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3. DATES COVERED (From - To) 9 September 2003 -12 September 2003

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2003 Intemational Conference on Antenna Theory and Techniques (ICATT)

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The Final Proceedings for 2003 Intemational Conference on Antenna Theory and Techniques (ICATT), 9 September 2003 -12 September 2003 This is an interdisciplinary conference. Topics include: 1. General antenna theory; 2. Reflector, lens and hybrid antennas; 3. Antenna an-ays; 4. Adaptive antennas, signal processing; 5. Broadband and multi-frequency antennas; 6. Low-gain, printed antennas 7. Antennas for mobile communication; 8. Antennas for remote sensing; 9. Antenna measurements; 10. Analytical and numerical methods; 11. Microwave components and circuits fiber-optics links; 12. Industrial and medical applications of microwave technologies 13. Electromagnetic compatibility; 14. Antenna radomes and absorbers; 15. Electromagnetism at the high school 15. SUBJECT TERMS \ BOARD, Electromagnetics, Antennas

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Antenna Theory and Techniques Volume 1 Organizers National Antenna Association (NAA) of Ukraine Kharkiv National University of Radio Electronics Karazin Kharkiv National University Sevastopol National Technical University National Technical University of Ukraine "Kyiv Polytechnic Institute" Academy of Sciences of Applied Radio Electronics IEEE AP/C/EMC/SP Kharkiv Joint Chapter of the Ukraine Section Ministry of Education and Science of Ukraine

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2003 International Conference on Antenna Theory and Techniques

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International Conference on Antenna Theor)- and Techniques, 9-12 September, 2003, Sevastopol, Ukraine

ICATT'03 CHAIRMAN Yakov S. Shifrin, Ukraine ICATT'03 TECHNICAL PROGRAM COMMIHEE Co-Chairmen Nicolay N. Kolchigin (Ukraine) Fedor F. Dubrovka (Ukraine)

Members Adrian Alden (Canada) Lev D. Bakhrakh (Russia) Alex B. Gershman (Canada) Nicolay N. Gorobets (Ukraine) David Jackson (USA) Peter Edenhofer (Germany) Elya Joffe (Israel) Victor Kravchenko (Russia) Kees van't Klooster (the Netherlands) Aleksander A. Konovalenko (Ukraine) David I. Lekhvitskiy (Ukraine) Anatoly S. Ilinskiy(Russia) Leo Ligthart (the Netherlands)

Lyubov M. Lobkova (Ukraine) Konstantin A. Lukin (Ukraine) Sergey A. Masalov (Ukraine) Josef Modelski (Poland) Germadiy A. Morozov (Russia) Dimitry M. Sazonov (Russia) Hiroshi Shigesawa (Japan) Tadashi Takano (Japan) Dimitry I. Voskresensky (Russia) Felix Yanovskiy (Ukraine) Yuriy Yukhanov (Russia) Vadim I. Zamyatin (Ukraine) Wen Xun Zhang (China)

ICATT'03 ORGANIZING COMMITTEE Co-Chairmen Mukhailo F. Bondarenko (Ukraine) Victor A. Karpenko (Ukraine)

Members Aleksander G. Luk'yanchuk Vladimir M. Shokalo (Ukraine) Victor A. Katrich (Ukraine) Vladimir I. Karpenko (Ukraine) Aleksander I. Dokhov (Ukraine) Gennadiy I. Churyumov (Ukraine) Mikhail B. Egorov (Ukraine) Marianna V. Ivashina (the Netherlands)

Vyacheslav V. Khardikov (Ukraine) Nina G. Maksimova (Ukraine) Vladimir I. Pravda (Ukraine) Yaroslav O. Rospopa(Ukraine) Anna V. Shishkova (Ukraine) Vladimir. G. Syrotyuk (Ukraine) Peter L. Tokarsky (Ukraine)

International Conference on Antenna Theor)^ and Techniques, 9-12 September, 2003, Sevastopol, Ukraine

