weight function v_D^K₂(n) is produced from the array weight func­

tion w^K_S(n) on ULA. . . . 58

4.3 The original weight function on MHA w_M^O(n) is designed from the shiftable Kaiser window w^K_S (n− k) by taking the function values from w_S^K(n− k) of n ∈ M. As we can see, w^O_M(n) is a Kaiser­like weight function with fewer sensors. . . 59 4.4 The corresponding DCA weight function from the designed original weight

function is v_D^O(n). Compared to w_M^O(n), v^O_D(n) is not a Kaiser­like weight function, w^O_M(n) is though. . . . 59 4.5 The magnitude of DCA weight function that depends on |w_M(9)|, i.e.,

|v_Ddep(9)(n)|, is designed from the Kaiser window on DCA domain, i.e., v_D^K₂(n), by taking the function values from v_D^K₂(n) of n∈ Ddep(9). As we can see,|vDdep(9)(n)| is a Kaiser­like weight function with fewer sensors on DCA. . . 61 4.6 The magnitude of DCA weight function which is independent of|wM(p)|,

i.e.,|v_Dindep(9)(n)|, is produced from |v_Ddep(9)(n)| with α1. . . 62 4.7 The weight function|v_D⁺⁺(n)| with α1 can be obtained by (4.25). As we

can see,|v_D⁺⁺(n)| is a Kaiser­like weight function. . . 63 4.8 We can obtain the DCA weight function|v_D(n)| with α1. Although the

function value|v_D(0)| is way off the function value v_D^K₂(0), it only influ­

ence the DC value in beampattern. . . 63 4.9 The magnitude of DCA weight function which is independent of|wM(p)|,

i.e.,|vDindep(9)(n)|, is produced from |vDdep(9)(n)| with α2. . . 64 4.10 The weight function|vD⁺⁺(n)| with α2 can be obtained by (4.25). As we

can see,|vD⁺⁺(n)| is a Kaiser­like weight function. . . 64 4.11 We can obtain the DCA weight function |vD(n)| with α2. Although the

function value|vD(0)| is way off the function value v_D^K₂(0), it only influ­

ence the DC value in beampattern. . . 65 4.12 The green dot and the blue dots are correspond to|v_Dindep(9)(4)| and |v_Dindep(9)(n)| for n∈ Dindep(9)\{4}, respectively. . . 66

4.13 The corresponding weight function|v_D⁺⁺(n)| of |vDindep(9)(n)| can be ob­

tained by (4.25). As we can see,|vD⁺⁺(n)| is a Kaiser­like weight function

with five function values equal to that of Kaiser­window. . . 66

4.14 We can obtain the DCA weight function|v_D(n)| of |v_Dindep(9)(n)|, and we can see that, |v_D(n)| is a Kaiser­like weight function with ten function values equal to that of Kaiser­window. . . 67

4.15 The red dots represent∠v_Ddep(4)(n) for n∈ Ddep(4). . . 68

4.16 The blue dots represent∠vDindep(4)(n) for n ∈ Dindep(4). . . 69

4.17 We can obtain∠vD⁺⁺(n) from (4.45). . . . 69

4.18 We can obtain ∠vD(n) from ∠vD⁺⁺(n). Besides, the magenta dot repre­ sents∠v_D(0) = 0. . . 70

4.19 The normalized beampattern is shown, i.e., _max(B^BSK(θK^s) S (θs)), and B_S^K(θ_s) rep­ resents the correspond beampattern of w^K_S(n). Through the simulation, we can obtain the main beamwidth is 0.139487(rad) = 7.992^◦, and the corresponding ripple ratio and sidelobe roll­off factor are−43.9582(dB) and 15.0875(dB) which are shown above, respectively. . . . 71

4.20 The normalized beampattern is shown, i.e., _max(B^BOM(θO^s) M(θs)), and B_M^O(θ_s) rep­ resents the correspond beampattern of w^O_M(n). Through the simulation, we can obtain main beamwidth is 0.200434(rad) = 11.484^◦, and the cor­ responding ripple ratio and sidelobe roll­off factor are−1.39556(dB) and 9.01585(dB) which are shown above, respectively. . . . 72

