design and analysis of non-uniform linear array for …
TRANSCRIPT
DESIGN AND ANALYSIS OF NON-UNIFORM
LINEAR ARRAY FOR DOA ESTIMATION OF
NON-CIRCULAR SIGNALS
PAYAL
CENTRE FOR APPLIED RESEARCH IN ELECTRONICS
INDIAN INSTITUTE OF TECHNOLOGY – DELHI
INDIA
April 2021
© Indian Institute of Technology Delhi (IITD), New Delhi, 2021
DESIGN AND ANALYSIS OF NON-UNIFORM
LINEAR ARRAY FOR DOA ESTIMATION OF
NON-CIRCULAR SIGNALS
by
PAYAL
Centre for Applied Research in Electronics
Submitted
in fulfillment of the requirements of the degree of
Doctor of Philosophy
to the
INDIAN INSTITUTE OF TECHNOLOGY – DELHI
INDIA
April 2021
Dedicated to
My Family
Certificate
This is to certify that the thesis entitled “Design and Analysis of Non-Uniform
Linear Array for DoA Estimation of Non-circular Signals” being submitted
by Ms. Payal to the Centre of Applied Research in Electronics, Indian Institute
of Technology Delhi, for the award of the degree of Doctor of Philosophy is the
record of the bona-fide research work carried out by her under my supervision. In
my opinion, the thesis has reached the standards fulfilling the requirements of the
regulations relating to the degree.
The results contained in this thesis have not been submitted either in part or in
full to any other university or institute for the award of any degree or diploma.
(Prof. Monika Aggarwal)
Centre of Applied Research in Electronics
Indian Institute of Technology Delhi
Hauz Khas, New Delhi 110016
India
i
Acknowledgements
I was able to complete this doctoral dissertation with the support of many people.
I would like to express my deep sense of gratitude and tribute to all of them. First
of all, I would like to express my sincere gratitude to my dissertation advisor Prof.
Monika Aggarwal for her valuable guidance, support and consistent encouragement
throughout my doctoral studies. It was their encouragement that enabled me to come
up with my own original ideas which helped me in formulating meaningful research
problems. Her profound technical knowledge, passion towards research, attention to
detail, and diligence helped me in shaping up my vision for future research. I am deeply
indebted for their inspiration, motivation and guidance.
I would also like to thank my student research committee members Prof. S. D
Joshi, Prof. Arun Kumar, Prof. Rajendar Bahl for their useful interactions, invaluable
comments and suggestions. I am also thankful to all the professors at IIT Delhi from
whom I had the opportunity in understanding the fundamentals during the courses.
I would also like to thank the staff members of CARE office for taking care of all
the paperwork and other logistics.
My stay at IIT Delhi would not have been so memorable without the friends that
I have made along the journey making my life joyful and thankful for being a con-
stant source of encouragement. Thanks go out to all PhD scholars and my friends of
signal processing group. They have always been around to provide useful suggestions,
companionship, and created a peaceful research environment.
Most importantly, I would like to acknowledge my family for their unconditional
love, continuous support, and blessings. Their high moral values, principles, and em-
ii
phasis on quality education made a pivot role in shaping me up the person I am today.
Although, they were physically far away from me, their immense faith and wish is
gratefully acknowledged. Finally, sincere thanks go to my family, who have always
been there for me through thick and thin times of my journey.
Payal
iii
Abstract
The Direction of Arrival (DoA) estimation becomes a fascinating research area in sensor
array signal processing due to its enormous application in the field of communication,
radar, radio astronomy, sonar, navigation, and emergency assistance, and so forth. Tra-
ditional DoA estimation algorithms such as Multiple Signal Classification (MUSIC),
EStimation of Parameters by Rotational Invariant Techniques (ESPRIT), Maximum
likelihood (ML) etc. are able to resolve only N − 1 sources using N sensors. The limit
of identifiability of an array has been increased using the virtual array concept. The
virtual sensors in the array come about as a result of the spatial covariance/ spatial
cumulants between the sensor output of the array. The position of virtual sensor is the
function of physical sensor position. Therefore, many non-uniform linear arrays such
as MRA, Nested array, Coprime array, ML-NA, etc. have been designed to increase
the degrees of freedom. Most of these structures have been designed by utilizing the
circular statistics of data.
In today’s environment of 4G/5G technology, Internet of things, etc. non-circular sig-
nals like AM, MASK, BPSK, or UQPSK, M-PSK, etc. are so much prevalent. The
non-circularity property of signals provide additional valuable statistical information
though non-circular/pseudo moments. In many practical fields, the demand of detec-
tion and estimation of these signals within the limited physical resources such as power,
bandwidth, number of sensors, etc. has increased. Therefore, in this dissertation, we
present the solution of estimating the parameter such as DoA of non-circular signals
using minimum number of sensors in an array.