Welcome to ICATT'2003! During four days (9-12 September, 2003) Sevastopol will be a host of the International Conference on Antenna Theory and Techniques (ICATT'03). Since 1995 it is the fourth meeting of such a kind in Ukraine. To a certain degree, ICATT is a successor of the regularly held earlier in the USSR antenna conferences organized and unchangeably chaired by Aleksander A.Pistol'kors - the widely recognized head of the Soviet antenna school. It should be noted that notwithstanding hard times that Ukraine and Russia go through now, the antenna science keeps actively advancing. The evidence of this is the present conference, whose scale remarkably exceeds that of previous ICATTs in the number of presented papers, the number of participants and the number of guests from "far and near abroad". Over 220 papers have been submitted to the ICATT'03 Program Committee from Ukraine, Russia, Mexico, the Netherlands, Japan, Italy, USA, Canada Belarus, Azerbaijan, Ireland, Denmark, France, Austria, Poland, China, Corea. These papers are to be presented at three morning plenary sessions and at afternoon meetings of 14 sessions. Special attention was paid to selection of invited speakers among the recognized antenna scientists from different countries. Undoubtedly, their review presentations in a number of the modern applied electromagnetics directions will be met by the ICATT'03 participants with a great interest and will be especially useful for young scientists. I am also sure that with a great interest the conference participants will attend on tlie 12'^ of September the Ukraine National Centre of Space Vehicles Control and Tests in Eupatoria, where they can become acquainted with a number of unique large antennas. All this gives me a confidence that ICATT'03 will be fruitful and stimulative for a further progress of antenna science and engineering. ICATT03 is being held in Sevastopol rich with many monuments reflecting ancient and heroic history of this wonderful maritime city. The ICATT'03 attendees will take also a chance to seeing sights of the Southern Coast of the Crimea. There are plamied also other exciting social events. All this will additionally adorn your stay in Sevastopol. I would like to thank all organizers, many my colleagues who performed a bulky work on the conference preparation and publishing its Proceedings, and of course all the participants who have made ICATT'03 possible. I would like also to thank our sponsors (especially European Office of Aerospace Research and Development of the USAF) for their financial contribution to the ICATT'03 organization and support of young scientists. I wish all the ICATT'03 participants a successful work at the conference meetings, pleasant contacts with colleagues and nice time in the sunny Crimea on the shore of the warm Black Sea. ij- «#"*,„ sin 9 A And the normalized radius is, 27rr P = A computer code was applied to compute the secondary pattern characteristic produced by a uniform, cosine raised to a power n , cosine on a pedestal P, and parabolic raised to power n distributions. The results shown in Table 2. The important radiation values of half power beamwidth, null to null beamwidth and position of the first sidelobes relative to the main lobe are found from the three parameters (Ki, K-j and Kz) in this Table by formulas similar to (4) through (6), but replacing the linear aperture dimension in wavelengths with the circular aperture diameter in wavelengths. That is:

D,JX'

(11)

On ''si —

«-•

(10)

D.JX'

(12)

D,„ / A = Aperture diameter in wavelengths.

5. Fig. 3. Coordinate system used to analyze circular aperture of diameter D„

MODERN FULL-WAVE METHODS There are aperture antennas that can be advantageously addressed with analysis approaches known as full-wave methods. The application of such methods

International Conference on Antenna Theory and Techniques, 9-12 September, 2003, Sevastopol, Ukraine

27

D. J. Kozakoff and V. Tripp

Table 2. Radiation characteristics of various circular aperture distributions Type of Comments Parameter Parameter Parameter distribution K2 Uniform Cosine raised to power " n "

Cosine on pedestal" P "

Parabolic raised to power" n "

n= 1 n= 2 n= 3 n= 4 n= 5 P = 0.0 P = 0.1 P= 0.2 P = 0.3 P = 0.4 P = 0.5 n = 0 n=r 1 n= 2 n= 3 n = 4

59.33 74.67 88.00 99.33 110.00 120.00 74.67 70.67 68.67 66.00 64.67 63.33 59.33 72.67 84.67 94.67 104.00

140.00 194.67 250.00 306.67 362.67 420.00 194.67 183.33 174.00 166.00 159.60 154.67 140.00 187.33 232.67 277.20 320.33

has rapidly expanded with the explosion of high power PC computers in recent decades. Analysis methods are called full-wave when they start with the fundamental equations of electromagnetics and discretize them such that they can be reduced to linear matrix equations suitable for solving by a computer. The advantage is that there are no approximations in principle, only the size of the discrete interval, which is usually between 10 and 20 intervals per wavelength. There are three primary full-wave methods used in electromagnetics; the finite element method (FEM) [15-18], the method of moments (MOM) [19-21], and the finite difference time domain (FDTD) method [22, 23]. The MOM discretizes Maxwell's wave equations in their integral form, the FDM discretizes the equations in the differential form, and the FEM method discretizes the equations after casting them in a variational form. All three techniques have been applied to aperture antenna analysis [24-26]. The MOM method finds natural application to antennas because it is based on surfaces and currents, whereas the other two methods are based on volumes and fields. This means that for MOM, only the antenna aperture surface structure must be discretized and solved, whereas for FEM and FDTD, all volumes of interest must be discretized. For antenna radiation, the far field would require an inordinate amount of space were it not for the recent development of absorbing boundary conditions. These boundary conditions approximate the radiation conditions of infinite distance in the space very near the radiating structure. The MOM works by solving for currents on all surfaces in the presence of a source current or field. The radiated field is then obtained by integration of these currents, much like it was obtained in the physical optics approaches. Thus, the MOM can be applied to any aperture antenna that the PO technique can be applied to, unless the problem is too large for the 28

93.67 119.33 145.50 173.00 200.30 228.17 119.33 112.67 107.83 104.17 99.47 97.83 93.67 116.33 138.67 160.17 181.33

1" Sidelobe Level, dB. -17.66 -26.07 -33.90 -41.34 -48.51 -55.50 -26.07 -25.61 -24.44 -23.12 -21.91 -20.85 -17.66 -24.67 -30.61 -35.96 -40.91