4.21 The normalized beampattern is shown, i.e., ^{Bα1,RM (θs)} max(B_M^α1,R(θs)), and B_M^α1,R(θ_s) represents the correspond beampattern of w_M^α1,R(n). Through the simula­ tion, we can obtain main beamwidth is 0.173416(rad) = 9.936^◦, and the corresponding ripple ratio and sidelobe roll­off factor are−1.39556(dB) and 9.01585(dB) which are shown above, respectively. . . . 72

4.22 The normalized beampattern is shown, i.e., ^{Bα2,RM (θs)}

max(B_M^α2,R(θs)), and B_M^α2,R(θ_s) represents the correspond beampattern of w_M^α2,R(n). Through the simula­

tion, we can obtain main beamwidth is 0.116239(rad) = 6.66^◦, and the corresponding ripple ratio and sidelobe roll­off factor are−1.39556(dB) and 9.01585(dB) which are shown above, respectively. . . . 73 4.23 The normalized beampattern is shown, i.e., _max(B^BMR(θR^s)

M(θs)), and B_M^R(θ_s) rep­

resents the correspond beampattern of v_D(n) with real weight function de­

sign. Through the simulation, we can obtain main beamwidth is 0.115611(rad) = 6.624^◦, and the corresponding ripple ratio and sidelobe roll­off factor are

−3.37649(dB) and 3.24866(dB) which are shown above, respectively. . . 73 4.24 The normalized beampattern is shown, i.e., _max(B^BCM(θC^s)

M(θs)), and B_M^C(θs) rep­

resents the correspond beampattern of v_D(n) with complex weight func­

tion design. Through the simulation, we can obtain main beamwidth is 0.114354(rad) = 6.552^◦, and the corresponding ripple ratio and sidelobe roll­off factor are shown above, respectively. . . 74 4.25 The original weight function w_M^α1,R(n) is shown above. . . . 75 4.26 The original weight function w_M^α2,R(n) is shown above. . . . 76 4.27 The original weight function from approach 2, i.e., w^R_M(n) and|w^C_M(n)|,

is shown above. . . 77 4.28 The angle of the original weight function∠w_M^C(n) is shown above. . . . . 78 4.29 The array weight function w^K_S(n) is produced from Kaiser window func­

tion w^K(n) for 0≤ n ≤ (NK − 1) with NK = 99 and β = 6. . . . 79 4.30 The DCA weight function v_D^K₂(n) is produced from the array weight func­

tion w^K_S(n) on ULA. . . . 79 4.31 We can obtain the DCA weight function |v_D(n)|, and we can see that,

|v_D(n)| is a Kaiser­like weight function with many values equal to that of Kaiser­window. . . 81 4.32 The corresponding w_M(n) from the designed DCA weight function|v_D(n)|

is shown above. . . 81

4.33 The original weight function on MHA w_M^O(n) is designed from the shiftable Kaiser window w^K_S (n− k) by taking the function values from w_S^K(n− k) of n ∈ M. As we can see, w^O_M(n) is a Kaiser­like weight function with fewer sensors. . . 82 4.34 The corresponding DCA weight function from the designed original weight

function is v_D^O(n). Compared to w_M^O(n), v^O_D(n) is not a Kaiser­like weight function, w^O_M(n) is though. . . . 82

List of Tables

2.1 Minimum Hole Array of N from 4 to 10. . . . 18 3.1 Properties of array gain in four cases. . . 36 3.2 The important equations for the analysis of the array gain of MVDR beam­

former. . . 38 4.1 Comparison of weight function design on original array domain and DCA

domain. . . 83

Chapter 1 Introduction

An array system is a system composed of multiple antennas. Compared with a single­

antenna system, we can place multiple antennas in different locations to collect infor­

mation about signals at different locations and use signal processing to extract statistical information related to the signals or filter the signals. The signal processing of such an array system is beamforming. Compared with temporal filtering, which can only perform frequency filtering on signals, the advantage of beamforming is that it can spatially filter signals. The use of Beamforming has a significant impact on research in many fields, for example, hearing aids [1], radar [2], navigation [3], wireless communications [4].

According to different usage environments, beamforming can be classified into non­

adaptive beamforming and adaptive beamforming. Non­adaptive beamforming is usually data­independent, and the design of beamformer is not related to the statistical information of the received signal. The design of non­adaptive beamformer is a weight function design problem based on a specified beampattern, and is also referred to as pattern synthesis problem. In [5, section 3.1], the similarity between the array weight function and the window function is discussed, and the array weight function design by window function is presented. In addition to the utilization of window function on weight function design, [6]

designs an interactive second­order cone programming (SOCP), which provides a sub­

optimal solution to the non­convex beam graph synthesis problem on conformal arrays.