To begin with, we developed the underdetermined mathematical model to estimate the
DoAs of non-circular signal using the complete second-order statistics of data. The pro-
iv
posed model generated a virtual array, called Sum-Difference Co-Array (SDCA), with
O(4N2) elements which may be able to resolve the O(4N2) sources using N sensor
array. A novel non-uniform linear array, termed asNested Array with Displaced Sub-
array (NADiS), has been proposed to maximize the aperture of SDCA. The stability
of the designed array is being studied using weight function. The analytical expression
of weight function for the proposed design is derived. Also, the Cramer Rao Bound
(CRB), a universal tool for evaluating the performance of any parameter estimation
algorithm, is derived for DoA estimation using the underdetermined model. Numerical
results demonstrate that the proposed array provided better performance gain over
the existing arrays with different algorithms such as SS based MUSIC, Compressive
Sensing (CS), Direct Augumenatble Approach (DAA).
Second-order statistics based methods have some limitations, like sensitive to modelling
error, colored Gaussian noise, limited resolvability, etc. To overcome these limitations
and to increase the resolvability of an array, we proposed a mathematical framework
corresponding to 2qth (q > 1) order/ higher-order non-circular statistics. The proposed
framework generate a virtual array, known as Pseudo Cumulant along with Cumulant
Virtual Array (PCCVA), which provide the O((2N)2q) degrees of freedom using the N
sensor array. The proposed mathematical model increases the identifiability capability
of linear arrays in context of non-circular signals. Further, we provide a generalized
method to compute the weight function corresponding to any virtual array.
The aperture of 2qth order non-circular virtual array is the function of physical array ge-
ometry. Therefore, we designed a non-uniform linear array such a way the fourth-order
non-circular virtual array has maximum aperture. The higher-order direction find-
ing methods reduces the number of sensors of the array and, thus reduce the receiver
hardware, which drastically deduces the overall cost of DoA estimation. Therefore, we
proposed an extension of proposed non-uniform linear array to an arbitrary even order
q > 1 giving rise to the 2qth order Cumulant based Nested Array (QoCNA), which
leads to the longer consecutive virtual sensors segment in 2qth non-circular virtual
array. The virtual array corresponding to proposed design provides a large degrees
of freedom which increase both the resolution power and identifiability capabilities of
v
array. Further, the stability of virtual array corresponding to the proposed array is
studied with the weight function analysis.
In the real situation, most of time the assumption such as uncorrelatedness are not
fulfilled. We analyze the non-uniform linear array, ULA and the virtual array per-
formance in presence of multipath and mutual coupling. The performance of ULA,
non-uniform linear array and virtual array in multipath environment and mutual cou-
pling is analyzed via the computer simulations as well as the finite element analysis
tool, i.e., COMSOL Multiphysics.
vi
सार
आगमन की दिशा (द.ओ.ए) का आकलन ससर सरणी दसगनल परोसदसिग म एक आकरषक अनसिधान कषतर बन जाता ह
सिचार, रडार, रदडयो खगोल दिजञान, सोनार, नदिगशन और आपातकाल क कषतर म इसक दिशाल अनपरयोग क कारण
सहायता,और आग। पारिपररक द.ओ.ए आकलन एलगोररिम जस दक मलटीपल दसगनल कलादसदिकशन
(मयसी),अनमान परामीटसष ऑि रोटशनल इनिररक तकनीक (ईएसपीआरआईटी), अदधकतम सिभािना (एमएल)
आदि किल एन 1 को हल करन म सकषम ह N ससर का उपयोग करन िाल सरोत। िचषअल सरणी अिधारणा का
उपयोग करक दकसी सरणी की पहचान की सीमा बढाई गई ह। आभासी सरणी क ससर आउटपट क बीच सथादनक
सहसियोजक / सथादनक कमयलटस क पररणामसवरप सरणी म ससर आत ह। िचषअल ससर की ससथदत भौदतक ससर
की ससथदत का कायष ह। इसदलए, कई गर-समान रसखक सरदणयो ि जस एमरए क रप म, नसटड ऐर, कोपराइम एर,
एमल-नए आदि को सवतितरता की दडगरी बढान क दलए दडजाइन दकया गया ह। इनम स जयािातर सिरचनाओि को डटा
