2012 IEEE International

Short Courses

 

The registration fee to attend short courses was significantly reduced thanks to the IEEE Antennas and Propagation Society that offered to subsidize the honorarium for all the instructors. The registration fee is $125 for a half day course and $200 for a full day course.

This page shows a temporary schedule for the short courses. It is possible to contact by email the instructors of each course by clicking on their name. For detailed information about the content of the courses, please refer to the course descriptions underneath the summary tables.

 

Full day courses

Date Time Title Instructors
July 8
8:00 - 17:00 Electromagnetic Band Gap (EBG) Structures in Antenna Engineering: From Fundamentals to Recent Advances Yaya Rahmat Samii (UCLA)
Fan Yang, (Tsinghua University, China)
July 8
8:00 - 17:00 Dielectric Resonator Antennas, Theory, Design and Applications with Recent Advancement Ahmed Kishk (Concordia University, Canada)
July 14
8:00 - 17:00 Advanced preconditioning techniques for computational electromagnetics 
Eric Michielssen (University of Michigan)
Francesco Andriulli (ENST Bretagne, France)

 

 

Half day courses

 Date Time Title Instructors
July 8 Morning OTA Measurements in Reverberation Chamber Per-Simon Kildal (Chalmers University, Sweden)

Charlie Orlenius (Bluetest)

July 8 Morning Phased Array for Microwave and Millimeter-wave Applications
Mohammad Fakharzadeh (Peraso Technologies Inc., Canada)
Safieddin Safavi-Naeini (University of Waterloo, Canada)

July 8 Morning Metamaterials  and Plasmonic  Materials: Theory and Electromagnetic Applications Andrea Alu (University of Texas at Austin)
July 8 Morning Use of the Principle of Analytic Continuation for the generation of phase and for interpolation/extrapolation of amplitude only data Tapan K Sarkar (Syracuse University)
Magdalena Salazar Palma (University of Carlos III of Madrid, Spain)

July 8
Morning Antenna Design and Challenges  in Cognitive Radio Christos Christodoulou, Youssef Tawk
(University of New Mexico)

July 8
Afternoon Reflectarray Antennas: Design, Reconfigurability And Potential Applications
Jose A. Encinar (Universidad Politécnica de Madrid, Spain)
Sean V. Hum (University of Toronto, Canada)
Julien Perruisseau-Carrier (Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland)

July 8
Afternoon Adaptive Arrays Randy L. Haupt, Mark Leifer (Ball Aerospace)
July 8
Afternoon Internal Handheld Device Antennas: Recent Advances, Future Perspectives, and Design Considerations for Bio-Compatible Handheld Device Antennas Kin-Lu Wong (National Sun Yat-sen University, Taiwan)
July 8
Afternoon
EM Design of Small Antennas for Industrial Applications Winfried Simon (IMST)
Dirk Manteuffel (University of Kiel, Germany)
Cyril Luxey (University Nice-Sophia Antipolis, France)
July 8
Afternoon
Advanced impedance matching and impedance analysis for antenna applications
Jussi Rahola (Optenni Ltd, Finland)
July 8
Afternoon
Antenna Design for Mobile Terminals
Zhijun Zhang (Tsinghua University, China)
July 8
Afternoon
Antenna Applications of Metamaterials and Transformation Electromagnetics
Yang Hao (Queen Mary University, UK)
Raj Mittra (Pennsylvania State University)

July 14
Morning
History, Design and Theory of Directive Fabry-Pérot Cavity Antennas as Leaky-Wave Radiators Filippo Capolino (University of California Irvine)
David Jackson (University of Houston)

July 14 Morning Advances in the Design of Electrically Small Antennas Steven R. Best (MITRE)
July 14
Afternoon
Multibeam Antennas and Beamforming Networks
Piero Angeletti, Giovanni Toso (European Space Research and Technology Centre, ESTEC), European Space Agency (ESA)
July 14
Afternoon
Natural and metamaterial antennas with emphasis on realization of wideband characteristics
H. Nakano (Hosei University
Koganei, Tokyo, Japan)

 

 

 

Detailed description

FULL DAY COURSES

Sunday, July 8, 2012

 

Electromagnetic Band Gap (EBG) Structures in Antenna Engineering: From Fundamentals to Recent Advances

