Sparse Arrays for Radar, Sonar, and Communications E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Specialized resource providing detailed coverage of recent advances in theory and applications of sparse arrays Sparse Arrays for Radar, Sonar, and Communications discusses various design approaches of sparse arrays, including those seeking to increase the corresponding one-dimensional and two-dimensional virtual array apertures, as well as others that configure the arrays based on solutions of constrained minimization problems. The latter includes statistical bounds and signal-to-interference and noise ratio; in this respect, the book utilizes the recent strides made in convex optimizations and machine learning for sparse array configurability in both fixed and dynamic environments. Similar ideas are presented for sparse array-waveform design. The book also discusses the role of sparse arrays in improving target detection and resolution in radar, improving channel capacity in massive MIMO, and improving underwater target localization in sonar. It covers different sparse array topologies, and provides various approaches that deliver the optimum and semi-optimum sparse array transceivers. . Edited by a world-leading expert in Radar and Signal Processing and contributed to by world-class researchers in their respective fields, Sparse Arrays for Radar, Sonar, and Communications covers topics including: * Utilizing sparse arrays in emerging technologies and showing their offerings in various sensing and communications applications * Applying sparse arrays to different environments and obtain superior performances over conventional uniform arrays * Solving the localization, beamforming, and direction-finding problems using non-uniform array structures for narrowband and wideband signals * Designing sparse array structures for both stationary and moving platforms that produce physical and synthesized array apertures. * Using deep neural networks that learn the underlying complex nonlinear model and output the sparse array configuration using representations of the input data spatio-temporal observations * Solving for optimum sparse array configurations and beamforming coefficients in sensing using iterative convex optimization methods Providing complete coverage of the recent considerable progress in sparse array design and configurations, Sparse Arrays for Radar, Sonar, and Communications is an essential resource on the subject for graduate students and engineers pursuing research and applications in the broad areas of active/passive sensing and communications.
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Dedication
About the Editor
List of Contributors
Preface
1 Sparse Arrays: Fundamentals
1.1 Introduction
1.2 Basics of Array Processing
1.3 What Are Sparse Arrays?
1.4 How Sparse Arrays Identify
O
(
N
2
) Sources
1.5 Identifying DOAs from Correlations
1.6 Coarray MUSIC
1.7 Examples of Sparse Arrays
1.8 Examples of Optimal Sparse Arrays
1.9 Coprime DFT Beamformers
1.10 Directions for Further Reading
Acknowledgment
References
Notes
2 Sparse Array Interpolation for Direction-of-Arrival Estimation
2.1 Introduction
2.2 Virtual Array Interpolation for Gridless DOA Estimation
2.3 Physical Array Interpolation for Off-grid DOA Estimation
2.4 Prospective Research Directions
Acknowledgments
References
Note
3 Wideband and Multi-frequency Sparse Array Processing
3.1 Introduction
3.2 Wideband DOA Estimation
3.3 Multi-frequency DOA Estimation
3.4 Wideband SBL for Beamforming
3.5 Suggested Further Reading
3.6 Conclusion
References
4 Sparse Arrays in Sample Starved Regimes: Algorithms and Performance Analysis
4.1 Introduction
4.2 Background on Correlation-Aware Sparse Support Recovery with Sparse Arrays
4.3 Universal Recovery Guarantees for OOSA: The Role of Non-negativity
4.4 Support Recovery with High Probability: How Many Snapshots Suffice?
4.5 Single-Snapshot Virtual Array Interpolation: Deterministic Guarantees
4.6 Concluding Remarks and Future Directions
References
Notes
5 Sparse Sensor Arrays for Two-dimensional Direction-of-arrival Estimation
5.1 Introduction
5.2 Two-Dimensional DOA Estimation Essentials
5.3 Sparse Array Geometries for 2D-DOA Estimation
5.4 Comparative Evaluation
5.5 Summary
References
Note
6 Sparse Array Design for Direction Finding Using Deep Learning
6.1 Introduction
6.2 General Design Procedures
6.3 Cognitive Sparse Array Design for DoA Estimation
6.4 TL for Sparse Arrays
6.5 Large Planar Sparse Array Design with SA-Assisted DL
6.6 DL-Based Sparse Array Design for Hybrid Beamforming
6.7 Deep Sparse Arrays for ISAC
6.8 Summary
Acknowledgments
References
Notes
7 Sparse Array Design for Optimum Beamforming Using Deep Learning
7.1 Motivation
7.2 Contributions
7.3 Problem Formulation
7.4 Efficient Generation of Training Data for Optimum Beamforming
7.5 Machine-Learning Methods for Sparse Array Design
7.6 Simulation Results
7.7 Future Directions
7.8 Conclusions
References
8 Sensor Placement for Distributed Sensing
8.1 Data Model
8.2 Distributed Estimation
8.3 Distributed Detection
8.4 Conclusions
References
Note
9 Sparse Sensor Arrays for Active Sensing: Models, Configurations, and Applications
9.1 Introduction
9.2 Active Sensing Signal Model
9.3 Sparse Array Configurations
9.4 Beamforming
9.5 Applications
9.6 Conclusions
Acknowledgment
References
Notes
10 Sparse MIMO Array Transceiver Design in Dynamic Environment
10.1 Review of MIMO Arrays and Sparse Arrays
10.2 Sparse MIMO Transceiver Design for MaxSINR with Known Environmental Information
10.3 Cognitive-Driven Optimization of Sparse Transceiver for Adaptive Beamforming
10.4 Sparse MIMO Transceiver Design for Multi-source DOA Estimation
10.5 Conclusion
References
11 Generalized Structured Sparse Arrays for Fixed and Moving Platforms
11.1 Introduction
11.2 Generalized Coprime Array Configurations
11.3 Synthetic Structured Arrays Exploiting Array Motions
11.4 Structured Arrays Design for Moving Platforms
11.5 DOA Estimation Exploiting Array Motions
11.6 Other Structured Arrays for Fixed and Moving Platforms
11.7 Conclusion
References
Notes
12 Optimization and Learning-Based Methods for Radar Imaging with Sparse and Limited Apertures
12.1 Introduction
12.2 SAR Observation Model
12.3 Model-Based Imaging and the Role of Sparsity
12.4 Learning-Based SAR Imaging
12.5 Conclusion
References
Notes
13 Sparse Arrays for Sonar
13.1 Introduction
13.2 Active Sonar Processing
13.3 Passive Sonar Processing
13.4 Experimental Sonar Examples
13.5 Further Reading on Sparse Sonar
Acknowledgements
References
14 Unconventional Array Architectures for Next Generation Wireless Communications
14.1 Introduction
14.2 Sparseness-Promoting Techniques for the Design of Unconventional Architectures
14.3 Co-design of Unconventional Architectures and Radiating Elements
14.4 Capacity-Driven Synthesis of Next Generation Base Station Phased Arrays
14.5 Final Remarks and Envisaged Trends
Acknowledgments
References
Notes
15 MIMO Communication with Sparse Arrays
15.1 Introduction
15.2 Fully Digital Architectures with Sparse Arrays
15.3 Hybrid Analog–Digital Architectures with Sparse Arrays
15.4 Conclusion and Future Directions
References
Notes
Index
End User License Agreement
List of Tables
Chapter 1
Table 1.1 Examples of minimum redundancy arrays.