क पररपतर आकडो ि का उपयोग करक दडजाइन दकया गया ह। आज क पररिश म 4 जी / 5 जी परौदयोगोदगकी, चीजो ि
की इिटरनट, इतयादि जस गर-पररपतर दसगनल जस एएम, एमएएसक, बीपीएसक ,या यपीसक, एम-पीएसक, आदि
बहत परचदलत ह। सिकतो ि की गर-गोलाकार सिपदि अदतररकत मलयिान सािसिकीय जानकारी परिान करती ह हालािदक
गर-पररपतर / छदम कषण। कई वयािहाररक कषतरो ि म, इन सिकतो ि का पता लगान और अनमान लगान की मािग सीदमत
भौदतक सिसाधन जस दक दबजली, बडदिडथ, ससर की सििा, आदि म िसि हई ह। इसदलए, इस शोध परबिध म,हम
ससर की नयनतम सििा का उपयोग करक गर-पररपतर सिकतो ि क द.ओ.ए जस परामीटर का आकलन करन का
समाधान परसतत करत ह एक सरणी म।
आरिभ करन क दलए, हमन डटा क सिपणष िसर करम क आिकडो ि का उपयोग करक गर-पररपतर दसगनल क द.ओ.ए
का अनमान लगान क दलए कम-स-कम गदणतीय मॉडल दिकदसत दकया। परसतादित मॉडल न 𝑂(4𝑁2) ततो ि क
साथ सम-अितर सह-ऐर (स.द.सी.ए) नामक एक आभासी सरणी उतपनन की, जो N ससर सरणी का उपयोग करक
𝑂(4𝑁2) सरोतो ि को हल करन म सकषम हो सकता ह। एक उपनयास गर-समान रखीय सारणी, दजस दिसथादपत सबर
(न.ए.डी.इ.स) क साथ नसटड एर क रप म कहा जाता ह, को स.द.सी.ए क एपचषर को अदधकतम करन का परसताि
दिया गया ह। िजन फि कशन का उपयोग करक दडजाइन की गई सरणी की ससथरता का अधययन दकया जा रहा ह।
परसतादित दडजाइन क दलए िजन समारोह की दिशलरणातमक अदभवयसकत वयतपनन ह। इसक अलािा, ि करमर राि
बाउिड (सी.र.बी), जो दकसी भी परामीटर अनमान एलगोररिम क परिशषन का मलयािकन करन क दलए एक सािषभौदमक
उपकरण ह, को अिडरएटडषमाइिड मॉडल का उपयोग करक द.ओ.ए आकलन क दलए परापत दकया जाता ह।
सििातमक पररणाम परिदशषत करत ह दक परसतादित सरणी न दिदभनन एलगोररिम जस एसएस आधाररत सिगीत, कि परदसि
सदसिग (सीएस), डायरकट ऑगमनबल अपरोच (डीएए) क साथ मौजिा सरदणयो ि पर बहतर परिशषन लाभ परिान दकया।
िसर करम क आकडो ि क आधार पर कछ सीमाए होती ह, जस मॉडदलिग तरदट क दलए सिििनशील, रिगीन गॉदसयन
शोर, सीदमत परदतधवदन, आदि। इन सीमाओि को िर करन क दलए और एक सरणी की अननमान को बढान क दलए,
हमन 2𝑞𝑡ℎ(𝑞 > 1) क अनरप एक गदणतीय रपरखा परसतादित की उचच-करम क गर-पररपतर आकड ऑडषर कर।
परसतादित ढािचा एक आभासी सरणी उतपनन करता ह, दजस कयसलट िचषअल ऐर (पी.सी. सी. वी. ए) क साथ सयडो
कयमलट क रप म जाना जाता ह, जो 𝑂((2𝑁)2𝑞 सवतितरता परिान करत ह। न ससर सरणी का उपयोग करना।
परसतादित गदणतीय मॉडल गर-पररपतर सिकतो ि क सििभष म रसखक सरदणयो ि की पहचान कषमता बढाता ह। इसक
अलािा, हम दकसी भी िचषअल सरणी क अनरप भार फि कशन की गणना करन क दलए एक सामानयीकत दिदध परिान
करत ह।
2𝑞𝑡ℎ ऑडषर नॉन-सकष लर िचषअल ऐर का एपचषर भौदतक सरणी जयादमदत का कायष ह। इसदलए, हमन एक गर-समान
रसखक सरणी को इस तरह स दडजाइन दकया ह दक चौथ करम क गर-पररपतर आभासी सरणी म अदधकतम एपचषर
ह। उचच-करम दिशा खोजन क तरीक, सरणी क ससर की सििा को कम कर ित ह और इस परकार ररसीिर हाडषियर
को कम कर ित ह, जो डीओए अनमान की समगर लागत को कािी कम कर िता ह। इसदलए, हमन परसतादित गर-
समान रसखक सरणी क दिसतार को मनमाना करन क दलए 𝑞 > 1 का आिश दिया, जो 2𝑞𝑡ℎ ऑडषर को जनम ि
रहा ह। 2𝑞𝑡ℎ गर-पररपतर आभासी सरणी। परसतादित दडजाइन क अनरप आभासी सरणी सवतितरता की एक बडी
दडगरी परिान करता ह जो ररजॉलयशन पािर और सरणी की पहचान कषमताओि िोनो ि को बढाता ह।
इसक अलािा, आभासी सरणी की ससथरता परसतादित सरणी क अनसार िजन फि कशन दिशलरण क साथ अधययन
दकया जाता ह। िासतदिक ससथदत म, जयािातर समय जस दक असिबिदधत धारणा परी नही ि होती ह। हम गर-समान
रसखक सरणी, य.ल. ए और मलटीपलथ और आपसी यगमन की उपससथदत म आभासी सरणी परिशषन का दिशलरण
करत ह। य.ल. ए, गर-समान रसखक सरणी और बहपदित िातािरण और आपसी यगमन म आभासी सरणी का
परिशषन कि पयटर दसमलशन क साथ-साथ पररदमत तत दिशलरण उपकरण, यानी, .ओ.म.स.ओ.ल. मलटीदफदजकस।
Table of Contents
Certificate i
Acknowledgements ii
Abstract iv
List of Figures xi
List of Tables xix
Abbreviations xxi
1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Underdetermined Estimation . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Second-order statistics based virtual array . . . . . . . . . . . . 3
1.2.2 Higher-order statistics based virtual array . . . . . . . . . . . . 6
1.3 Non-Circular Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Array Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Key Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Second-Order Statistics Based Array : NADiS 16
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Signal Model and virtual array generation . . . . . . . . . . . . . . . . 17
vii