Yahya Rahmat-Samii, University of California, Los Angeles, USA, and Fan Yang, University of Mississippi, USA
rahmat@ee.ucla.edu, fan_yang@tsinghua.edu.cn

This comprehensive and application-oriented short course on the state-of-the-art in electromagnetic band gap (EBG) engineering explains the theories, designs, and antenna applications of EBG structures. The course will start with an overview of the EBG research history and important discoveries. An accurate and efficient FDTD/PBC algorithm will be presented for general periodic structures analysis. Next, detailed presentations will be provided on the unique electromagnetic features and diverse functional designs of EBG structures. Furthermore, a wealth of practical antenna examples with design details will be presented to illustrate promising applications of EBG structures in antenna engineering.

 

Dielectric Resonator Antennas, Theory, Design and Applications with Recent Advancement

Ahmed Kishk, Concordia University, Canada
kishk@encs.concordia.ca

An overview for the development of the dielectric resonator antennas will be provided. Theory of operation will be discussed to provide basic understanding. Excitation mechanisms such as probe, slot, image line and waveguides are presented. Mutual coupling and wide scanning capabilities are considered. Techniques for broadband and size reduction of the DRA are presented. Recent developments of DR as a multifunction device for antennas and microwave circuits as a low and high Q-factor simultaneously will be provided.

 

Saturday, July 14, 2012

Advanced preconditioning techniques for computational electromagnetics

Francesco P. Andriulli, ENST Bretagne, France and Eric Michielssen, The University of Michigan, USA
francesco.andriulli@telecom-bretagne.eu, emichiel@eecs.umich.edu

This course reviews the state of the art in effective preconditioning techniques for integral equations pertinent to the analysis of electromagnetic boundary value problems. The techniques covered permit the construction of rapidly convergent iterative solvers for electric and combined field integral equations and as such are a perfect complement to fast multipole and related accelerators. Applications of these techniques range from antenna analysis to the characterization of microwave devices and circuits, the analysis of electromagnetic compatibility phenomena, and the synthesis of metamaterials. The course will cover theoretical and practical issues related to the development and implementation of several preconditioners, including those that derive from Calderon identities. Moreover, the course will detail the incorporation of the presented techniques into integral equation codes and their interaction with fast matrix-vector multiplication schemes.

 

HALF DAY COURSES

Sunday, July 8, 2012

 

OTA Measurements in Reverberation Chamber

Per-Simon Kildal, Chalmers University of Technology, Sweden, Charlie Orlenius, Bluetest AB
Swedenper-simon.kildal@chalmers.se, Charlie.orlenius@bluetest.se

The reverberation chamber has during the last 10 years been developed into a fast, accurate and cost-effective instrument for emulating rich isotropic multipath and thereby characterizing small antennas and wireless terminals Over-The-Air (OTA) during fading. The course will explain how the chamber works, and give formulas for the average transfer function as well as the uncertainty by which we can measure efficiency, diversity gain, maximum available MIMO capacity, total radiated power and receiver sensitivity. The latter can be obtained both as total isotropic sensitivity (TIS) and average fading sensitivity (AFS), the latter measured during continuous fading. Most of the course will be devoted to throughput measurements of complete systems with MIMO capability, such as for WLAN 802.11n, LTE and WiMAX. The course will also explain how the time and frequency domain characteristics of the chamber (Doppler spread, time delay spread, coherence bandwidth) can be determined, and controlled to become similar to those in real-life environments, so that both frequency-selective and frequency-flat fading can be tested. A new simple theoretical model for throughput of LTE device with MIMO and OFDM frequency diversity will be presented, showing perfect agreement with measurements. The course will also describe why the rich isotropic environment with Rayleigh fading is representative also for characterizing performance in non-rich multipath environments, even with the presence of Line-Of-Sight (LOS), provided the statistics of the user is taken into account.

 

Phased Array for Microwave and Millimeter-wave Applications

Mohammad Fakharzadeh, Peraso Technologies Inc., Canada, Safieddin Safavi-Naeini, University of Waterloo, Canada
Mohammad@perasotech.com, safavi@maxwell.uwaterloo.ca

Despite its impressive potentials and properties, phased array antennas (PAA) have not become a commercial product yet. Cost and complexity of PAA are beyond the scales of consumer electronics devices. In this short course, we discuss the advantages and disadvantages of different PAA configurations for portable micro and millimeter wave application. Furthermore, the methodology of designing an effective beamforming algorithm and the required hardware components will be discussed. This course demonstrates that a powerful, robust beamforming algorithm, integrated in efficient single-receiver architecture, constitutes the core of a low-cost PAA. It is shown, that an efficient beamforming algorithm can compensate for the hardware errors and imperfections of the low-cost components of the system. Finally, some samples of the state-of-art phased arrays will be presented.