Chapter 2
Table 2.1 List of notations.
Chapter 3
Table 3.1 SBL pseudocode: input consists of data , dictionary , and noise ...
Table 3.2 The four sub-arrays used for processing NC09 experiment data.
Chapter 5
Table 5.1 Weight functions of parallel coprime arrays.
Table 5.2 Weight functions of parallel nested and other arrays.
Table 5.3 Weight functions of different sparse arrays.
Table 5.4 Comparison between physical sensors and achievable degrees of free...
Table 5.5 Comparison between array geometries in terms of computational comp...
Chapter 6
Table 6.1 Number of classes and the reduced number of classes for a UCA ...
Table 6.2 The accuracy percentages for training and validation datasets in 1...
Table 6.3 Training validation accuracy (%) for different TL scenarios.
Table 6.4 2-D DOA estimation simulation parameters.
Chapter 7
Table 7.1 MLP and CNN accuracy with considered binary switching strategies f...
Table 7.2 MLP and CNN accuracy with considered binary switching strategies w...
Chapter 8
Table 8.1 Summary of metrics for Gaussian probability distributions.
Chapter 11
Table 11.1 Comparison of coarray aperture, number of unique and consecutive ...
Table 11.2 Number of unique and consecutive lags for moving coprime arrays....
Table 11.3 Number of unique and consecutive lags for two-level nested arrays...
Table 11.4 Number of unique lags of a moving MRA.
Table 11.5 Number of unique lags of a moving MHA.
Table 11.6 Theoretical expressions of achievable DOF and consecutive lags fo...
Chapter 12
Table 12.1 Proximal operators associated with the previously described objec...
Table 12.2 Computation times, error, and cost-ratios for FERM and hybrid ADM...
Table 12.3 Average SNR and SSIM values for different noise and 2D-rectangula...
List of Illustrations
Chapter 1
Figure 1.1 A plane wave arriving at angle and impinging on a -sensor line...
Figure 1.2 Example of a MUSIC spectrum. See the text for details.
Figure 1.3 A digital beamformer with weights or taps working on a linear a...
Figure 1.4 (a) A standard uniform linear array (ULA) and (b) a sparse array ...
Figure 1.5 Example of a coarray MUSIC spectrum (Section 1.6) for the nested ...
Figure 1.6 Difference coarrays for the eight-element arrays in Fig. 1.4. (a)...
Figure 1.7 (a) A sparse linear array. (b) Its coarray, with the central ULA ...
Figure 1.8 Dividing the correlation estimates into overlapping blocks for ...
Figure 1.9 (a) The nested array with sensors. The standard ULA part has ...
Figure 1.10 The two sparse ULAs making up a basic coprime array with senso...
Figure 1.11 The two sparse ULAs making up the extended coprime array. There ...
Figure 1.12 Basic coprime array with sensors (). (a) The two sparse ULAs ...
Figure 1.13 The extended coprime array, with . (a), (b) The sparse ULAs, (c...
Figure 1.14 Coarray MUSIC spectrum for an extended coprime array with sens...
Figure 1.15 Weight functions for several arrays with sensors. (a) ULA, (b)...
Figure 1.16 Beamformer based on the sparse ULA .
Figure 1.17 (a)–(e) Magnitude responses of the filter and the filters ....
Figure 1.18 (a)–(d) Magnitude responses of the filter and the filters ....
Figure 1.19 The filter bank of filters defined as in (1.83). Magnitude r...
Figure 1.20 The coprime DFT filter bank. (a) The beamformer gains from t...
Figure 1.21 Adopted from [27]. Magnitude responses of the filters for th...
Figure 1.22 Plots of the CRB for sparse arrays identifying more DOAs than se...
Figure 1.23 Direct MUSIC on the coprime array (1.94) and a ULA both with s...
Chapter 2
Figure 2.1 Illustration of a coprime array. (a) A coprime pair of sparse ULA...
Figure 2.2 Illustration of various array representations with an example of
Figure 2.3 Phase offsets among the virtual measurements of each subarray.
Figure 2.4 Resolution comparison in terms of the normalized spatial spectrum...
Figure 2.5 Resolution comparison in terms of the normalized spatial spectrum...
Figure 2.6 DOFs comparison in terms of the spatial spectrum for sources. (...
Figure 2.7 DOFs comparison in terms of the spatial spectrum for sources. (...
Figure 2.8 RMSE performance comparison with single incident source. (a) RMSE...
Figure 2.9 RMSE performance comparison with multiple incident sources. (a) R...
Figure 2.10 Computation time comparison with different sampling interval.
Figure 2.11 Illustration of array configurations. (a) The nonuniform coprime...
Figure 2.12 DOFs illustration of the proposed physical array interpolation-b...
Figure 2.13 Comparison of DOA estimation performance. (a) RMSE versus SNR wh...
Figure 2.14 Difference lags obtained from six-element coprime array, nested ...
Figure 2.15 Illustration of multi-dimensional sparse arrays.
Figure 2.16 Development of tensor signal processing.
Chapter 3
Figure 3.1 DOA estimation results obtained by the narrowband method and the ...
Figure 3.2 RMSEs versus input SNR obtained for a single frequency by “Narrow...
Figure 3.3 DOA estimation results obtained for a single frequency and multip...
Figure 3.4 RMSEs versus .
Figure 3.5 Difference coarray at various operational frequencies.
Figure 3.6 Dual-frequency coarray. (a) and . (b) and .
Figure 3.7 Dual-frequency sparse reconstructions results under proportional ...
Figure 3.8 Dual-frequency sparse reconstructions results for nonproportional...
Figure 3.9 Multi-frequency group sparse reconstructions results with nonprop...
Figure 3.10 NC09 experiment map showing location of VLA2 (star) and trajecto...
Figure 3.11 NC09: Evolution of DOAs with time as the ship moves. (a) Distanc...
Figure 3.12 NC09: Evolution of DOAs with time (15–30 minutes portion from Fi...
Figure 3.13 versus for the time step indicated by the verticle line arou...
Chapter 4
Figure 4.1 Mean Normalized Power Estimation Error (NE) and the Upper Bound (...