Adaptive beamforming is usually data­dependent. Therefore, the design of the adaptive beamformer is related to the statistical information of the received signal. For example, in the poor snapshot number scenarios which only have fewer data for estimation of the correlation matrix, [7] proposed a solution to adjust the zero position of suboptimal MVDR beamformer. [8] presents an approach for array response adjustment problem by modified

the optimal weight function of MVDR beamformer.

In the case of using a linear array system, it can be divided into two kinds of array systems base on the array location set: Uniform Linear Array and Sparse Array. Sparse array has many types, for example, minimum redundancy array(MRA), minimum hole array(MHA or Golomb Array), co­prime array, etc. In recent years, literature states that non­adaptive beamforming on sparse array usually has better beampattern performance than ULA [9], for instance, narrower main beamwidth, smaller sidelobe level, and deeper null, etc. For adaptive beamforming [7, 8], we usually expect more statistical informa­

tion of received signals to improve the design of beamformer. In [5, section 3.9.2], we can obtain more statistics information of correlation matrix of received signals by taking the advantage of difference co­array(DCA) location set on minimum redundancy array.

DCA location set composed of elements that are differences of any two different sensor locations. By using a sparse array, we can obtain more statistical information about the received signal while reducing the redundancy of the same statistical information. Al­

though sparse array has many advantages, literature about MVDR beamforming [7, 8]

barely discusses the performance of MVDR beamformer using sparse array.

In this thesis, we consider the analysis of array gain of MVDR beamformer, and weight function design base on Difference Co­array (DCA) set D on MHA. We found that, for different values of|a^H(θ_s)a(θ_i)|², the performance of array gain of MVDR beamformer has four different cases. Besides, different array location set leads to different value of

|a^H(θs)a(θi)|². With the understanding of the importance of|a^H(θs)a(θi)|²to array gain, the discussion about different values of|a^H(θ_s)a(θ_i)|² produce from different array lo­

cation settings is also included. Regarding the pattern synthesis problem, we propose an algorithm for designing DCA weight function by taking advantage of the special structure of DCA on MHA. Numerical simulations show that compared to the weight function de­

sign on original array, the designation of weight function on DCA has more chances to get better beampattern performance.

In the following context, Chap. 2 briefly reviews the basic signal model of beamform­

ing and introduction to DCA and MHA. Chap. 3 discusses the importance of|a^H(θ_s)a(θ_i)|²

in MVDR beamforming and the importance of array configuration for the value of|a^H(θs)a(θi)|². Chap. 4 analysis the structure of DCA weight function on MHA, and presents the pro­

posed algorithm for weight function design on DCA. Chap. 5 makes a summary of this thesis.

In this thesis, we use boldface on lower case letters to represent a vector and use bold­

face on uppercase letters to represent a matrix. We use blackboard bold on uppercase letters to represent a set. The superscript H and T are correspond to the Hermitian oper­

ation and the transpose operation of a vector or a matrix, respectively. The superscript * represents the conjugate operation.

Chapter 2 Review of beamforming

2.1 Array Model

We use a linear array system to receive multiple single­frequency signals from different directions. These signals usually contain desired signals, interference signals, or both.

For a linear array system with N sensors, we choose one of them on the boundary as a reference point in which location d0 = 0 on the digital axis. We use d as a unit distance, and denote the location of all sensors on the linear array by d0d, d₁d, d₂d,· · · , dN−1d. We can denote the array location set with unit distance d by

S = {d0, d₁, . . . , d_N−1} (2.1)

={0, d1, . . . , d_N−1} . (2.2)

The received single­frequency signals can be modeled as

r(t) =



r_d0(t) r_d1(t) · · · rdN−1(t)

T

(2.3)

= XM

k=1

x_k(t; θ_k) + n(t). (2.4)

with

x_k(t; θ_k) =



x_k(t− τd0(θ_k)) · · · xk(t− τdN−1(θ_k))

T

. (2.5)

xk(t; θk) is the single­frequency signal vector arrived at array system from θk, and M is the number of arrival signals. The delay time of k­th signal at sensor location l is

τ_l(θ_k) = dcos(θ_k)l

c , (2.6)

and c = 3× 10⁸ with unit (m/s) is the propagation speed of an electromagnetic wave.