2.2.1 Signal Model of non-circular signals . . . . . . . . . . . . . . . . 17
2.3 Proposed array configuration . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 Weight Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 Cramer Rao Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6 Comparison of different nested array structure . . . . . . . . . . . . . . 40
2.6.1 In terms of aperture of VULA . . . . . . . . . . . . . . . . . . . 40
2.6.2 In terms of weight function . . . . . . . . . . . . . . . . . . . . . 41
2.6.3 In terms of CRB . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.7 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.7.1 Spatial Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.7.2 MSE Versus SNR . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.7.3 MSE Versus Sensors . . . . . . . . . . . . . . . . . . . . . . . . 48
2.7.4 MSE Versus Snapshots . . . . . . . . . . . . . . . . . . . . . . . 49
2.7.5 MSE Versus Number of Sources . . . . . . . . . . . . . . . . . . 49
2.7.6 MSE Versus Separation Between Sources . . . . . . . . . . . . . 50
2.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Generation of virtual array using the 2qth order non-circular statistics 54
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2 Proposed mathematical framework . . . . . . . . . . . . . . . . . . . . 55
3.2.1 Data Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2.2 2qth order non-circular statistics of received data . . . . . . . . 57
3.2.3 Generation of Virtual Array from 2qth order cumulants matrix . 60
3.2.4 Determination the virtual sensor position . . . . . . . . . . . . . 62
3.3 Weight Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.1 Virtual array corresponding to circular statistics . . . . . . . . 69
3.3.2 Virtual array corresponding to non-circular statistics . . . . . . 70
3.4 Comparison of CVA and PCCVA aperture . . . . . . . . . . . . . . . . 72
3.5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.5.1 Spatial Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
viii
3.5.2 Probability of Resolution vs SNR . . . . . . . . . . . . . . . . . 78
3.5.3 MSE vs SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.5.4 MSE versus separation between sources . . . . . . . . . . . . . 81
3.5.5 MSE versus Number of sources . . . . . . . . . . . . . . . . . . 82
3.5.6 Effect of weight function on estimation performance . . . . . . 83
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4 Higher-Order cumulant based NULA: QoCNA 88
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.2 Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.3 Virtual Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3.1 Fourth Order Statistics based Virtual Array . . . . . . . . . . . 92
4.3.2 Virtual Array corresponding 2qth order statistics . . . . . . . . 94
4.4 Proposed array structure . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.5 Proposed Array structure for 2qth order non-circular statistics . . . . . 103
4.6 Comparison of proposed array structures with existing array structure . 106
4.6.1 In terms of degrees of freedom . . . . . . . . . . . . . . . . . . . 106
4.6.2 In terms of virtual array stability . . . . . . . . . . . . . . . . . 108
4.7 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.7.1 Probability of Resolution vs SNR . . . . . . . . . . . . . . . . . 114
4.7.2 Performance Analysis in terms of Mean Square Error . . . . . . 115
4.7.3 Performance comparison of Proposed NULA, ULA and Proposed
virtual array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.7.4 Performance comparison corresponding HOS crucial parameters 126
4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5 Performance analysis of NULA in presence of multipath and mutual
coupling 129
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2 Signal model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
ix
5.3 DoA estimation Algorithm using Sparse Bayesian Learning based Rele-