 

Metamaterials and Plasmonic Materials: Theory and Electromagnetic Applications

Andrea Alù, University of Texas, USA
alu@mail.utexas.edu

This short course will provide a broad introduction to the field of metamaterials and plasmonics, covering a wide range of topics, from the theoretical approaches to study anomalous wave propagation in periodic arrays, to their application in a variety of fields of research and over a wide spectrum of frequencies. The course will focus on various exciting properties of metamaterials and plasmonic materials, from their theoretical and numerical modeling and rigorous homogenization principles, to their practical application to realize electromagnetic cloaks, negative-index materials, sub-diffractive waveguides and antennas, enhanced nonlinearities, sensing, imaging and energy harvesting devices, optical nanocircuits and nanoantennas and various other exciting applications. I will discuss in detail how the metamaterial concepts may be applied to various electromagnetic problems of interest to overcome current technological challenges and provide a breakthrough in applied fields related to electromagnetics, radio-science and optics.

 

Use of the Principle of Analytic Continuation for the generation of phase and for interpolation/extrapolation of amplitude only data

Tapan K Sarkar, Syracuse University, USA, Magdalena Salazar Palma, University of Carlos III, Spain
tksarkar@syr.edu, salazar@tsc.uc3m.es

The objective of this course is to illustrate the application of a well known but not often used property of a linear time invariant system, which is causality. In nature all, time domain responses are causal. This powerful principle can be utilized in application of numerical techniques to improve the performance of numerical methods or enhance data processing using this important principle. For example, the causality principle implies that the real and the imaginary parts of a transfer function are not really independent but dependent on each other and related by the Hilbert transform. This principle can be used for example to generate the non minimum phase function from an amplitude only data. However, the reconstruction of the phase from amplitude only data has a non unique solution as the starting point of the transient pulse can be delayed without changing its amplitude spectrum. In addition, this principle can also be used to interpolate/extrapolate amplitude only data without knowing explicitly the phase. Now that if one can generate the phase function from amplitude only data, it is then possible to generate the time domain response from amplitude only data. Numerical techniques with examples will be presented to illustrate the applicability of these principles to practical problems in antenna characterization and improving the computational efficiency of numerical methods.

  

Antenna Design and Challenges in Cognitive Radio

Christos Christodoulou, Youssef Tawk, University of New Mexico, USA
christos@ece.unm.edu, yatawk@ece.unm.edu

A Cognitive Radio (CR) system, built on a Software Defined Radio (SDR) platform, aims to improve spectral and power utilization by dynamically interacting with any RF environment. The two main objectives of cognitive radios are to ensure highly reliable communication links whenever and wherever needed (in space and terrestrial applications), and to efficiently utilize the radio spectrum. In order to meet these requirements, cognitive radio systems are expected to use “machine learning” techniques to software-control not only the antenna structure but the entire radio system to sense the channel activity over a wide range of frequencies, tune the radiation and communication characteristics accordingly, and self-learn from the past experience. The aim of this short course is to introduce the subject of cognitive radio and the role that antennas can play now and in the future. The outline of the course is as follows: 1- Introduction to cognitive radio and its difference from software defined radio 2- Classification of different cognitive radio communication systems 3- Antenna design requirements and limitations 4- Wideband sensing antennas 5- Use of reconfigurable antennas 6- Different reconfiguration techniques (electrical/optical/mechanical/material based) 7- FPGA and software controlled reconfigurable antennas 8- Reconfigurable antennas versus reconfigurable filtennas (filter+antenna) 9- Machine learning algorithms for cognitive radio 10- Concluding remarks

 

Reflectarray Antennas: Design, Reconfigurability And Potential Applications

Jose A. Encinar, Polytechnic University of Madrid, Spain, Sean V. Hum, University of Toronto, Canada, Julien Perruisseau-Carrier, EPFL, Switzerland
Jose.encinar@upm.es, svhum@waves.utoronto.ca, julien.perruisseau-carrier@epfl.ch