The local noise vector in array system is

n(t) =



nd0(t) · · · ndN−1(t)

T

. (2.7)

Because the received signals are single frequency, we have

x_k(t− τl(θ_k)) = x_k(t)· e^{−jwsτl(θk)} (2.8)

for k = 1,· · · , M and l ∈ S. wsis the manipulating frequency of received signals, and

w_sτ_l(θ_k) = 2πf_sdcos(θk)l

c (2.9)

= 2πdcos(θ_k)l

c fs

(2.10)

= 2π d

λ_scos(θk)l. (2.11)

x_k(t, θ_k) can be rewritten as

xk(t; θk) =



x_k(t)e^{−j2πλsdcos(θk)d0} · · · xk(t)e^{−j2πλsdcos(θk)dN−1}

T

(2.12)

= x_k(t)a(θ_k) (2.13)

with steering vector(or array manifold vector) of θ_kas

a(θ_k) =



e^{−j2πλsdcos(θk)d0} · · · e^{−j2πλsdcos(θk)dN−1}

T

. (2.14)

The received signals vector can finally be modeled as

r(t) = XM

k=1

x_k(t)a(θ_k) + n(t). (2.15)

2.2 Beamforming

The received signals of each sensor are filtered by the weight function tap and then com­

bined together. This filtering operation is called beamforming and the output of the beam­

former can be modeled as

y(t) = w^Hr(t) (2.16)

= XM k=1

x_k(t)



w^Ha(θ_k)

 +



w^Hn(t)



(2.17)

with weight function vector as

w^H =



w_d0 w_d1 · · · wd_N−1



, (2.18)

w_lis the weight function tap at sensor location l. We can convert the vector representation of y(t) to

y(t) = w^Hr(t) (2.19)

=X

l∈S

w(l)r_l(t) (2.20)

=X

l∈S

w(l)



X^M

k=1

x_k(t)e^{−j2πλsdcos(θk)l}+ n_l(t)



 (2.21)

= XM k=1

x_k(t)



X

l∈S

w(l)e^{−j2πλsdcos(θk)l}



 +



X

l∈S

w(l)n_l(t)



 (2.22)

with w(l) = w_l for l∈ S.

For example, if M = 1, the received signal vector can be modeled as

r(t) = x₁(t; θ₁) + n(t), (2.23)

and the beamforming operation is shown in Fig. 2.1.

Figure 2.1: The operation of beamforming can be illustracted above.

2.2.1 Array Response and Beampattern

In the beamforming operation, we define

A(θ) = w^Ha(θ) (2.24)

=X

l∈S

w(l)e^{−j2πλsdcos(θ)l} (2.25)

=X

l∈S

R(l) (2.26)

as array response for signal come from θ. The sensor response of weight tap wlat sensor location l is

R(l) = w(l)e^{−j2πλsdcos(θk)l}. (2.27)

As we can see, the mathematics operation of A(θ) is similar to the discrete­time fourier transform(DTFT) operation in discrete­time signal processing. We can rewrite array re­

sponse as the DTFT operation below:

A(θ) = DT F T 

w(l) 

Ω=2π_λs^dcos(θ)l (2.28)

=X

l∈S

w(l)e^−jΩl

Ω=2π_λs^dcos(θ)l

(2.29)

= W (Ω)

Ω=2π_λs^dcos(θ)l

. (2.30)

DT F T w(l)

is the DTFT operation of discrete­time signal w(l), and W (Ω) is the fre­

quency response of w(l). By the representation of array response, y(t) can be rewritten as

= XM k=1

x_k(t)A(θ_k) +



w^Hn(t)



. (2.31)

For an array response, beampattern can be modeled as

B(θ) =|A(θ)|² (2.32)

=



X

l∈S

R(l)







X

l∈S

R(l)





∗

(2.33)

=



X

l∈S

w(l)· e^{−j2πλsdcos(θ)l}







X

m∈S

w(m)· e^{−j2πλsdcos(θ)m}





∗

(2.34)

=X

l∈S

X

m∈S

w(l)w^∗(m)· e^{−j2πλsdcos(θ)(l−m)} (2.35)

=|W (Ω)|²

Ω=2π_λs^dcos(θ)l

, (2.36)

and beampattern can also refer to the square of magnitude response of W (Ω).