vance Vector Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.4 DoA estimation in presence of coherent multipath . . . . . . . . . . . . 134
5.4.1 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.5 In presence of Mutual Coupling . . . . . . . . . . . . . . . . . . . . . . 139
5.5.1 Finite Element Study . . . . . . . . . . . . . . . . . . . . . . . . 141
5.5.2 Simulation setup in COMSOL MULTIPHYSICS . . . . . . . . . 141
5.5.3 Effect of mutual coupling on acoustic wavefront . . . . . . . . . 143
5.5.4 Spatial Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6 Conclusion 149
6.1 Future Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Appendix 151
Bibliography 159
Work Published Based on this Dissertation 173
Technical Biography of Author 176
x
List of Figures
1.1 Nested Array configuration and corresponding difference co-array . . . 5
1.2 Coprime Array (a) Physical sensor placement (b) Difference co-array . 6
1.3 (a) Scatter plot of radar data (b) Scatter plot of wind data (c) Co-
variance and complementary co-variance function plot of wind data (d)
Co-variance and complementary co-variance function plot of wind data
[ T. Adali, P. J. Schreier and L. L. Scharf, “Complex-Valued Signal Pro-
cessing: The Proper Way to Deal With Impropriety,” in IEEE Trans-
actions on Signal Processing, vol. 59, no. 11, pp. 5101-5125, Nov.
2011.] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 A basic model for array signal processing. . . . . . . . . . . . . . . . . 12
2.1 Proposed NLA Structure : NADiS . . . . . . . . . . . . . . . . . . . . . 24
2.2 An example of NADiS configuration where N = 8, N1 = 4, N2 = 4 and
L = 10 D = 7. (a) The NADiS configuration (b) The virtual array . . . 27
2.3 An example of NADiS configuration where N = 8, N1 = 5, N2 = 3 and
L = 13 D = 9. (a) The NADiS configuration (b) The virtual array . . . 30
2.4 Sum and Difference co-array of Array Ω (Note: The number in the curly
brackets . denotes the frequency of corresponding entry.) . . . . . . . 33
2.5 Cross Sum Co-array and Cross Difference Co-array between ULA-I and
ULA-II of NADiS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.6 Weight function of NADiS whose physical sensors located at 0, 1, 2, 3, 4, 13, 22, 31 36
2.7 Weight Function of proposed array structure for 6 physical sensors and
SDCC structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
xi
2.8 The performance of the proposed CRB with SNR for ADCA, CPASD,
SDCC and NADiS. The number of physical sensors array is 6. The num-
ber of snapshots is K = 600. For M = 17 sources uniformly distributed
in −60 to 60 with step of 7.3. . . . . . . . . . . . . . . . . . . . . . . 42
2.9 Spatial Spectra estimated using Compressive sensing (CS) for both con-
figuration with 8 sensors and M = 27 sources located uniformly between
−60 to 60 with the step of 4.5, 30 dB SNR and 600 snapshots. . . . 44
2.10 Spectrum of Direct Augmented Approach (DAA) at SNR = 30dB for 8
sensors NADiS and CPASD 600 snapshots and with 27 sources. . . . . 45
2.11 Mean Square Error versus SNR for 1000 Monte Carlo experiments with
N = 8 physical sensors array. The number of snapshots is K = 600. For
M = 27 sources uniformly distributed in −60 to 60 with step of 4.5. 46
2.12 Mean Square Error versus Signal-to-Noise Ratio. Graph compared for
the six sensors configuration with 600 snapshots and for 17 sources. . . 47
2.13 Mean Square Error versus SNR for 1000 Monte Carlo experiments with
N = 6 physical sensors with the location of 0, 1, 2, 3, 10, 17. The
number of snapshots is K = 400. For M = 5 sources located at
−60,−33,−6, 21, 48. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.14 Mean Square Error versus Number of sensors for 1000 Monte Carlo
experiments with M = 27 sources uniformly distributed in −60 to 60
with step of 4.5. The number of snapshots is K = 600 and SNR = 20dB. 49
2.15 Mean Square Error versus snapshot for 1000 Monte Carlo experiments
with N = 6 physical sensors with the location of 0, 1, 2, 3, 10, 17. For
M = 17 sources uniformly distributed in −60 to 60 with step of 7.3.