Peak Gain Relative to Uniform 0.0 -1.42 -2.89 -4.04 -4.96 -5.73 -1.42 -0.98 -0.66 -043 -0.27 -0.17 0.0 -1.24 -2.55 -3.58 -4.43

available computer resources. Ensemble [27] is commercially available software package that is a 2.5dimensional MOM program used primarily for patch antennas or antennas that can be modeled as layers of dielectrics and conductors. If the top layer is a conductor with radiating holes, the holes are aperture antennas, which this program is designed to analyze. There is another full-wave commercial software packages that are widely used for aperture antenna problems, the High Frequency Structure Simulator (HFSS) [28]. This is a 3-dimensional FEM software package with extensive modeling and automatic meshing capability. It is best for horn antennas or other kinds of antennas formed by apertures in various non-layered structures. The latest version uses the "perfectly matched layer" type of absorbing boundary' conditions. In practice, full-wave methods cannot be directly applied to high-gain aperture antennas like reflectors or lenses without difficulties because these structures are usually many wavelengths in size which requires large amount of computational resources. Often, however, if there is symmetry in the problem that can be exploited, the number of unknowns for which to solve can be greatly reduced. For instance, a high gain reflector antenna that has circular symmetr>' allows for body-of-revolution (BOR) symmetry [29. 30] simplifications in the modeling. Similarly, a large lens requires a computer program with dielectric capability [31] in addition to BOR symmetry modeling. REFERENCES

D. J. Kozakoff, Analysis of Radome Enclosed Antennas, Artech House Publishers, Norwood, MA, 1997. S. A. Schelkunoff, Some equivalence theorems of electromagnetics and their application to radiation

Internarional Conference on Antenna Thcorj' and Techniques, 9-12 September, 2003, Sevastopol, Ukraine

Aperture Antenna Radiation problems, Bell System Technical Journal, 15: 92112,1936. 3. C. Huygens, Traite de La Lumiere, Leyden, 1690. Translated into English by S. P. Thompson, Chicago, IL, University of Chicago Press, 1912. ^^ 4. J. D. Kraus and K. R. Carter, Electromagnetics, 2" edition, New York, McGraw-Hill Book Company, pp.464-467. 5. A. Sommerfeld, Theorie der Beugung, in P. Frank and R. von Mises (Editors), Die Differential und Integralgleichungen der Mechaniik und Physik, Braunschweig, Germany: Vieweg, 1935. 6. C. A. Balinis, Antenna Theory Analysis and Design, New York: Harper and Row Publishers, 1982. 7. A. D. Oliver, Basic Properties of Antennas, in A. W. Rudge, et al, (Editors), The Handbook of Antenna Design, lEE Electromagnetic Wave Series UK, London: UK, Peter Peregrinum Publishers, 1986, Chapter 1. 8. H. Jasik, Fundamentals of Antennas, in R. C.Johnson (Editor), Antenna Engineering Handbook, 3"* Edition, New York: McGraw Hill Book Company, 1993. 9. J.D. Kraus, Antennas, 2"'' Edition, New York: McGraw Hill Book Company, 1988. lO.H. G.Booker and P. C. Clemmow, The Concept of an Angular Spectrum of Plane Waves and its Relation to That of Polar Diagram and Aperture Distribution, Proceedings of the IEEE, London, UK,Ser.3,97: 11-17,1950. ll.D. R. Rhodes, The Optimum Line Source for the Best Mean Sqauare Approximation to a Given Radiation Pattern, IEEE Transactions on Antennas and Propagation, AP-11, 440-446, 1963. 12. R. S. Elliot, Antenna Theory and Design, Englewood Cliffs, NJ: Prentice Hall Book Company, 1987. 13.1. S. Sokolnikoff and R. M. Redhefer, Mathematics of Physics and Modern Engineering, New York: McGraw Hill Book Company, 1958. 14. S. Silver, Microwave Antenna Theory and Design, New York: McGraw Hill Book Company, 1949. 15. J. L. Volakis, A. Chatterjee and L. C. Kempel, Finite Element Methods for Electromagnetics, IEEE Press, New York (ISBN 0-7803-3424-6) and Oxford University Press, London (0-10-850479-9) 1998. 16. P. P. Sylvester and G. Pelosi, Editors, Finite Elements for Wave Electromagnetics: Methods and Techniques, IEEE Press, New York, 1994. 17.P. P. Sylvester and R. L. Ferrari, Finite Elements for Electrical Engineers, Cambridge University press. New York, 1992.