This course gives a comprehensive overview of reflectarrays and provides valuable guidance in designing these antennas for practical applications. First, an overview of reflectarray antennas is presented, including various implementations and their most significant features. Several design techniques will be covered for reflectarray antennas, addressing bandwidth improvement, contoured beams and dual-reflector configurations. Different concepts will be shown for electronically reconfiguring or scanning the beam in reflect- and transmit-array antennas, by using unit-cells with MEMS, PIN or varactor diodes. The control elements can be either embedded in the resonant elements or in delay-lines coupled to the radiating elements. Both approaches will be extensively covered, including their accurate modeling and practical implementations. Finally, some recent developments for passive and reconfigurable reflectarrays will be illustrated. Examples such as stringent contour-beam antennas for Direct Broadcast Satellite, multi-beam reflectarrays for point-to-multi-point applications, dual-reflector configurations, beam scanning reflectarrays and transmitarrays will be presented in detail.

 

Adaptive Arrays

Randy L. Haupt, Mark Leifer, Ball Aerospace, USA
rhaupt@ball.com, mleifer@ball.com

Adaptive arrays improve the reception of desired signals in the presence of interference signals in radar, sonar, seismic, and communications systems. They automatically sense the presence of interference and suppress them while simultaneously enhancing desired signal reception without prior knowledge of the signal/interference environment. Adaptive arrays are designed to complement other interference suppression techniques, such as low sidelobes, spread-spectrum techniques, and high directivity. This short course is broken into three parts: (1) Fundamentals – arrays, digital beamforming, signals, terminology, (2) Algorithms – gradient based, direction inversion of the covariance matrix, random search, and non-digital beamforming approaches, and 3) Advanced concepts – array calibration and compensation, multipath, MIMO, and reconfigurable arrays. The material in the course is based on the two books: R. A. Monzingo, R. L. Haupt, and T.W. Miller, Introduction to Adaptive Arrays, 2ed., SciTech Publishing, 2010. R.L Haupt, Antenna Arrays: A Computational Approach, New York: Wiley, 2010.

 

Internal Handheld Device Antennas: Recent Advances, Future Perspectives, and Design Considerations for Bio-Compatible Handheld Device Antennas

Kin-Lu Wong, National Sun Yat-sen University, Taiwan
wongkl@mail.nsysu.edu.tw

Internal handheld device (handset and tablet PC) antennas have recently shown great advances for WWAN operation including the GSM850/900/1800/1900/UMTS bands. The traditional antenna design was mainly based on the λ/4 PIFA. Recently, a variety of planar antennas including the printed λ/8 PIFA, λ/4 loop, λ/8 half-loop, and λ/4 slot antennas have been demonstrated to be promising for achieving much smaller antenna size yet wideband operation to cover five-band WWAN or eight-band WWAN/LTE operation. Compact integration of the internal WWAN/LTE antennas with nearby system ground plane and electronic elements in the handset is also promising. Small-size LTE MIMO array can also be disposed in a very limited space with good isolation inside the handset. These recent advances and the applied techniques in achieving wideband/multiband operation with small antenna size will be presented. The internal antennas with high efficiency yet low near-field radiation to achieve acceptable HAC (hearing-aid compatibility) values and Head/Hand or Body SAR (specific absorption rate) values have been demanded for practical applications. Such antennas with acceptable HAC/SAR values or even much lower near-field radiation with respect to human exposure are treated as bio-compatible handheld device antennas. The related near-field radiation theory will be addressed, and the design techniques for the bio-compatible internal handheld device antennas will be presented. Future trends for handheld device antennas will also be discussed.

 

EM Design of Small Antennas for Industrial Applications

Winfried Simon, IMST, Germany, Dirk Manteuffel, University of Kiel, Cyril Luxey, University Nice-Sophia Antipolis, France
simon@imst.com, manteuffel@tf.uni-kiel.de, Cyril.LUXEY@unice.fr

The design of antennas for small devices like PDA phones, laptop computers and WLAN objects is driven by more issues than only the antenna concept. The antenna design is one part in a whole chain of developments. During the design stages of the product, the antenna designer should be able to anticipate the influence of possible changes in the product specifications and be flexible to adapt the antenna concept accordingly. EM simulations enable the antenna designer to investigate these issues at an early stage where no hardware prototypes are available. This short course aims on preparing the participants to use numerical EM software for the efficient design of antennas for commercial applications. The course will consist of a well balanced amount of lectures and practical design work using state of the art design software.