Here SNR = 30dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.16 Mean Square Error versus sources for 1000 Monte Carlo experiments
with N = 6 physical sensors with the location of 0, 1, 2, 3, 10, 17. The
number of snapshots is K = 800 and SNR = 30dB. . . . . . . . . . . . 51
xii
2.17 Mean Square Error versus sources for 1000 Monte Carlo experiments
with N = 8 physical sensors with the location of 0, 1, 2, 3, 4, 13, 22, 31.The number of snapshots is K = 800 and SNR = 30dB. . . . . . . . . 51
2.18 Mean Square Error versus Separation between Sources for 1000 Monte
Carlo experiments with N = 8 physical sensors with the location of
0, 1, 2, 3, 4, 13, 22, 31. The number of snapshots is K = 600 and SNR
= 30dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1 Physical sensors placement . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 4th order virtual array (i) with CVA approach (ii) with PCCVA approach 67
3.3 6th order virtual array (i) with CVA approach (ii) with PCCVA ap-
proach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.4 Weight Function of HO-NA with 6 sensor with PCCVA and CVA ap-
proaches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.5 Compared the weight function of 4th-order PCCVA and CVA of (a)
array S26 (b) array S35 (c) Nested Array (d) Co-Prime Array . . . . . . 74
3.6 MUSIC Spectrum comparison between CVA and PCCVA approaches for
Number of sensors 4 and Number of sources 8. The number of snapshots
is K = 20, 000 and SNR is 0 dB. . . . . . . . . . . . . . . . . . . . . . . 75
3.7 MUSIC Spectrum comparison between CVA and PCCVA for array S16
and 11 8-PSK sources. The number of snapshots is K = 20, 000 and
SNR is 0 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.8 Spatial Spectrum obtained using compressive sensing of PCCVA and
CVA. There are 6 sensors and M=39 sources located uniformly between
−60 to 60 with the step 3.1. The number of snapshots is K = 20, 000
and SNR is 0 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.9 Probability of resolution versus SNR for the array S16, S26 and S35 with
9 sources located at −60 to 60 with the step of 14 at 2000 snapshots. 78
xiii
3.10 Mean square error (MSE) vs. SNR for array S16 and M = 8 sources from
the direction −60,−43,−26,−9, 8, 25, 42, 59, using spatial smoothing
based MUSIC algorithm and compressive sensing. The number of snap-
shots is K = 5000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.11 Mean Square Error (MSE) vs. SNR for S16, S26 and S35 array config-