Figure 4.2 Probability of successful support recovery as a function of . He...
Figure 4.3 Probability of successful support recovery as a function of spars...
Figure 4.4 Comparative probability of successful support recovery for and ...
Figure 4.5 Phase transition of success rate as function of sparsity and nu...
Figure 4.6 Log–log plot of as a function of , for fixed . Here and a...
Figure 4.7 Success rate of M-SBL, M-FOCUSS and SPICE as a function of sparsi...
Figure 4.8 Comparison of beamforming on the nested array and interpolated vi...
Figure 4.9 (a) MSE in DOA vs. SNR for sources with a nested array with s...
Chapter 5
Figure 5.1 2D-DOA signal model with a twenty-element URA.
Figure 5.2 Classification of 2D sparse antenna arrays.
Figure 5.3 Timeline for previous work on 2D sparse array geometries.
Figure 5.4 Examples of 2D sparse arrays constructing using parallel linear a...
Figure 5.5 2D-DOA estimation for parallel arrays (a) PCA, (b) PCA with modif...
Figure 5.6 Examples of nonparallel sparse arrays. (a) Two NAs, (b) Four NAs,...
Figure 5.7 Examples of 2D sparse arrays constructed using planar subarrays o...
Figure 5.8 The estimated and the actual normalized angles for interleaved ar...
Figure 5.9 Examples of 2D sparse arrays constructed using planar subarrays o...
Figure 5.10 Examples of 2D sparse arrays constructed using VSPA. (a) VSPA-CA...
Figure 5.11 The estimated and the actual normalized angles for (a) VSPA-CA-C...
Figure 5.12 Additional examples of 2D sparse arrays. (a) HOBA-2, (b) Hourgla...
Figure 5.13 2D-DOA estimation for (a) HOBA-2, (b) Hourglass, (c) Thermos, (d...
Figure 5.14 uDOF as a function of number of sensors for (a) parallel arrays,...
Chapter 6
Figure 6.1 Illustration of DL-based antenna selection.
Figure 6.2 Design procedures for DL-based antenna selection for the applicat...
Figure 6.3 Structure of the CNN for sensor selection.
Figure 6.4 Placement of antennas for (a) UCA with elements, (b) RDA with
Figure 6.5 Performance of test dataset using CNN with respect to for a UCA...
Figure 6.6 DoA estimation performance with respect to SNR. dB. The antenna...
Figure 6.7 (a) Antenna selection percentage over trials. (b) MSE of DoA fo...
Figure 6.8 The representation of the source and target domains for knowledge...
Figure 6.9 DoA estimation performance for for different array geometries....
Figure 6.10 Performance of , and versus the number of DoA angles. Senso...
Figure 6.11 DoA estimation performance when source domain has URA; target do...
Figure 6.12 DoA estimation performance when source domain has UCA; target do...
Figure 6.13 Improved SA algorithm for random 2-D sparse subarray generation ...
Figure 6.14 The array configurations of (a) parent -sensor URA, (b) the con...
Figure 6.15 RMSE performance versus (a) SNR and (b) number of snapshots for ...
Figure 6.16 Receiver architectures with antenna selection for single user mm...
Figure 6.17 The CNN architecture for antenna selection and RF beamformer des...
Figure 6.18 Performance of CNN. (a) Validation loss versus number of channel...
Figure 6.19 ISAC hybrid beamforming scenario with antenna selection.
Figure 6.20 ISAC hybrid beamforming performance with antenna selection.
Chapter 7
Figure 7.1 Block diagram of adaptive switched sensor beamformer.
Figure 7.2 Overview of the proposed sparse array design approach using machi...
Figure 7.3 Eight element sparse array configuration.
Figure 7.4 Lag redundancy of the sparse array shown in Fig. 7.3.
Figure 7.5 DFT of the lag redundancy.
Figure 7.6 Power spectrum of the desired signal.
Figure 7.7 Explanation of the proposed objective criterion for the optimum a...
Figure 7.8 Explanation of the proposed objective criterion for the worst pos...
Figure 7.9 Plot of the proposed objective criterion in ascending order.
Figure 7.10 Performance comparison of proposed SBSA algorithm with other sta...
Figure 7.11 Architecture of the dual neural network (DNN). Both the data cor...
Figure 7.12 (a) Binary switching Strategy-1: Adjacent antennas; (b) Binary s...
Figure 7.13 MLP architecture.
Figure 7.14 CNN architecture.
Figure 7.15 Performance comparison for Scenario 1 considering one interferen...
Figure 7.16 Performance comparison for Scenario 2 considering two interferen...
Figure 7.17 Performance comparison for Scenario 3 considering a mix of one o...
Figure 7.18 Confusion matrix for Scenario 3 considering a mix of one or two ...
Figure 7.19 Confusion matrix for Scenario 3 considering a mix of one or two ...
Figure 7.20 Symmetric sparse arrays (Dark circles shows antenna selected, Li...
Figure 7.21 Two sparse array combinations with nonoverlapping single missing...
Chapter 8
Figure 8.1 Illustration of the running example that will be used throughout ...
Figure 8.2 Comparison of the non-Boolean selection obtained by optimizing th...
Figure 8.3 Selection obtained by greedily optimizing the D-optimality criter...
Figure 8.4 (a) Sensor placements obtained from 8.25 for different and corr...
Figure 8.5 (a) Sensor placements obtained from 8.25 for different and corr...
Figure 8.6 Performance of the optimal selection with conditionally independe...
Figure 8.7 Comparison of the selection of the convex (a) and submodular (b) ...
Figure 8.8 Comparison of the selection of the convex and submodular optimiza...
Figure 8.9 (a) Sensor selection of the submodular strategy for different num...
Figure 8.10 (a) Sensor selection of the submodular strategy for different nu...
Chapter 9
Figure 9.1 Active sensing signal model. A linear Tx array with sensors ill...
Figure 9.2 Categorization of (aligned) active array configurations based on ...
Figure 9.3 Active MRAs with physical (transceiving) sensors [10]. Most of ...
Figure 9.4 The (a) NA generator and shift parameter define the (b) symme...
Figure 9.5 NA and symmetric NA configurations using minimum-redundancy param...
Figure 9.6 The effective Tx–Rx beampattern of an active array equals the pro...
Figure 9.7 Transmit beampattern (magnitude squared) of phased array (steered...
Figure 9.8 Beamforming weights and corresponding beampatterns of ULA and MRA...
Figure 9.9 Tx–Rx beampattern synthesis using image addition. Both the ULA an...
Chapter 10
Figure 10.1 Schematic diagram of distributed MIMO Radar.
Figure 10.2 Schematic diagram of colocated MIMO radar.
Figure 10.3 Virtual array and virtual aperture of MIMO array.