 

Advanced impedance matching and impedance analysis for antenna applications

Jussi Rahola, Optenni Ltd, Finland
jussi.rahola@optenni.com

For many antenna applications, the impedance matching can be done much faster by designing matching circuits than by modifying the antenna geometry. This course starts by reviewing the basics of impedance matching using inductors, capacitors and transmission lines. Two definitions of waves in microwave engineering are reviewed: the traveling waves for the analysis of physical wave amplitudes and the power waves for the analysis of the propagation of power. Practical issues for the design of efficient and wideband matching circuits are discussed, including component losses and tolerances. In addition, the concept of bandwidth potential and antenna Q estimation are presented for the estimation of the obtainable bandwidth of antennas through matching circuits. Finally, the concept of electromagnetic isolation is presented for studying the worst-case isolation (independent of antenna matching) in two-antenna systems.

 

Antenna Design for Mobile Terminals

Zhijun Zhang, Tsinghua University, Beijing, China
zjzh@tsinghua.edu.cn

The lecturer is an antenna engineer turned professor who has worked at Apple, Nokia and Amphenol. This is a comprehensive course for fresh and intermediate engineers involved in antenna design. The course instructs attendees through all aspects of real world antenna designs, which includes how to make an stable antenna fixture, designing various types of antennas, designing an antenna with good manufacturability, using various matching technique to improve antenna performance, setting up production measurement for mass manufacturing, and making antenna SAR and HAC Compliant. Most popular antenna categories, such as internal PIFA, integral IFA, internal folded monopole, ceramic antennas, stubby antennas and whip stubby antennas, are introduced in the course. The course focuses on the basic principle of each kind of antenna and emphasizes on key parameters of antenna optimization. Complimentary matching software, which accompanies with the course, is provided so readers can practice various antenna matching technique and design matching circuits for real projects.

 

Antenna Applications of Metamaterials and Transformation Electromagnetics

Yang Hao, Queen Mary University, UK, Raj Mittra, Pennsylvania State University, USA
yang.hao@eecs.qmul.ac.uk, mittra@engr.psu.edu

Transformation electromagnetics (TE), an idea behind the invisibility cloak, has caught both the scientific and public imagination, and has stimulated a huge growth in related research around the world. The potential of the underlying TE approaches however have much wider applicability than cloaking alone, in arguably more important applications in electromagnetics and microwave engineering. However, theory and concepts are outstripping practical demonstration and testing, leading to a mismatch in what may be theorised and computed and what can be realised for impact in society and commerce. In this short course, we will introduce fundamentals of the TE approach, modeling tools, design examples and experimental results with a clear focus on the reduction to practice. Specifically, we will extend these techniques into the design of novel antennas and controlling surface wave propagation In particular, the tradeoff in the performance of the proposed designs with or without metamaterials will be discussed.

 

Saturday, July 14, 2012

History, Design and Theory of Directive Fabry-Pérot Cavity Antennas as Leaky-Wave Radiators

Filippo Capolino, University of California, Irvine, USA, David Jackson, University of Houston, USA
f.capolino@uci.edudjackson@uh.edu

Fabry-Pérot Cavity (FPC) antennas are receiving increasing attention as a possible design solution for applications that require very directive antennas. They were first conceived of decades ago, and have more recently been related to leaky-wave antennas, as well as to certain EBG and metamaterial antennas. They offer significant advantages for some applications in terms of being planar and somewhat low-profile, having simplicity of design and fabrication, and achieving a high radiation efficiency. They exhibit moderate to large gain values (typically 15-30 dB) but usually have a fairly narrow bandwidth (with the bandwidth typically decreasing as the gain increases). A significant design advantage relies on the fact that they are fed by a single or perhaps a handful of elements, even for very large gain values, with a planar guiding surface doing the beamforming. Current research trends are aimed at enlarging the bandwidth and well as decreasing the height of the structure to make it lower in profile. This short course is devoted to the FPC antenna, providing a complete history of the antenna as well as explaining the principles of operation and deriving simple CAD formulas for initial designs. Recent results from various researchers will be shown, illustrating the current research trends regarding bandwidth enhancement, miniaturization, and feeding strategies.