urations and M = 9 8-PSK sources signal from the direction between
−60 to 60 with the step of 15. The number of snapshots is K = 2000. 80
3.12 Mean Square Error (MSE) vs. SNR for S16, S26 and S35 array config-
urations and M = 27 sources from the direction between −60 to 60
with the step of 4.6. The number of snapshots is K = 2000. . . . . . . 81
3.13 Mean square error (MSE) vs. SNR for S16, S26 and S35 array configura-
tions and M = 11 sources from the direction between −60 to 60 with
the step of 11.3. The number of snapshots is K = 2000. . . . . . . . . 81
3.14 Mean square error (MSE) vs. Separation between the 10 sources for S16,
S26 and S35 array configurations at 20 dB SNR using spatial smoothing
based MUSIC algorithm. The number of snapshots is K = 2000. . . . . 82
3.15 Mean square error (MSE) vs. Number of sources for S16, S26 and S35 ar-
ray configurations at 20 dB SNR using spatial smoothing based MUSIC
algorithm. The number of snapshots is K = 3000. . . . . . . . . . . . . 83
3.16 Weight function of an array whose physical sensors located at 0 1 2 3 10 17 84
3.17 MUSIC Spectrum for Number of sensors 6 and Number of sources 15.
The number of snapshots is 2000 and SNR is 20 dB. (a) Full aperture
virtual array; (b) with −25 to 25 virtual array aperture. . . . . . . . . 85
3.18 MSE vs Aperture of virtual array with 6 sensor array. The number
of snapshots is 2000. (a) Overdetermined scenario with 5 sources; (b)
Underdetermined scenario with 15 sources. . . . . . . . . . . . . . . . . 86
4.1 Proposed Array configuration . . . . . . . . . . . . . . . . . . . . . . . 97
4.2 Proposed array structure for 6 physical sensors (a) Physical sensor array
(b) virtual array (c) VULA . . . . . . . . . . . . . . . . . . . . . . . . . 101
xiv
4.3 Optimized proposed array structure for 6 physical sensors (a) Physical
sensor array (b) virtual array (c) VULA . . . . . . . . . . . . . . . . . 103
4.4 Aperture of fourth-order VULA vs number of sensors . . . . . . . . . . 107
4.5 Weight function of fourth-order virtual array corresponding to 6 physical
sensor arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.6 Spatial Spectra for 6 sensors proposed array withM = 41 sources located
uniformly between −60 to 60 with the step of 3 at 20dB. The number
of snapshots is K = 2000. . . . . . . . . . . . . . . . . . . . . . . . . . 110
4.7 Normalized spatial spectrum of different array geometries for 6 sensors
array and M = 41 sources located uniformly between −60 to 60 with
the step of 3 at 20dB SNR using SS based MUSIC method. The number
of snapshots is 2000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.8 Spatial Spectra using SS based MUSIC algorithm of different 6 sensors
array at 20dB with 800 snapshots for 10 sources directions −56,−32,−27,−13, 0, 11, 23, 36, 45, 54, 5.(a) 2LNA (b) 3L-FONA (c) 4L-FONA (d) Proposed array (FoCNA). . 112
4.9 Spatial Spectra using compressive sensing methods of different 6 sensors
array structures at 20dB with 800 snapshots for 10 sources directions
−56,−32,−27,−13, 0, 11, 23, 36, 45, 54, 5. (a) 2LNA (b) 3L-FONA (c)
4L-FONA (d) Proposed array (FoCNA). . . . . . . . . . . . . . . . . . 113
4.10 Spatial Spectra for 6 sensors proposed array withM = 26 sources located
uniformly between −55 to 55 with the step of 3 at 20dB. The number
of snapshots is K = 20000. . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.11 Probability of resolution versus SNR for the different 6 sensor arrays with
25 sources located at −60 to 60 with the step of 5 at 2000 snapshots. 115
4.12 Mean Square Error (MSE) vs. SNR for N = 6 physical sensors array
and M = 31 sources uniformly distributed in −60 to 60 with step of
4 using different algorithm. The number of snapshots are 1500 and 2000.116
4.13 MSE vs. Number of Sources for N = 6 physical sensors FoCNA at 20dB
SNR using different algorithm. The number of snapshots is 2000. . . . 117
xv
4.14 MSE vs. SNR with N = 6 physical sensors for FoCNA and M = 25
sources uniformly distributed in −60 to 60 with step of 5 using SS
based MUSIC algorithm. The number of snapshots is K = 2000. . . . . 118
4.15 MSE vs. SNR with array S2 and M = 49 sources uniformly distributed
in −60 to 60 with step of 2.5 using SS based MUSIC algorithm. The
number of snapshots is K = 3000. . . . . . . . . . . . . . . . . . . . . . 118
4.16 MSE vs. SNR with N = 6 physical sensors for FoCNA and M = 17
sources uniformly distributed in −60 to 60 with step of 7 using SS
based MUSIC algorithm. The number of snapshots is K = 2000. . . . . 119
4.17 MSE vs. SNR with N = 6 physical sensors for FoCNA and BPSK AND
8-QAM signal with M = 9 DoAs using SS based MUSIC algorithm. The
number of snapshots is K = 800. . . . . . . . . . . . . . . . . . . . . . 120
4.18 MSE vs. Number of snapshot with N = 8 physical sensors arrays and
M = 49 sources uniformly distributed in −60 to 60 with step of 2.5
using SS based MUSIC algorithm at 20dB SNR. . . . . . . . . . . . . . 120
4.19 MSE vs. Number of Sources for N = 8 physical sensors arrays at 20
dB SNR using 2000 snapshots using SS based MUSIC algorithm. The
sources are uniformly distributed in −60 to 60. . . . . . . . . . . . . . 121
4.20 MSE vs. Number of Sources for N = 6 physical sensors arrays at
20dB SNR using 2000 snapshots using SS based MUSIC algorithm. The
sources are uniformly distributed in −60 to 60. . . . . . . . . . . . . 122
4.21 MSE vs Number of Sources for array S3 at 20 dB SNR. The number of
snapshots is 2000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.22 (b) MSE vs. Angular Separation between 7 sources which are uniformly
distributed in −60 to 60 for array S3 at 20 dB SNR. The number of
snapshots is 2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.23 MSE vs. Number of Sensors for M = 25 sources uniformly distributed
in −60 to 60 with step of 5 using SS based MUSIC algorithm. The
number of snapshots is 2000 and 20 dB SNR. . . . . . . . . . . . . . . 124
xvi
4.24 MSE vs. SNR withN = 6 physical sensors with the location of 0, 1, 2, 33, 46, 59.The number of snapshots is K = 2000. For M = 5 sources located at
−60,−33,−6, 21, 48 . . . . . . . . . . . . . . . . . . . . . . . . . 124
4.25 Probability of Resolution vs. SNR with N = 6 physical sensors with the
location of 0, 1, 2, 33, 46, 59. The number of snapshots is K = 2000.