Figure 10.4 A special MIMO virtual array: sum-coarray.
Figure 10.5 Relationship between output SINR and target angle with the direc...
Figure 10.6 Optimal MIMO transceiver configuration when the target is from
Figure 10.7 Relationship between the output SINR and the target angle when t...
Figure 10.8 Schematic diagram of cognitive MIMO Radar.
Figure 10.9 Division of the large -antenna uniform linear array into grou...
Figure 10.10 Schematic diagram of virtual sensor for cognitive MIMO radar an...
Figure 10.11 Response of cognitive MIMO radar to environmental changes.
Figure 10.12 Relationship between output SINR and target angle with the dire...
Figure 10.13 Relationship between the output SINR and the target angle when ...
Figure 10.14 The influence of the total number of sensors and the antenna sp...
Figure 10.15 Optimal MIMO transceiver configuration selected by the proposed...
Figure 10.16 Relationship between MSE and input SNR for different transceive...
Chapter 11
Figure 11.1 The prototype coprime array configuration.
Figure 11.2 An example of prototype coprime configuration coarrays, where ...
Figure 11.3 The CACIS configuration.
Figure 11.4 An example of CACIS configuration coarrays, where , and . (a...
Figure 11.5 The CADiS configuration.
Figure 11.6 An example of CADiS configuration coarrays, where , , and ....
Figure 11.7 CACIS configuration coarrays, for different compression factor
Figure 11.8 CADiS configuration coarrays with displacement , corresponding ...
Figure 11.9 DOA estimation exploiting a moving sparse array.
Figure 11.10 Difference coarray of original coprime array. (a) , ; (b) ,
Figure 11.11 Difference coarray of synthetic coprime array. (a) , ; (b) ,...
Figure 11.12 Difference coarrays of original sparse arrays. (a) Two-level ne...
Figure 11.13 Difference coarray of synthetic sparse arrays. (a) Two-level ne...
Figure 11.14 The array configuration of the dilated nested array.
Figure 11.15 Non-negative part in the difference coarray of the synthetic di...
Figure 11.16 Estimated DOAs for original sparse arrays using sparsity-base t...
Figure 11.17 Estimated DOAs for synthetic sparse arrays using sparsity-base ...
Figure 11.18 Estimated DOAs using coarray MUSIC. (a) The original coprime ar...
Chapter 12
Figure 12.1 Ground plane geometry for spotlight-mode SAR sensing. The radar ...
Figure 12.2 Typical processing steps for conventional SAR image formation.
Figure 12.3 Data selection matrix configurations with white and black colo...
Figure 12.4 Illustration of different feature enhanced SAR imaging approache...
Figure 12.5 Reference image from SARPER™ – airborne SAR system developed by ...
Figure 12.6 Minimum-norm reconstruction from 3.2% of reference samples in Fo...
Figure 12.7 Reconstruction with the ADMM-based method from 3.2% of reference...
Figure 12.8 Root-mean-square error (RMSE) in phase error estimation in radia...
Figure 12.9 Reference image reconstructed using full-data followed by PGA.
Figure 12.10 Reconstruction results from phase-corrupted data with 25% 1D-ra...
Figure 12.11 Reconstruction results for 2D-rectangular data limitation with ...
Figure 12.12 Reconstruction results for 2D-rectangular data limitation with ...
Figure 12.13 Reconstruction results for 2D-random data limitation. (a, e, i)...
Figure 12.14 Reconstruction results for 1D-linear data limitation. (a, e, i)...
Chapter 13
Figure 13.1 Two uniform line arrays and their associated beampatterns. (a) D...
Figure 13.2 One-dimensional active sonar example using a three-element trans...
Figure 13.3 Mills cross configuration for multibeam echo-sounding. The trans...
Figure 13.4 Mills cross configuration. (a) Shows a 15-element transmitter ar...
Figure 13.5 A filled ULA and its difference coarray support.
Figure 13.6 The difference coarray weightings for the ULA in Fig. 13.5.
Figure 13.7 Sparse arrays and their beampatterns. (a) Two sparse arrays with...
Figure 13.8 Sparse arrays and their difference coarray supports. From the to...
Figure 13.9 Difference coarray weightings for the three sparse arrays in Fig...
Figure 13.10 A coprime array with undersampling factors and , the self di...
Figure 13.11 A nested array, its self difference coarray support, and cross ...
Figure 13.12 Difference coarray weightings of the coprime array in Fig. 13.1...
Figure 13.13 Difference coarray weightings of the nested array in Fig. 13.11...
Figure 13.14 Block diagrams of the predominant processors. (a) CBF block dia...
Figure 13.15 Coprime and nested arrays, each with sensors. Subarray and ...
Figure 13.16 CBF, Product, and Min Beampatterns for (a) the nested array fro...
Figure 13.17 Subarray , Subarray , product, and Min beampatterns for (a) t...
Figure 13.18 Coprime array with , , and and its self difference coarray ...
Figure 13.19 Scanned responses of different processors for a coprime array. ...
Figure 13.20 plot corresponding to the CBF for a -sensor ULA.
Figure 13.21 Comparison of the predominant processors for the coprime (a) an...
Figure 13.22 Comparison of CBF, augmented, product, and min processors for t...
Chapter 14
Figure 14.1 Sketch of the (a)
conventional
or
fully-populated
array architec...
Figure 14.2
Sparseness-Promoting Techniques
(
ST/MT-BCS Synthesis Method
,
Fla
...
Figure 14.3
Sparseness-Promoting Techniques
(
D-CS Synthesis Method
, , , )...
Figure 14.4
Sparseness-Promoting Techniques
(
D-CS Synthesis Method
,
Taylor P
...
Figure 14.5
Sparseness-Promoting Techniques
(
TV-CS Synthesis Method
, , ) –...
Figure 14.6
Sparseness-Promoting Techniques
(
TV-CS Synthesis Method
,
Taylor
...
Figure 14.7
Co-Design Strategy
(
Antenna Element Synthesis
, , ) – (a) Plots...
Figure 14.8
Co-Design Strategy
(
Array Clustering
, , , , ) – (a) Plots of...
Figure 14.9
Capacity-Driven Synthesis
(
Domino Tiles
, , , , , ) – Cluste...
Figure 14.10
Capacity-Driven Synthesis
(
P-shaped Tile
s, , , , , , ) – ...
Chapter 15
Figure 15.1 Block diagrams of the fully digital architecture with a sparse a...
Figure 15.2 Illustration of a full switching network.
Figure 15.3 Illustration of a binary switching network.
Figure 15.4 Illustration of binary switching on adjacent antennas.
Figure 15.5 Illustration of binary switching on separate antennas.
Figure 15.6 Illustration of binary switching on random antenna pairs.