18. J. Jin, The Finite Element Method in Electromagnetics, John Wiley InterScience, New York, 1993. 19.R. F.Harrington, Field Computation by Moment Methods, Macmillan Company, New York, 1968. 20. E. K. Miller, L. Medgyesi-Mitschang, E. H. Newman, Editors, Computational Electromagnetics, Frequency Domain Method of Moments, IEEE Press, New York, 1992. 21.R. C. Hansen, Editor, Moment Methods in Antennas and Scattering, Artech House, Norwood, MA (ISGN 0890064660), 1990. 22.A. Taflove, Computational Electromagnetics: The Finite Difference Time Domain Method, Artech House, Norwood, MA, 1995. 23. K. S. Kunz, R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics, CRC Press, Cleveland, OH (ISBM 0949386578), 1993. 24. D.Chun, R.N.Simons and L.P. B. Kotehi, Modeling and Characterization of Cavity Backed Circular Aperture Antenna with Suspended Stripline Probe Feed, IEEE Antennas and Propagation Society International Symposium, 2000 Digest, Salt Lake City, UT, 2000. 25. J. Y. Lee, T. S. Homg, N. G. Alexopoulos, Analysis of Cavity Backed Aperture Antennas with a Dielectric Overlay, IEEE Transactions on Antennas and Propagation, Vol. 42, No.ll, November 1994, pp. 1556-1562. 26. D.Sullivan and J. L. Young, Far-Field TimeDomain Calculation from Aperture Radiators Using the FDTD Method, IEEE Transactions Antennas and Propagation, Vol.49, No.3, March 2001, pp.464-469. 27. Ensemble Software, Version 6.1, Ansoft Corporation, Four Station Square, Suite 200, Pittsburgh, PA 15219, Tel. 412-261-3200. 28 HFSS, Ansoft Corporation, Four Station Square, Suite 200, Pittsburgh, PA 15219, Tel. 412-261-3200. 29. Z. Altman and R. Mittra, Combining an Extrapolation Technique with the method of Moments for Solving Large Scattering Problems Involving Bodies of Revolution, IEEE Transactions on Antennas and Propagation, Volume 44, No. 4, April 1996, pp. 548-553. 30. A. D. Greenwood and J.M.Jin, Finite Element Analysis of Complex Axisymmetric Radiating Structures, IEEE Transactions on Antennas and Propagation, Vol.47, No.8, August 1999, p.l260. 31. J. M. Putnam and L. N. Medgyesi-Mitschang, Combined Field Integral Equation for Inhomogeneous Two- and Three-Dimensional Bodies: The Junction Problem, IEEE Transactions on Antennas and Propagation, Vol.39, No.5, May 1991, pp.667-672.

International Conference on Antenna Theory and Techniques, 9-12 September, 2003, Sevastopol, Ukraine

29

International Conference on Antenna Thcoty and Techniques, 9-12 September, 2003, Sevastopol, Ukraine

pp. 30-35

ROBUST ADAPTIVE BEAMFORMING: AN OVERVIEW OF RECENT TRENDS AND ADVANCES IN THE FIELD A. B. Gershman Smart Antenna Research Team (SmART) Department of Communication Systems Gerhard-Mercator University, Duisburg. Germany (on leave from the Communication Research Laborator>' Department of Electrical and Computer Engineering McMaster University, Hamilton, Ontario. Canada)

ABSTRACT In recent decades, adaptive arrays have been widely used in sonar, radar, wireless communications, microphone array speech processing, medical imaging and other fields. In practical array systems, traditional adaptive bcamforming algorithms are known to degrade if some of exploited assumptions on the environment, sources, or antenna array become wrong or imprecise. Therefore, the robustness of adaptive beamforming techniques against environmental and array imperfections and uncertainties is one of the key issues. In this paper, we present an overview of recent trends and advances in the field of robust adaptive bcamforming. Keywords: Robust adaptive beamforming, worst-case performance optimization, diagonal loading, array respon.sc mismatch, uncertainty set.

1.

INTRODUCTION

The traditional approach to the design of adaptive beamformers assumes that no components of the desired signal are present in the beamformer training data [1], [2]. In such a case, adaptive beamforming is known to be sufficiently robust against errors in the array response to the desired signal and limited training sample size and a variety of rapidly converging techniques have been developed for this case [1]. Although the assumption of signal-free training snapshots may be relevant in certain specific cases (e.g., in some radar and active sonar problems), there are many applications where the interference and noise observations are always "contaminated" by the signal component. Typical examples of such applications include wireless communications, passive sonar, microphone array speech processing, and medical imaging. It is well known that even in the ideal case where the signal steering vector (array response) is precisely known at the receiving sensor array, the presence of the desired signal in the training data snapshots can lead to Supported in parts by the Wolfgang Paul Award Program of the Alexander von Humboldt Foundation (Gemiany) and German Ministr\- of Education and Research, the Premier Research Excellence Award Program of the Minislr>' of Energy, Science, and Technology (MEST) of Ontario, and the Natural Sciences and Engineering Research Council (NSERC) of Canada. 0-7803-7881-4/03/$17.00 ©2003 IEEE.

essentially reduced convergence rates of adaptive beam-forming algorithms relative to the signal-free training data case [3], [4], This may cause a severe performance degradation of adaptive beamforming techniques in scenarios with a small training sample size. In practical situations, the performance degradation of adaptive beamforming techniques may become even more substantial because of a possible violation of underlying assumptions on the environment, sources, or sensor array. One of typical causes of such a performance degradation is a mismatch between the nominal (presumed) and actual array responses to the desired signal. Adaptive array techniques are known to be very sensitive even to slight errors of this type because in the presence of such errors adaptive beamformers tend to misinterpret the desired signal components in array observations as an interference and to suppress these components by means of adaptive nulling instead of maintaining distortionless response towards them [3], [4]. This phenomenon is of^en referred to as signal self-nulling. Errors in the array response to the desired signal frequently occur in practice because of look direction errors, imperfect array calibration (distorted array shape), as well as unknown environmental wavefront distortions, local scattering, and sensor mutual coupling. Another typical cause of array response errors in wireless communications is a restricted amount of