For M = 5 sources located at −60,−33,−6, 21, 48 . . . . . . . . 125
4.26 MSE vs. Number of snapshot with N = 6 physical sensors fourth-order
and second-order NULA and M = 11 sources using SS based MUSIC
algorithm at 20dB SNR. . . . . . . . . . . . . . . . . . . . . . . . . . . 127
4.27 MSE vs Second Source power using SS based MUSIC algorithm of 6
sensors array with 800 snapshots for 2 sources directions 35, 40 at
different power level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.1 Spatial Spectra estimated using SBL-RVM and beamforming algorithm
for 6 sensors ULA and three coherent sources from the direction 38, 50
and 60, at 20 dB SNR and 2000 snapshots. . . . . . . . . . . . . . . . 135
5.2 Spatial Spectra estimated using SBL-RVM and beamforming algorithm
for 17 sensors ULA and three coherent sources from the direction 38,
50 and 60, at 20 dB SNR and 2000 snapshots. . . . . . . . . . . . . . 136
5.3 Spatial Spectra estimated using SBL-RVM and beamforming algorithm
for 6 sensors NADiS and corresponding virtual array. Consider three
coherent sources from the direction 38, 50 and 60, at 30 dB SNR and
600 snapshots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.4 Sound speed variation with depth . . . . . . . . . . . . . . . . . . . . . 138
5.5 Sound Pressure level and ray profile with distance . . . . . . . . . . . . 139
5.6 Impulse response of the given channel . . . . . . . . . . . . . . . . . . 139
5.7 DoA estimation using 6 sensors ULA for COMSOL data . . . . . . . . 140
5.8 DoA estimation using 17 sensors ULA for COMSOL data . . . . . . . 140
5.9 DoA estimation using 6 sensors NULA for COMSOL data . . . . . . . 141
5.10 Model Setup in COMSOL . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.11 Model mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
xvii
5.12 ULA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.13 ULA with mutual coupling . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.14 NULA with MC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.15 Different NULAs structures for 6 physical sensors. (a) ULA (b) Nested
Array (c) CoPrime Array (d) CADiS (e) NADiS . . . . . . . . . . . . . 145
5.16 Spatial Spectrum of 6 sensor ULA and NULAs with no mutual coupling
effect. The direction of sources are [128, 135, 145]. The number of
snapshots is K = 1500 (a) 6 sensor ULA (b) CADiS (c) Nested Array
(d) CoPrime Array (e) NADiS (f) 18 sensor ULA. . . . . . . . . . . . . 146
5.17 Spatial Power Spectrum of ULA with 6 sensor, NULA and ULA with
18 sensor with mutual coupling effect. The direction of sources are
[128135]. The number of snapshots is K = 1500. . . . . . . . . . . . . 148
xviii
List of Tables
2.1 Comparison of VULA aperture of various array configuration . . . . . . 40
3.1 Co-arrays corresponding to column of matrix B4,v(θm) . . . . . . . . . 66
3.2 Comparison of virtual array aperture with proposed method . . . . . . 72
4.1 Virtual Array Corresponding to ULA . . . . . . . . . . . . . . . . . . . 96
4.2 Comparison of the consecutive lags for different array structures based
on the 2q-th order virtual array . . . . . . . . . . . . . . . . . . . . . . 107
5.1 Simulation parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.2 COMSOL Simulation Parameter . . . . . . . . . . . . . . . . . . . . . . 141
xix
Abbreviations
ASK Amplitude Shift Keying
AWGN Additive White Gaussian Noise
BPSK Binary phase-shift keying
CRB Cramer Rao bound
CS Compressive Sensing
DoA Direction of Arrival
ESPRIT Estimation of Signal Parameters by Rotational Invariant Techniques
GMSK Gaussian MSK
MVDR Minimum Variance Distorsionless Response
ME Maximum Entropy
ML Maximum Likelihood
MuSiC Multiple Signal Classification
MSK Minimum Shift Keying
MSE Mean Square Error
NULA Non-Uniform Linear Array
NC Non-Circular
PSK Phase-Shift Keying
SS Spatial Smoothing
SNR Signal-to-noise ratio
SBL-RVM Sparse Bayesian Learning based Relevance Vector Machine
ULA Uniform Linear Array
UQPSK Unbalanced Quaternary Phase-Shift Keying
VULA Virtual Uniform Linear Array
WSS Wide Sense Stationary
xxi