Figure 15.7 Multi-user MIMO capacities, achieved by the DPC, when an 128-ant...
Figure 15.8 Block diagram of the fully connected hybrid analog/digital trans...
Figure 15.9 Block diagram of the array of sub-arrays hybrid analog/digital t...
Figure 15.10 Block diagrams of two switches-based analog beamforming/combini...
Figure 15.11 Spectral efficiency attained by different hybrid design methods...
Figure 15.12 Spectral efficiency attained by various hybrid design methods f...
Figure 15.13 Spectral efficiency exhibited by various hybrid design methods ...
Guide
Cover
Title Page
Copyright
Dedication
About the Editor
List of Contributors
Preface
Table of Contents
Begin Reading
Index
End User License Agreement
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IEEE Press445 Hoes LanePiscataway, NJ 08854
IEEE Press Editorial BoardSarah Spurgeon, Editor in Chief
Jón Atli Benediktsson Anjan Bose James Duncan Amin Moeness Desineni Subbaram Naidu
Behzad Razavi Jim Lyke Hai Li Brian Johnson
Jeffrey Reed Diomidis Spinellis Adam Drobot Tom Robertazzi Ahmet Murat Tekalp
Sparse Arrays for Radar, Sonar, and Communications
Edited by
Moeness G. Amin
Villanova UniversityVillanova, PA, USA
Copyright © 2024 by The Institute of Electrical and Electronics Engineers, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey.Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data:
Names: Amin, Moeness G., editor. | John Wiley & Sons, publisher.
Title: Sparse arrays for radar, sonar, and communications / [edited by] Moeness G. Amin.
Description: Hoboken, New Jersey : Wiley-IEEE Press, [2024] | Includes index.
Identifiers: LCCN 2023040085 (print) | LCCN 2023040086 (ebook) | ISBN 9781394191017 (cloth) | ISBN 9781394191024 (adobe pdf) | ISBN 9781394191031 (epub)
Subjects: LCSH: Antenna arrays. | Antenna arrays–Design and construction. | Signal processing.
Classification: LCC TK7871.67.A77 S63 2024 (print) | LCC TK7871.67.A77 (ebook) | DDC 621.382/2–dc23/eng/20230907
LC record available at https://lccn.loc.gov/2023040085
LC ebook record available at https://lccn.loc.gov/2023040086
Cover Design: WileyCover Image: © Lars Poyansky/Shutterstock; Yuichiro Chino/Getty Images; enot-poloskun/Getty Images
I dedicate this book to my family whose love and support have been instrumental for all accomplishments in my academic career.
About the Editor
Moeness G. Amin received his PhD degree from the University of Colorado, Boulder, in 1984. Since 1985, he has been with the Faculty of the Department of Electrical and Computer Engineering, Villanova University, Villanova, PA, USA, where he became the Director of the Center for Advanced Communications, College of Engineering, in 2002.
Dr. Amin is a Life Fellow of the Institute of Electrical and Electronics Engineers (IEEE); Fellow of the International Society of Optical Engineering (SPIE); Fellow of the Institute of Engineering and Technology (IET); and a Fellow of the European Association for Signal Processing (EURASIP). He is the recipient of the 2022 IEEE Dennis Picard Gold Medal in Radar Technologies and Applications; the 2017 Fulbright Distinguished Chair in Advanced Science and Technology; the 2016 Alexander von Humboldt Research Award; the 2016 IET Achievement Medal; the 2014 IEEE Signal Processing Society Technical Achievement Award; the 2009 Technical Achievement Award from the European Association for Signal Processing; the 2015 IEEE Aerospace and Electronic Systems Society Warren D. White Award for Excellence in Radar Engineering. He is also the recipient of the 2010 Chief of Naval Research Challenge Award; the 1997 IEEE Philadelphia Section Award, the 1997 Villanova University Outstanding Faculty Research Award. He is the recipient of the IEEE Third Millennium Medal.
Dr. Amin served as Chair/Member of the Electrical Cluster of the Franklin Institute Committee on Science and the Arts (2001–2016). He served on the Editorial Board of the Proceedings of the IEEE and is currently a member of the Editorial Board of the IEEE Press. Dr. Amin has over 900 journal and conference publications in signal processing theory and applications, covering the areas of wireless communications, radar, satellite navigations, ultrasound, healthcare, and RFID. He is the editor of four books on radar.
List of Contributors
Elias AboutaniosSchool of Electrical Engineering and TelecommunicationsUniversity of New South WalesKensingtonNSWAustralia
Kaushallya AdhikariElectrical, Computer, and Biomedical Engineering DepartmentUniversity of Rhode IslandKingstonRIUSA
Fauzia AhmadDepartment of Electrical and Computer EngineeringTemple UniversityPhiladelphiaPAUSA
Saleh A. AlawshCenter for Communication Systems and SensingKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia
Ahmed AlkhateebSchool of ElectricalComputer and Energy EngineeringArizona State UniversityTempeAZUSA
M. Burak AlverFaculty of Engineering & Natural SciencesMaltepe UniversityIstanbulTurkey
Moeness G. AminCenter for Advanced Communications (CAC)College of EngineeringVillanova UniversityVillanovaPAUSA
Nicola AnselmiDICAM - Department of Civil, Environmental, and Mechanical EngineeringELEDIA Research CenterUniversity of TrentoTrentoItaly
Müjdat ÇetinDepartment of Electrical and Computer Eng. & Georgen Institute for Data ScienceUniversity of RochesterRochesterNYUSA
Sundeep Prabhakar ChepuriDepartment of Electrical Communication Engineering (ECE)Indian Institute of ScienceBengaluruIndia
Mario CoutinoNetherlands Organisation for Applied Scientific Research (TNO)The HagueThe Netherlands
Ahmet M. ElbirInterdisciplinary Centre for SecurityReliability and Trust (SnT)University of LuxembourgEsch-sur-AlzetteLuxembourg
Xiang GaoNational Key Laboratory of Science and Technology on CommunicationsUniversity of Electronic Science and Technology of ChinaSichuanChina
Peter GerstoftNoiseLabUniversity of California San DiegoLa JollaCAUSA
Sotirios GoudosELEDIA Research CenterAristotle University of ThessalonikiThessalonikiGreece
Yujie GuAdvanced Safety & User ExperienceAptivAgoura HillsCAUSA
Alper GüngörAselsan Research CenterAselsanAnkaraTurkey
Emre GüvenCognizen Arge Mühendislik ve Yazlm A.Ş.AnkaraTurkey
Syed A. HamzaElectrical Engineering DepartmentSchool of EngineeringWidener UniversityChesterPAUSA
Koichi IchigeDepartment of Electrical and Computer EngineeringYokohama National UniversityYokohamaJapan
Kyle JuretusECE DepartmentCollege of EngineeringVillanova UniversityVillanovaPAUSA
Visa KoivunenDepartment of Information and Communications EngineeringAalto UniversityEspooFinland
Pranav KulkarniDepartment of Electrical EngineeringCalifornia Institute of TechnologyPasadenaCAUSA
Geert LeusFaculty of Electrical Engineering, Mathematics and Computer Science (EEMCS)Delft University of TechnologyDelftThe Netherlands
Wei LiuSchool of Electronic Engineering and Computer ScienceQueen Mary University of LondonLondonUK