Robust Adaptive Beamforming: an Overview of Recent Trends and Advances in the Field

pilot symbols/intervals. In such cases, robust approaches to adaptive beamforming are required [3, 57]. Besides array response errors, performance degradation of adaptive beamforming can be additionally caused by a nonstationary character of the beamformer training data [8-10]. This phenomenon can be caused by a nonstationary behavior of the propagation channel, by interferer and antenna motion, as well as antenna vibration. There are several implications of such a nonstationarity. First of all, it naturally restricts the training sample size and leads to a degraded performance of adaptive beamforming algorithms even in the case of signal-free training snapshots. Furthermore, if the desired signal is present in the beamformer training snapshots, this type of degradation becomes much stronger than in the signal-free training data case [3, 4]. Finally, in the case of rapidly moving interferers the performance can break down because the array weights are not able to adapt fast enough to compensate for the interferer motion. Therefore, interference cancellation may be insufficient in such cases where robust approaches to adaptive beamforming are required [3, 5-7]. The same situation occurs in the case of moving antenna arrays, e.g., towed arrays of hydrophones in sonar [11] or moving antenna platforms in airborne applications [10].

2.

TRADITIONAL APPROACHES

The beamformer output is given by y{k) = w^ x{k),

(1)

where k is the time index, x(A;) is the M x 1 complex vector of array observations, w is the M xl complex vector of beamformer weights, M is the number of array sensors, and (o^ is the Hermitian transpose. The training snapshot vector is given by x(0 = s,(i) + i(i)+n(i), (2) where s,(i), i{t), and n(i) are the statistically independent components of the desired signal, interference, and sensor noise, respectively. In the particular case when the desired signal is a point source and has a time-invariant wavefront, we have s.,(i) = s(t)a,si as where s(i) is the complex signal waveform and as is the M X 1 signal steering vector. The optimal weight vector can be obtained through maximizing the Signal-to-Interference-plus-Noise Ratio (SINK) [1] SINR =

i^^R.w

w^Ri+„w'

(5)

are the M x M signal and interference-plus-noise co-variance matrices, respectively, and E{-} denotes the statistical expectation. Generally, the matrix Rj can have an arbitrary rank, i.e., l

(9)

where p{} is the operator which returns the principal eigenvector of a matrix. In the rank-one signal case, the solution (9) can be rewritten in a more familiar form [ I ] Wopt = aRf+n as,

(10)

wherea = (af Rf+„as)~^ In practical applications, the matrix Ri+„ is unavailable and is replaced in (9) and (10) by the sample covariance matrix [1]

^ - :^EL-W-^W = ^xx^ (11) where X = [x(l),x(2),...,x(Ar)] is the MxN matrix of the beamformer training data and N is the number of snapshots available (training sample size). One of the most popular approaches to robust adaptive beamforming in the cases of arbitrary signal array response errors and small training sample size is the diagonal loading method [5], [12], [13], [14]. Its key idea is to regularize the solution for the weight vector [13] by adding a quadratic penalty term to the objective function in (8) as min w^ Rw + 7w^ w s. t. w^ R^ w = 1, (12) w

(3)

where R3^E{s,(i)s/(i)}

Ri+n=E{[i(0 + n(0][i(t) + n(i)f}

(4)

where 7 is the loading factor. The solution to (12) is given by WDL=>{(R + 7l)-'Rs},

(13)

and

International Conference on Antenna Theory and Techniques, 9-12 September, 2003, Sevastopol, Ukraine

31

A. B. Gershman where I is the identity matrix. In the rank-one signal case, this solution can be written in a more familiar form [5], [13], [14] WDL

=(R + 7ir'a,.

(14)

It is well-known that diagonal loading can improve the adaptive beamforming performance in scenarios with an arbitrary signal array response mismatch [3], [5], [13]. However, the main shortcoming of this method is that there is no easy and reliable way of choosing the parameter 7 . Another popular approach to robust adaptive beamforming is the eigenspace-based beamformer [4], [15]. This approach is only applicable to the point signal source case. The key idea of this method is to use, instead of the presumed steering vector as, the projection of as onto the sample signal-plus-interference subspace. Write the eigendecomposition of (11) as R-EAE^-l-GfG^,

(15)

where the M x{L +1) matrix E contains the L + 1 signal-plus-interference subspace eigenvectors of R, and the (L -|-1) x (L -f-1) diagonal matrix A contains the corresponding eigenvalues of this matrix. Similarly, the M x{M - L-l) matrix G contains the {M - L -1) noise-subspace eigenvectors of R, while the (M - Z, - 1) x (M - L - 1) diagonal matrix r is built from the corresponding eigenvalues. The number of interfering sources L is assumed to be known. The weight vector of the eigenspace-based beamformer is given by = R-'v, where

v

PE«S

and

(16)