Kaiser Window

Because of the relation between beampattern and DTFT operation, we can use the window function mentioned in discrete­time signal processing as the array weight function design.

In [5, section 3.1], Kaiser window function can adjust the peak height of sidelobe and main beamwidth by parameter β, and these two factors are important for beamformer to mitigate the impact of interference signals. The Kaiser window can be modeled as

w^K(l) =













I0



β vu ut1−"

2(l−(NK −1) 2 ) NK

#2



I0(β) , 0≤ |l| ≤ (NK− 1)

0, otherwise

(2.37)

with a modifiable parameter β and the weight tap number of Kaiser window N_K. The Kaiser window can be referred to the array weight function on ULA with array location set

S =



1, 2, · · · , (NK− 1)



(2.38)

and array weight function w(l) = w^K(l). With the knowledge of mathematical correspon­

dence of Kaiser window and array weight function, we can obtain the corresponding array response A(θ) = W^K(Ω)

Ω=2π_λs^dcos(θ)l

and beampattern B(θ) = |W^K(Ω)|²

Ω=2π_λs^dcos(θ)l

.

2.3 Minimum Variance Distortionless Response Beamform­

ing

In this section, we consider x₁(t) as the desired signal and the remaining x_k(t) for k = 2,· · · , M are interference signals. Moreover, xk(t) for k = 1,· · · , M are all random

signals. We can separate the desired signal and the interference signals as

r(t) = x1(t)a(θ₁) +



X^M

k=2

x_k(t)a(θ_k) + n(t)



 . (2.39)

The output of the beamformer is

y(t) = x1(t)



w^Ha(θ1)

 + w^H



X^M

k=2

xk(t)a(θk) + n(t)



 . (2.40)

The goal of MVDR beamformer is, forming unit gain on DOA of the desired signal while minimizing the output power of interference signals and local noise. We can model the output of MVDR beamformer as

y(t) = x₁(t)



w^H(t)a(θ₁)



+ w^H(t)



X^M

k=2

x_k(t)a(θ_k) + n(t)



 (2.41)

= x₁(t) + y_in(t) (2.42)

with minimum average power of y_in(t). y_in(t) is the beamformer output of interference signals and local noise. In addition, we add time index t to the weight vector for adaptive beamforming.

Based on the beamformer design problem mentioned above, we can model the design problem as

minimize

w(t) Eny_in(t)²o

(2.43)

subject to w^H(t)a(θ₁) = 1. (2.44)

The average power of y_in(t) is

Eny_in(t)²o

(2.45)

= w^H(t)R_in(t; θ₂,· · · , θM)w(t) (2.46)

with the auto­correlation matrix

R_in(t; θ₂,· · · , θM)

=E









X^M

k=2

x_k(t)a(θ_k) + n(t)







X^M

k=2

x_k(t)a(θ_k) + n(t)





H



, (2.47)

and we can notice that R_in(t; θ₂,· · · , θM) is a Hermitian matrix, i.e., R_in(t; θ₂,· · · , θM) = R^H_in(t; θ2,· · · , θM). We rewrite the optimization problem as

minimize

w(t)

w^H(t)R_in(t; θ₂,· · · , θM)w(t) (2.48)

subject to w^H(t)a(θ₁) = 1 (2.49)

and then obtain the corresponding optimal weight vector of MVDR beamformer [5, section 6.5.1] as

w^H_opt(t; θ₁,· · · , θM) = Λ(t; θ₁,· · · , θM)a^H(θ₁)R⁻¹_in (t; θ₂,· · · , θM) (2.50)

with constant Λ(t; θ₁,· · · , θM) = a^H(θ₁)R⁻¹_in (t; θ₂,· · · , θM)a(θ₁)
₋₁

. By the Hermitian property of Rin(t; θ2,· · · , θM), we can know that Λ(t; θ1,· · · , θM) is real value, i.e.,

Λ(t; θ₁,· · · , θM) = Λ^H(t; θ₁,· · · , θM). (2.51)

More concepts or properties of MVDR beamformer can be found in [5, section 6.2.1], [10, 11]. With the basic knowledge of MVDR beamformer, we will analyze the array gain of MVDR beamformer in detail in Chap. 3.