Andrea MassaDICAM - Department of Civil, Environmental, and Mechanical EngineeringELEDIA Research CenterUniversity of TrentoTrentoItalySchool of Electronic Science and EngineeringELEDIA Research CenterUniversity of Electronic Science and Technology of ChinaChengduChinaELEDIA Research CenterTsinghua UniversityBeijingHaidianChinaSchool of Electrical EngineeringTel Aviv UniversityTel AvivIsraelandELEDIA Research CenterUniversity of Illinois ChicagoChicagoILUSA
Kumar Vijay MishraComputational and Information Sciences Directorate (CISD)United States DEVCOM Army Research LaboratoryAdelphiMDUSA
Ali H. MuqaibelElectrical Engineering DepartmentKing Fahd University of Petroleum and MineralsDhahranSaudi ArabiaandCenter for Communication Systems and SensingKing Fahd University of Petroleum and MineralsDhahranSaudi Arabia
Giacomo OliveriDICAM - Department of Civil, Environmental, and Mechanical EngineeringELEDIA Research CenterUniversity of TrentoTrentoItaly
Piya PalDepartment of Electrical and Computer EngineeringUniversity of California San DiegoLa JollaCAUSA
Lorenzo PoliDICAM - Department of Civil, Environmental, and Mechanical EngineeringELEDIA Research CenterUniversity of TrentoTrentoItaly
Heng QiaoUniversity of Michigan-Shanghai Jiao Tong University Joint InstituteShanghai Jiao Tong UniversityShanghaiChina
Guodong QinSchool of Electronic EngineeringXidian UniversityXi'anShaanxiChina
Si QinMicrosoft ChinaBeijingChina
Robin RajamäkiElectrical and Computer Engineering DepartmentUniversity of California San DiegoLa JollaCAUSAandDepartment of Information and Communications EngineeringAalto UniversityEspooFinland
Paolo RoccaDICAM - Department of Civil, Environmental, and Mechanical EngineeringELEDIA Research CenterUniversity of TrentoTrentoItalyandELEDIA Research CenterXidian UniversityXi'anShaanxi ProvinceChina
Ammar SaleemFaculty of Engineering & Natural SciencesSabanci UniversityIstanbulTurkey
Marco SalucciDICAM - Department of Civil, Environmental, and Mechanical EngineeringELEDIA Research CenterUniversity of TrentoTrentoItaly
Zhiguo ShiKey Laboratory of Collaborative Sensing and Autonomous Unmanned Systems of Zhejiang ProvinceZhejiang UniversityHangzhouChinaandCollege of Information Science and Electronic EngineeringZhejiang UniversityHangzhouChina
Palghat P. VaidyanathanDepartment of Electrical EngineeringCalifornia Institute of TechnologyPasadenaCAUSA
Kathleen E. WageElectrical and Computer Engineering DepartmentGeorge Mason UniversityFairfaxVAUSA
Xiangrong WangSchool of Electronic and Information EngineeringBeihang UniversityBeijingChina
Xianghua WangSchool of Artificial IntelligenceBeijing University of Posts and TelecommunicationsBeijingChina
Weitong ZhaiSchool of Electronic and Information EngineeringBeihang UniversityBeijingChina
Yimin D. ZhangDepartment of Electrical and Computer EngineeringTemple UniversityPhiladelphiaPAUSA
Chengwei ZhouState Key Laboratory of Industrial Control TechnologyZhejiang UniversityHangzhouChinaandCollege of Information Science and Electronic EngineeringZhejiang UniversityHangzhouChina
Preface
The timing of this book is motivated by recent advances in sparse arrays design. The book builds on strides made over the last decade in convex optimizations and machine learning, and covers various emerging applications in sensing and communications. It is intended as a reference book for students, faculty, and scientists pursuing research in the broad areas of active and passive sensing and wireless technologies. Also, benefiting from this book are the engineers who engage in system developments design, and manufacturing using multi-sensor platforms.
Sensor arrays have endowed transmitters and receivers with far more capabilities in radar, sonar, and communications to achieve system objects compared to a single sensor deployment. Sparse arrays are a type of sensor arrays which have been around for many years. It was recognized early on that sensors can be arranged in random or nonuniform ways, different from the conventional uniform array. Different array structures, in essence, can provide different tradeoffs comprising performance, aperture size, mutual coupling, cost, and implementation.
Recent research findings in sparse arrays has shown increased capacity in wireless communications and improved sensing in radar and sonar using both narrowband and wideband signal models. Sparse arrays, in lieu of conventional uniform arrays, allow dealing with more targets and radio frequency (RF) emitters than the number of physical sensors. Optimum Sparse arrays for direction finding and beamforming have been developed which provide superior performance in target detection and tracking compared to uniform arrays of the same aperture or number of sensors. Sparse array applications have broadened to include synthetic aperture radar (SAR), multiple-input multiple-output (MIMO) radar, Massive MIMO communications, 5G and Beyond, RF surveillance and sensing, target localization, wideband urban radar processing, and space–time adaptive processing.
This book captures new important applications of sparse arrays and delineates their utilities. It describes various array configurations for stationary and moving sensor platforms and examines transceiver array design under fixed and dynamic environments. The book includes array designs that seek to increase the corresponding one-dimensional and two-dimensional virtual array apertures and others that configure and synthesize the sparse arrays based on solutions of constrained minimization problems. Optimality is also considered for combined sparse array-waveform design. The book presents a view of the Perception–Action Cycle in cognitive sensing through the lens of sparse arrays and antenna switching technology for direction finding and beamforming.
I am grateful to the authors of all chapters. They have provided the state-of-the-art coverage of sparse arrays supported by analytical rigor and validated by examples involving real and simulated data.
The book consists of 15 chapters. Each chapter addresses a different aspect of sparse array design and application. Chapter 1 introduces recent and conventional sparse arrays and explains their advantages. Using the notion of a difference coarray, the chapter delineates how sparse sensor arrays can identify more sources than the number of sensors. Chapter 2 considers sparse array interpolation. It addresses the interpolation problem of obtaining a fully augmented sparse array where the co-array (virtual array) has consecutive elements, that is, all spatial autocorrelation lags are available. Chapter 3 considers wideband and multi-frequency operations of sparse arrays for both beamforming and direction-of-arrival (DOA) estimation. It demonstrates enhanced sparse array degrees of freedom (DOF) beyond those obtained using narrowband signals.