Pj, = E(E'^E)-'E^ =

- EE" is the orthogonal projection matrix onto the estimated signal-plus-interference subspace. When used in adequate situations, the eigenspacebased beamformer is known to be one of the most powerful techniques applicable to arbitrary steering vector mismatch case [15]. However, very serious shortcomings of this approach are that it is entirely based on the low-rank stationary model of the training data and requires exact knowledge of Z,. Furthermore, this approach is limited to high Signal-to-Noise-Ratio (SNR) cases because at low SNRs the estimation of the projection matrix onto the signal-plus-interference subspace breaks down due to a high probability of subspace swaps [16]. In situations with nonstationary training data, several advanced methods have been developed to mitigate performance degradation of adaptive beamformers. In [10], a power series expansion based approach is used to obtain a class of robust solutions for nonstationary array weights. However, the algorithms obtained are restricted by scenarios with slowly moving jammers only. To preserve the adaptive array performance in situations with a higher mobility of in32

terferers, several authors exploited the idea of artificial broadening the adaptive beampattem nulls in unknown interferer directions using either point [8], [9], [11] or derivative [17]-[20] null constraints. The first approach is usually referred to as the Data-Driven Constraints (DDC) method, while the second one is called the Matrix Taper (MT) method. Relationships between the DDC and MT approaches have been studied in [21] where it has been shown that the DDC method can be interpreted and implemented as a specific form of matrix taper.

3. RECENT ADVANCES As follows from the previous section, all traditional approaches to robust adaptive beamforming are ad hoc techniques. Only recently have some theoretically rigorous robust adaptive beamforming approaches been proposed in this field [16,22-28]. A common idea used in these algorithms is to define the so-called uncertainty sets and optimize worst-case performance. First of all, let us consider the non-point signal source case and discuss the robust beamformer derived in [24-25]. Following this work, we assume that both the signal and interference-plus-noise covariance matrices are known with some errors. That is, there is always a certain mismatch between the actual and presumed values of these matrices. This yields R, =R,-i^A^1'

(17)

^i+l -Ri+n +^2'

(18)

where R^ and Rj^^„ are the presumed signal and interference-plus-noise covariance matrices, respectively, whereas R^, and Rj^,, are their actual values. Here, A| and A2 are the unknown matrix mismatches. In the presence of the mismatches A, and Aj, equation (3) for the output SINR of an adaptive array has to be rewritten as SINR-.

W"R,W

(19)

W"RH„W

Let A| and Aj be norm-bounded by some known constants [24] |A,| e||w| + l [16], [26]. Taking this into account, we can rewrite (29) as minw^Rw s.t. Iw'^a,-11> eW .

(30)

w

The solution to (30) can be found by means of Lagrange multiplier method and can be expressed in the following form: (R + Ae^l) 'a,.(31)

w= Aaf (R-fAe'^l) ^ a, - 1

Equation (31) shows that the robust beamformer in (29) belongs to the class of diagonal loading techniques. Note, however, that it is not possible to use (31) directly for computing the optimal weight vector because it is not clear how to obtain the Lagrange multiplier ^ in a closed form. Several extensions of this approach have been considered. In [26], the robust formulation (27) has been extended to the case where the uncertainty is anisotropic [26]. Furthermore, the authors of [26] and [27] derived several alternative algorithms based on Newton's method rather than SOC programming. In [22], the approach of [16] has been extended to the case in which, besides the steering vector mismatch, there is a nonstationarity of the training data (which may be caused by nonstationary interferers and propagation channel, as well as antenna motion or vibration). The approach of [22] suggests, instead of modelling uncertainty in the covariance matrix (as in

International Conference on Antenna Theor}' and Techniques, 9-12 September, 2003, Sevastopol, Ukraine

33

A. B. Gershman (18)), to model such an uncertainty by means of adding it directly to the data matrix X. To take into account nonstationarity of the training data, let us introduce the mismatch matrix A=X-X,

(32)

where X and X are, respectively, the actual and presumed data matrices at the beamforming sample (which is often referred to as the test cell). The presumed data matrix X corresponds to the actually received data. However, in on-line problems, this data corresponds to the measurements that are made prior to the test cell. Therefore, because of nonstationarity effects, such past data samples may inadequately model the test cell, where the actual data matrix is unknown and equal to X . Hence, in the nonstationary case, the actual sample covariance matrix can be written as \XX" =^ ^(X+f)(X + f)"ir ■Jv"" ji'-^fj'^-^^f'

(33)

It is important to stress that, according to (33), the matrix R is guaranteed to be Hermitian and nonnegative definite. However, this matrix is unknown because the type of nonstationarity (and, therefore, the mismatch A) are unknown. To combine the robustness against interference nonstationarity and steering vector errors, the authors of [22] use ideas similar to that proposed in [16] and [26]. That is, they assume that the norms of both the steering vector mismatch 5 and the data matrix mismatch A are bounded by known constants: ||8||l,