 

Advances in the Design of Electrically Small Antennas

Steven R. Best, MITRE, USA
sbest@mitre.org

Optimization of the performance properties of electrically small antennas represents a challenging design problem for the antenna engineer. As wireless devices decrease in size, there is an increasing demand for physically smaller antennas, yet the performance requirements are rarely relaxed. This 1/2-day short course provides a detailed discussion on the theory, challenges, performance trade-offs and design approaches associated with electrically small antennas. The short course begins with an overview of the fundamental theory and inherent limitations of small antennas. The presentation focuses on providing an understanding of small antenna performance in terms of impedance, radiation patterns, bandwidth, radiation efficiency, matching efficiency, and quality factor (Q). Techniques used to design self-resonant and impedance matched electrically small antennas are described and compared. The relationships between the small antenna’s performance properties and its physical characteristics are discussed in detail. The performance of the small antenna on small finite ground planes is considered with a particular emphasis on how the antenna’s location on the ground plane affects impedance, pattern and polarization properties. This short course also presents and describes practical approaches for the design of wireless device antennas. These discussions include an understanding of the basic theory of these designs, equivalent circuit analysis, and ground plane effects.

 

Multibeam Antennas and Beamforming Networks

Piero Angeletti, Giovanni Toso EUROPEAN SPACE AGENCY (ESA), The Netherlands
piero.angeletti@esa.int, giovanni.toso@esa.int

The objective of this course consists in presenting the state of the art and the on-going developments in Multi-Beam Antennas (MBAs) and Beam-Forming Networks (BFNs). MBAs find application in several fields including communications, remote sensing (e.g. radars, radiometers, etc.), electronic surveillance and defense systems, science (e.g. multibeam radio telescopes), RF navigation systems, etc. The BFN plays an essential role in any antenna system relaying on a set of radiating elements to generate a beam. The course will cover both theoretical and practical aspects for the following topics: 1. Overview of system requirements. 2. Multibeam Antennas 2.1 Linear and Planar Direct Radiating Arrays (based on Periodic or Aperiodic lattices) 2.2 Reflector-based architectures (Single-Feed-per-Beam, Multiple-Feed-per-Beam) 2.3 Lens-based architectures (free space and constrained) 3 Beamforming Networks 3.1 Analogue BFNs (Corporate, Blass, Nolen, Butler matrices) 3.2 Digital BFNs 4.RF Technology for Active Components 4.1 Low Noise Amplifiers (LNAs, High Power Amplifiers (HPAs), T/R Modules, etc. 5 Overview of some Operational Multibeam Antennas/BFNs 5.1 MBAs for spaceborne Narrowband and Broadband Satellite Communication Systems 5.2 MBAs for Wireless Communications 5.3 On-going European Developments 6. Current Design and Technological Challenges

 

Natural and metamaterial antennas with emphasis on realization of wideband characteristics

Hisamatsu Nakano, Hosei University Koganei, Japan
hymat@hosei.ac.jp

Electromagnetic properties in nature are right-handed. In contrast, a metamaterial (MTM) has both left- and right-handed properties. Chapter 1 describes the historical background of natural and MTM antennas, focusing on their wideband antenna characteristics. Chapter 2 summarizes the analysis methods for these antennas. Chapter 3 presents the radiation characteristics of several representative antennas, including (1) linearly- and circularly-polarized grid arrays, (2) a high-gain FSS antenna system, (3) a high-gain dielectric rod, (4) a light-weight high-gain cigar element, (5) curl and spiral with an EBG reflector, (6) an inverted F element with an EBG reflector, (7) an MTM strip-line, (8) an MTM curl, and (9) an MTM spiral. Note that antennas (1)-(4) belong to the natural antenna category and antennas (5)-(9) belong to the MTM antenna category. Note also that grids, the FSS system, and the MTM strip-line can have a tilted beam whose direction varies with the operating frequency; the rod and cigar elements operate as end-fire antennas; the curl, spiral, and inverted F elements above an EBG reflector have extremely low-profile structures; and the MTM curl and spiral are designed to be dual circularly-polarized antennas.

 

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