2.4 Difference Co­array location set

In this section, we define the difference co­array(DCA) location set as

D =



l l = i − j, i ∈ S, j ∈ S



. (2.52)

for any original array location setS. For example, the corresponding DCA array location sets of original array location set

S1 =



0, 1, 2, 3



(2.53)

and

S2 =



0, 1, 4, 6



(2.54)

is

D1 =



l l = i − j, i ∈ S¹, j ∈ S1



(2.55)

=



−3, −2, −1, 0, 1, 2, 3



(2.56)

and

D2 =



l l = i − j, i ∈ S2, j∈ S2



(2.57)

=



−6, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6



. (2.58)

2.5 Minimum Hole Array

In this section, we give a brief introduction to MHA. First of all, the array location set of an array system with sensor number N on ULA is

S =



0, 1, 2, · · · , (N − 1)



. (2.59)

Refer to the discussion of MRA in [5, section 3.9.2], we can know that, MHA is designed so that the number of sensor pairs that have the same spatial correlation lag except zero lag is equal to one, and the number of holes is made as small as possible. We can define the difference lag function as

c(γ) = X

l−m=γ

u(l)u^∗(m), (2.60)

which represents the number of difference lag γ. The weight function u(l) and u^∗(m) are uniformly weighting function, i.e., u(l) = u^∗(m) = 1 for l, m∈ S. For example, a N = 5 array system with array location set



0, 1, 2, 3, 4



(2.61)

has difference lag function

c(γ) =







5− |γ|, γ = 0, ±1, ±2, ±3, ±4

0, otherwise

(2.62)

shown in Fig. 2.2. As we can see, the number of difference lags γ = ±1, ±2, ±3 are all greater than one, which means the DCA of N = 5 ULA has redundancies on difference lags γ =±1, ±2, ±3.

0 1 2 3 4 5

-4 -3 -2 -1 0 1 2 3 4

Figure 2.2: The difference lag function with N = 5 on ULA is shown above.

For a N = 5 MRA, we have array location set as



0, 1, 4, 7, 4



, (2.63)

and the corresponding difference lag function is

c(γ) =













5, γ = 0

2, γ =±3

1, γ = ±1, ±2, ±4, ±5, ±6, ±7, ±8, ±9

. (2.64)

shown in Fig. 2.3. Compared to ULA, MRA only have redundancies on γ =±3.

0 1 2 3 4 5

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

Figure 2.3: The difference lag function with N = 5 on MRA is shown above.

For a N = 5 arbitrary array, we have array location set as



0, 1, 4, 9, 15



, (2.65)

and the corresponding difference lag function is

c(γ) =







5, γ = 0

1, γ =±1, ±3, ±4, ±5, ±6, ±8, ±9, ±11, ±14, ±15 (2.66)

shown in Fig. 2.4. Although the number of redundancy is zero, there are holes in DCA.

0 1 2 3 4 5

-15 -10 -5 0 5 10 15

Figure 2.4: The difference lag function with N = 5 on arbitrary array is shown above.

For a N = 5 MHA, we have array location set as



0, 1, 4, 9, 11



, (2.67)

and the corresponding difference lag function is

c(γ) =







5, γ = 0

1, γ = ±1, ±2, ±3, ±4, ±5, ±7, ±8, ±9, ±10, ±11 (2.68)

shown in Fig. 2.5. Compared to arbitrary array, MHA only has holes on γ =±6. In fact, MHA has the minimum number of holes while c(γ) = 1 for γ ∈ D\{0}. In Table 2.1, some MHA location set is shown. More detail about the definition of MHA can be found

in [5, section 3.9.2], [12, 13].

0 1 2 3 4 5

-15 -10 -5 0 5 10 15

Figure 2.5: The difference lag function with N = 5 on MHA is shown above.