Chapter 4 deals with sparse arrays in view of limited data. It characterizes array performance by deriving the interplay between the array geometry, number of snapshots and signal-to-noise ratio (SNR). Chapter 5 deals with two-dimensional (2D) sparse array design which is fundamental to source localization applications. The main objective is to increase the DOF which is a measure of the number of sources that can be correctly localized.
Chapters 6 and 7 apply deep learning (DL) for sparse array design. The two chapters, respectively, present an overview of DL applications of sparse array design for high resolution direction finding and maximum signal-to-interference and noise ratio (MaxSINR) beamforming. Chapter 8 deals with optimum placements of sensors in sensor networks. It considers both detection and estimation problems utilizing a limited number of network sensors. Chapter 9 focuses on active sensing based on sparse arrays. Using the notion of sum-coarray, it illustrates several advantages of sparse arrays over conventional uniform arrays, including improved resolution, fewer physical sensors, and the capability to identify a large number of scatterers.
Chapter 10 covers sparse arrays for MIMO array radar. In this regard, it deals with active sensing and complements Chapter 9. It examines joint sparse optimizations of MIMO radar transceiver under different task requirements. Chapter 11 introduces and analyzes generalized structured sparse arrays for fixed and moving platforms. It shows that both platforms yield increased DOF of the associated virtual arrays, thus improving DOA estimation performance. Chapter 12 deals with sparse observation synthetic aperture radar (SAR) and presents optimization and learning-based algorithms for SAR imaging. It applies compressed sensing-based imaging and deep learning-based regularizers within a physics-based observation model, and provides a supportive real SAR data example.
Chapter 13 tackles sparse arrays for sonar systems. It introduces fundamental concepts in passive and active sonar using sparse arrays and presents a unifying framework for understanding standard processing techniques and performance metrics. Chapter 14 discusses sparse arrays in the context of phased arrays and antenna technology. It reviews recent advances and trends in the synthesis of unconventional array architectures and presents the guidelines for the state-of-the-art strategies in phased array synthesis for next generation wireless communications. Chapter 15 provides an overview of the different sparse array architectures for MIMO communication systems and their design approaches. It draws insights into the potential gains achieved under fully digital and hybrid analog–digital architectures in various communication scenarios.
Moeness G. Amin
Villanova University
Villanova
Pennsylvania, USA
1Sparse Arrays: Fundamentals
Palghat P. Vaidyanathan and Pranav Kulkarni
Department of Electrical Engineering, California Institute of Technology, Pasadena CA, USA
1.1 Introduction
A sensor array or just “array” is a collection of sensors that performs spatial sampling of signals arriving from various directions. In general, the signals can be EM waves, sound waves, or other types of signals. In this chapter, we concentrate on plane monochromatic EM waves arriving from different directions and impinging on a sensor array. Arrays typically processes the signals received at the sensors to estimate the directions of arrival or DOAs. In some cases the arrays operate in the beamforming mode, by providing large gain in certain directions of arrival, compared to other directions.
An array is said to be linear if the sensors are on a straight line, and this is what we consider in this introductory chapter. More general arrays are discussed in Chapter 5 in the book. The most commonly used arrays are uniform linear arrays or ULAs, which have adjacent sensors separated by where is the wavelength of the impinging monochromatic waves. But there are also other types of arrays where a number of adjacent sensor pairs have larger separation. These are called sparse arrays, and they have a number of advantages.
In this chapter, we will introduce sparse arrays and discuss their advantages. In the world of array signal processing, such arrays have been known for many decades, especially in the context of imaging and beamforming. Sparse arrays received more attention in the last decade because of the introduction of nested and coprime arrays, which have certain advantages compared to theoretically optimal arrays that have been well-known for much longer.
We begin by reviewing the fundamentals of Array Signal Processing in Section 1.2, including DOA estimation and beamforming. Sparse arrays are introduced in Section 1.3. A unique property of some sparse arrays is that the number of DOAs that can be identified can exceed the number of sensors in the array. The reason for this counterintuitive property is explained in Section 1.4, and in greater detail in Section 1.5. The difference coarray, which is central to this property, is also introduced. Specific algorithms for DOA estimation with sparse arrays, such as coarray MUSIC, are introduced in Section 1.6. Well-known classes of sparse arrays such as nested arrays and coprime arrays are introduced in Section 1.7, and their properties are discussed. Then, in Section 1.8, we review classical optimal sparse arrays such as the minimum redundancy arrays or MRAs, minimum hole arrays or MHAs (also called Golomb rulers), and perfect arrays.
Section 1.9 explains how coprime arrays can be used to develop a novel type of beamformer, where the number of nonoverlapping spatial beams can be much larger than the number of sensors. This is another unique advantage of some sparse arrays. In Section 1.10, we conclude the chapter by providing directions for further reading.
Notations We use bold face letters for matrices and vectors, as in and , where usually uppercase letters denote matrices and lowercase letters denote column vectors. The notations , and denote, respectively, transpose, conjugate, and transpose-conjugate. The notation stands for the greatest common divisor (gcd) of the integers .
1.2 Basics of Array Processing
Figure 1.1 shows a linear array [1], that is, an array with sensors located on a straight line. There are sensors located at . The figure also schematically shows a plane EM wave impinging on the array. The direction of arrival (DOA) is indicated by the angle shown in the figure. The “plane wave” property arises from the assumption that the source which generates the wave is far away. With denoting the wavelength, we assume in this chapter that are at integer multiples of , that is,
where are integers. Such arrays are called integer arrays or integer linear arrays. Either one of the following sets
specifies such an array. When we say that the integer array is , we actually mean that the sensors are at (1.1). We will refer to the quantity as the aperture of the array. The array
is called a uniform linear array or ULA and has been widely used for decades. Arrays of the form , are also called ULAs, but they are “sparse” ULAs because . To distinguish, we sometimes refer to (1.3) as the standard ULA.
1.2.1 Expression for the Array Output
A plane monochromatic propagating wave can be expressed as
Figure 1.1 A plane wave arriving at angle and impinging on a -sensor linear array.
where is its amplitude and is the frequency. Here, is time, is the spatial coordinate along the line of propagation, and is the wavenumber, so that . The velocity of propagation is given by
In practice, the signal received is not monochromatic. A more reasonable assumption would be that it has a narrow bandwidth around the frequency . In this case, the signal received by the th sensor has the general form
where is a constant (i.e., independent of time) and is a narrowband lowpass signal.1 Its bandwidth is small compared to , or equivalently changes very slowly compared to . Note that (1.6) has Fourier transform where is the Fourier transform of .