(35) V||A|l'-

(24)

For condition (22b), such a sum, as is known [11], has the Erlang distribution p^, double SNR h (19c). On this bases, a sample of such T} = {K-M + 2)/{K +1) = (5 + 2)/{K + 1)(35) a volume is often considered practically sufficient for the mean value of the normalized to a maximum (19) an adaptive detector (10a), (12). However, such a sample can lead to prohibitively estimating (random) SINR large losses in its statistical characteristics of detection. v = filyi K,) is quite obvious. It is stipulated by the difference in the distribution laws (17) and (34) of pre-threshold statistics (8a) and (10a), (12) of optimal and adaptive detectors, by virtue of which, the equality of their energetic characteristics does not entail that of their statistical characteristics as well. This concerns also the detector (10b), (12) with non-coherent integration of output signals of the adaptive whitening filter, to whose analysis we now proceed. For statistics f., (10b) under conditions (12)-(15)

10

15

20

25

30

35

40

45

SO

where p^_ (x) is the density of the random value distribution T, ^^JK,

(40)

equal to [13] p^_ (X) —

B-' {M,S + \)x^'-' /(I + xf^' ,7 = 0, (41) /(■xy-iFiiM-hK + hM;-z),^ = l, B{M,6 + l)(i + ^ + ^f^i' Z = //X/(1 + fi+x). Threshold

.T„

fixing FAP F is equal to

xo = K -xi, where Xi is a root of the equation 1 (1+;-) 0

PD under condition (41) is determined by the integral J= (1+.,)

(43)

= J r{^-yf'-'2Fdl,K + \;M:z{y))dy 0

^(j/) = (l-2/)///(H-//.), 40

Fig. 5. which in general case, as well as in (39), is not expressed neither by elementary nor known special functions. Fig. 4 shows a family of characteristics D = D{fi) (42), (43) of the considered adaptive detector with NCI under the same condition (F = 10-'", M = 8), as in Fig. 3. A dashed curve here is calculated by (32) and con-esponds to its potentialities under these conditions. It is seen from comparison of Fig. 4 and Fig. 3 that the detector with NCI (10b) has significantly higher "statistical" speed than that with CI (10a). For example, at K = K,, = 13, it provides PD D = D„ = 0.9 already at SINR ji K 32 dB, i.e. at SNR h less by 10 dB. Two-fold (by 3 dB) growth h compensates losses of adaptive processing already a\ K = K, w Ri 28 = 3.5 ■ M, what is half as much as with CI. Fig. 5 visually illustrates the comparative efficiency of adaptive detectors with NCI and CI under considered conditions. Here, for F = lO"'', £) = 0.5 and D = 0.9, there are shown threshold values of product h-x for processing (10a) with CI (dashed curves) and (lOb) with NCI (solid curves) at different volumes of

International Conference on Antenna Theory- and Techniques, 9-12 September, 2003, Sevastopol, Ukraine

On Losses of Coherent Signal in the Adaptive Detector with Non-Coherent Integration

sample K. Horizontal straight lines show potentialities of corresponding detectors. It is seen that under considered conditions, at any /T < 45 « 5.5 • M, the detector with NCI is more effective, and the less is volume of sample K the more is a gain in SNR h. In particular, at K = K^ = 12, \t reaches « 9 dB. A formal reason of distinctions is connected with the difference in distribution laws (34) and (41) of pre-threshold statistics (10a) and (10b). Physically, the advantages of the latter statistics are originated by errors of interference estimation CM (12) due to finiteness of a volume of training sample. These errors change vectors t-, and p^ as compared with "ideal" vectors t^ and p^ both on length and direction. Statistics (10a) "feels" these both changes, whereas statistics (10b) depends only on length of vector p^, and does not "feel" its direction. Namely more "robustness" of statistics (10b) provides higher statistical speed and, hence, higher efficiency of adaptive detector with NCI at small yolumes of training sample.

5.

CONCLUSION There have been compared efficiencies of two types of detectors of Gaussian coherent signals against the background of Gaussian correlated interference - with coherent (CI) and non-coherent (NCI) integration of output signals of the adaptive canceller of interference. It has been shown that at the maximum likelihood estimate of adaptive canceller parameters, the detector with NCI has roughly a double statistical speed, and therefore, at small volume of training sample provides a gain in detecting characteristics. In connection with this, the steady notion about "losses" at non-coherent integration of coherent signal is valid only for training samples of "large" {K >{b -1)M) volume. For many practical situations, where samples of such a volume are unachievable (due to non-stationarity of interference, limited productivity of hardware, etc.) transition to NCI not only simplifies processing, but can enhance its efficiency. In this case, there is made easier also solution of the important problem of stabilization of false alarm level (it is planned to devote a special publication to substantiation in detail of this statement). Note in conclusion that the based on (35)-(37) universal energetical criterion of the adaptive processing speed, first introduced in [10], imposes the lowered requirements to volume of training sample, at which there can appear inadmissible large losses in statistical indices of detection. It is possible to speciiy the substantiated requirements to this volume owing to the introduced in [12,13] and applied here statistical criterion of speed.