Table 2.1: Minimum Hole Array of N from 4 to 10.

sensor number array location set

N = 4 •

n

0 1 4 6 o

N = 5

• n

0 1 4 9 11 o

• n

0 2 7 8 11 o

N = 6

• n

0 1 4 10 12 17 o

• n

0 1 4 10 15 17 o

• n

0 1 8 11 13 17 o

• n

0 1 8 12 14 17 o

N = 7

• n

0 1 4 10 18 23 25 o

• n

0 1 7 11 20 23 25 o

• n

0 1 11 16 19 23 25 o

• n

0 2 3 10 16 21 25 o

• n

0 2 7 13 21 22 25 o

N = 8 •

n

0 1 4 9 15 22 32 34 o

N = 9 •

n

0 1 5 12 25 27 35 41 44 o

N = 10 •

n

0 1 6 10 23 26 34 41 53 55 o

Chapter 3 Analysis of Array Gain on MVDR Beamformer and the Importance of Array Configuration

3.1 Introduction

In the field of array signal processing, MVDR Beamformer is often used for analysis because MVDR Beamformer is an adaptive beamformer, which is practical, and the opti­

mization problem corresponding to MVDR BF has an optimal solution. The related works on MVDR beamformer can be found in [14, 15, 16, 17]

In [18], mathematics derivation of array gain on beamformer is presented. In the en­

vironment of single interference signal and local noise, [18] states that the array response function is either increase or decrease as INR grows by the observation of numerical simu­

lation. We found that array response function has four different cases as INR grows, which is different from the discussion in [18]. In this thesis, we analyze the mathematics deriva­

tion of array gain and discuss the four cases of array gain function in detail. Moreover, we consider the array gain of MVDR beamformer on sparse array, which didn’t mention in [18]. We found that the performance of array gain on MVDR beamformer depends on the value of|a^H(θ_s)a(θ_i)|², besides, we can found that the value of|a^H(θ_s)a(θ_i)|²depends on array configuration(array location set). Numerical simulation shows that the performance of array gain on MHA is better than that on ULA in overall consideration.

With the knowledge of relation between |a^H(θ_s)a(θ_i)|² and the performance of array gain, using appropriate array configuration to obtain the desired|a^H(θ_s)a(θ_i)|² is impor­

tant. [19] found that MHA has optimum low peak sidelobe level, which means MHA has an opportunity to have the optimum performance of array gain on the location of low peak sidelobe level.

3.2 Data Model

As we discuss in section 2.3 about MVDR beamforming, we can refer (2.39) and obtain

r(t) = x(t)a(θ_s) + i(t)a(θ_i) + n(t)

. (3.1)

x(t) and i(t) correspond to desired signal and interfernce signal, respectively. The output of the MVDR beamformer is

y(t) = x(t)



w^H(t)a(θ_s)



+ w^H(t) i(t)a(θ_i) + n(t)

(3.2)

= x(t) + y_in(t) (3.3)

with minimum average power of y_in(t). The average power of y_in(t) is E

|yin(t)|²

= w^H(t)R_in(t; θ_i)w(t) with

R_in(t; θ_i) =En

i(t)a(θ_i) + n(t)

i(t)a(θ_i) + n(t)
Ho

(3.4)

In the following discussion, we will discuss the mathematics derivation of auto­correlation matrix in detail.

3.2.1 Auto­Correlation Matrix of Interferer Signal and Noise

General Definition

The general definition of auto­correlation matrix of interferer and noise is defined as

R_in(t₁, t₂, θ_i) (3.5)

=En

i(t₁)a(θ_i) + n(t₁)

i(t₂)a(θ_i) + n(t₂)
Ho

(3.6)

=E { i(t1)a(θ_i)i^∗(t₂)a^H(θ_i) + i(t₁)a(θ_i)n^H(t₂) (3.7) + n(t₁)i^∗(t₂)a^H(θ_i) + n(t₁)n^H(t₂)} (3.8)

=E

i(t1)i^∗(t2)

a(θi)a^H(θi) + a(θi)En

i(t1)n^H(t2) o

(3.9) +E

n(t1)i^∗(t₂)

a^H(θ_i) +E

n(t1)n(t₂)

. (3.10)

(3.11)

We can found that, R_in(t₁, t₂, θ_i) is composed of interferer signal, cross­correlation vec­

tor of interferer signal and noise, and auto­correlation matrix of noise. Auto­correlation function of interferer signal is

R_i(t₁, t₂) =E

i(t₁)i^∗(t₂)

. (3.12)

Cross­correlation vector of noise and interferer signal is

r_ni(t₁, t₂) (3.13)

=E

n(t₁)i^∗(t₂)

(3.14)

=E (

nd0(t1), nd1(t1), · · · , nd_N−1(t1)

T

i^∗(t₂) )

(3.15)

=



R_nd0i(t₁, t₂) R_nd1i(t₁, t₂) · · · Rn_dN−1i(t₁, t₂)

T

(3.16)