The EM wave has the form , where is the distance coordinate along the direction of propagation. Let us say at time the signal at sensor is exactly . Then the signal at sensor is where is the extra distance traveled by the wave to arrive at (as compared to ). It is clear that . So the signal at sensor at time is
where is independent of . Since and , the above can be written as , where . It therefore follows that the set of received signals has the form
where is the dimensionless quantity
called the directional frequency, and is a different way to represent the DOA . For simplicity, we often refer to also as DOA. We always assume to be in the range
so that . The high frequency signal can be removed by demodulation at the output of every sensor. Assuming that all sensors are omnidirectional and have identical physical properties, the constant in (1.8) is independent of sensor number . So we shall ignore and from now on, and write the output of the array in response to the plane wave as
where is the amplitude of the signal arriving from DOA (or more precisely DOA ). More generally, if we have sources with the same wavelength (frequency ) with DOAs
the array output is the superposition
where , and is the amplitude of the th source, assumed to be “narrowband” (i.e., bandwidth ).
1.2.2 Sampling the Array Outputs
In practice, the array outputs are uniformly sampled to obtain , where is the sample spacing. So the set of sampled outputs is
where . Since there is always noise in the received signal, a more accurate model for the array output samples has the form
where is the additive noise term. Here is the number of samples or snapshots. The matrix in (1.14) can be written as
where are column vectors:
is called the array manifold matrix, and is called the array manifold vector or the steering vector.
1.2.3 Covariance of the Array Output
We assume that the signal and noise are jointly wide sense stationary (WSS) random processes with the following properties:
A1. Signal and noise have zero mean: .
A2. Signal and noise are uncorrelated: .
A3. In words, noise components and at any two sensors are uncorrelated, with
equal variance
at all sensors.
A4. , that is, the signal covariance is positive definite. In particular, has rank .
These assumptions typically hold over short durations of time (a few hundred samples). Under these assumptions, the array output is a zero-mean WSS process with a covariance matrix
In expressions involving expectation operators, we often drop the snapshot index for simplicity. For example, we write , etc., instead of . This is because these expectations do not depend on , in view of the WSS assumption. The covariance (1.18) is usually estimated by computing the sample covariance matrix
Here, is the number of shapshots. Since is the time index, the WSS property mentioned above is called temporal WSS property, to distinguish it from something called spatial WSS property, which we shall encounter later.
1.2.4 The MUSIC Algorithm
For the case , the DOAs can be found using algorithms such as MUSIC [3] and ESPRIT [4] by starting from (an estimate of) the eigenvectors of . A sufficient condition for these algorithms to identify DOAs without ambiguity is that the array be a standard ULA as in (1.3). Assuming this is the case, we briefly mention how MUSIC works, as it is essential for an understanding of Section 1.6. With , the autocorrelation matrix in (1.18) is positive definite, and its eigenvalues can be ordered as
Since is Hermitian, we can always find an orthonormal set of eigenvectors , so that the eigenvector matrix
is unitary. Now, it can be shown that the subspace spanned by the first eigenvectors above is the same as the subspace spanned by the columns of . This is called the signal subspace because , where are the DOAs of the signals. The vectors are therefore orthogonal to the last eigenvectors in (1.21). Defining the noise subspace matrix
it then follows that for all DOAs . Under some conditions (which are surely satisfied for the ULA when ) it can also be shown that , only if is a DOA. We can therefore identify the DOAs by finding the values of such that .2 In practice, since can only be estimated from the estimate , one plots the quantity
Figure 1.2 Example of a MUSIC spectrum. See the text for details.
on a dense grid of and locates its peaks to find . is called the MUSIC spectrum. An example of the MUSIC spectrum is shown in Fig. 1.2. In this example, we have a 10-sensor ULA () and there are six signals () arriving from various angles, which are indicated by the vertical dashed lines. The number of snapshots used is , and the SNR is 5 dB.3 Indeed, the peaks of the MUSIC spectrum are very close to the actual source angles. It should be mentioned here that the peak heights are not necessarily proportional to the signal powers .
Variations of the above method lead to other algorithms such as root-MUSIC [5]. Since these algorithms are based on the properties of the signal and noise subspaces, they are called subspace algorithms. Even though the above description of MUSIC is for the ULA, it is possible to extend it to non-ULAs under certain conditions. One such example is described in Section 1.10.3.
1.2.5 Invertibility of the Array Manifold
We know that the output of an array in response to a single source coming from direction is proportional to , as defined in (1.17). Let the integer array be with as usual. Without loss of generality, let us assume , so that It is clear that we can identify uniquely from measurements of the array output only if is uniquely determined by . This property is called the array manifold invertibility.
Figure 1.3 A digital beamformer with weights or taps working on a linear array of sensors.
Thus, a necessary condition for unambiguous identifiability of DOAs using a sparse array is that the array manifold vector be “invertible.” It can be shown [6, 7] that the array satisfies this property if and only if the greatest common divisor of the integers is unity, that is, . For example, the sparse ULA has and does not satisfy invertibility whereas the sparse array satisfies it and so does . The array satisfies invertibility no matter what the elements are. Needless to say, the standard ULA satisfies invertibility. More extensive properties of the geometry of array manifolds can be found in [8].
1.2.6 Beamforming
In some scenarios, the array output covariance is analyzed using subspace methods in order to estimate the DOAs as explained above. In some other situations, the array is used as a beamformer. Figure 1.3 shows the schematics of a beamformer. Here, the outputs of the sensors are linearly combined using weights (or taps) to obtain the output . (The snapshot index is omitted in the figure for simplicity.) If a plane monochromatic wave with amplitude arrives at an angle , then the array output (at the sensor locations) is . So the beamformer output is
where as before, and
is called the beamformer gain or transfer function. If are designed such that is large in the neighborhood of some and small elsewhere, then the signals received at or near the DOA are admitted by the beamformer and other DOAs are attenuated. So the beamformer can receive signals in one direction in preference to others.
1.3 What Are Sparse Arrays?
The most commonly used integer linear array is the standard ULA, which satisfies Eq. (1.3). This is shown in Fig. 1.4a. In contrast if for some (or all) , the array is said to be sparse. Figure 1.4b shows an example of a sparse array, called the nested array (to be defined formally in Section 1.7.1). Sparse arrays have been known for a long time although they received significant attention only in the last decade or so. We will discuss the details later, but here are some basic points about sparse arrays:
Figure 1.4 (a) A standard uniform linear array (ULA) and (b) a sparse array called the nested array (Section 1.7.1).
More sources than sensors
. A well-designed sparse array with sensors can identify DOAs unambiguously, in fact