REFERENCES 1. p. A. Bakout, I. A. Bol'shakov et al. Questions of statistical radar theory. V.l.-M.:"Sovatskoe radio". Edited by G. P. Tartakovskiy 1963. -p. 424. (in Rus.) 2. Ya. D. Shirman, V. N. Manjos. Theory and techniques of proscessing radar information against the background of interference. M.: Radio i Svyaz', 1981.p.416.(inRus.) 3. Radio electronic systems: foundations of construction and theory. Handbook / Edited by Ya. D. Shirman. M.: CJSC "MAKVIS"; 1998.828 p. (in Rus.) 4. D. Middleton. Introduction to the statistical communication theory, 1960. 5. L. A. Vainshtein, V. D. Zubakov. Signal selection against the background of random interferences. M.: "Sovetskoe radio", I960.- 448 p. 6. Okhrimenko A. E., Tosev I. T. The analysis of the Detection Characteristics of Interperiod Processing Systems. Radiotehnika I elektronika, 1971, V.16,X2l,pp.67-75. 7. Proskurin V. I. Probability distribution for the square functional of Gaussian random process. Radiotekhnika i elektronika, 1985, V.30, Me 7, p. 1335-1340. 8. Fedinin V. V. Efficiency evaluation peculiarities of MTI systems with noncoherent pulse storage. Radiotekhnika i Elektronika, 1981, V.26, M 5, p. 955-961. (in Russian) 9. Kiselev A. Z. Theory of radar detection on the basis of using vectors of target scattering. M.: Radio i Svyaz, 2002,.272 p. (in Russian) lO.I. S. Reed, J. D. Mallett, L. E. Brennan. Rapid Convergence Rate in Adaptive Arrays, IEEE Trans. Aerosp. Electron. Syst., November 1974, Vol. AES-10, No. 6, pp. 853-863. 1 I.Hastings N. A. J. and Peacock J. B. Statistical Distributions. A Handbook for students and practitioners. London, Butterworths, 1975. I2.LekhovitskyD. L, FlekserP. M. Speed of RMB adaptive filters by statistical criteria. In print 13. Lekhovitsky D. I., Flekser P. M. Speed of adaptive whitening filters with non-coherent integration of adaptive of output signals. In print. 14. Handbook of mathematical functions with Formulas, Graphs and Mathematical Tables, Edited by M. Abramowitz and I. Stegun National Bureau of standards, Applied mathematics series - 55. Issued June, 1964.

International Conference on Antenna Theory and Techniques, 9-12 September, 2003, Sevastopol, Ukraine

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International Conference on Antenna Theory and Techniques, 9-12 September, 2003, Se%'astopol, Ukraine

pp. 42-46

FOCUSING OF THE SPATIALLY SEPARATED ADAPTIVE ANTENNA ARRAYS ON MULTIPLE RADIATION SOURCES BY METHOD OF CORRELATION IDENTIFICATION OF THE BEARINGS Yu. N. Sedyshev, P. Yu. Sedyshev, S. N. Rodenko

Abstract The problems of space-time signal processing in coherent passive radar multistatic system with the adaptive antenna arrays (AAAs) in the receiving points are discussed. Stated that application of pair of spatially separated adaptive antenna arrays in selffocusing mode, which are involved in two-stage procedure of measurement of multiple jamming sources coordinates by base-correlation method, enables to solve bearing identification problem and to optimize the parameters of space-time processing system according to adaptation time. The problem of receiving of two-dimensional responses, which unequivocally characterize space position of each source being located, in the plane of base line - target is solved. The results of simulation are presented.

1.

INTRODUCTION

Owing to multiplicity of jamming the signal detection is realized at high level of both correlated and uncorrelated noise. Radiation of electronic countermeasure systems, radio frequency unintended interference, ground clutter and natural noise sources background can be such interferences. High intensity of disturbing signals and their simultaneous influence are the strong reasons to use space-time processing systems on the basis of adaptive antenna arrays for the desired signals processing. Their main advantages [1] are the capability of jamming suppressing; multichanneiing of suppression out of a boresight; accurate direction finding of radiation sources and localization of each one with the minimal error even under their close spacing. For the last years the research of methods of application of adaptive antenna arrays in multistatic passive radar systems has become widely spread. The works [2,3,4] synthesize optimal and quasi-optimal algorithms of detection of both deterministic and stochastic signals against the background of space-correlated jamming. Also those works give the estimation of signal-to-noise output ratio loss compare to optimal processing [3,5] and of the estimation of efficiency of focusing on the located radiation source in the bistatic system with AAA in one of the receiving points [3,6]. These works does not consider issues regarding application of joint adaptive aperture and correlation inter-aperture processing of noise signals in each position in order to increase efficiency of space localization of multiple radiation sources under limited time of space scanning. Using of multistage measurement procedure in spacetime processing systems with a great number of degrees 0-7803-7881-4/03/$17.00 ©2003 IEEE.

of freedom provided in [9,10] enables to eliminate difficulties as for detection, measurement and identification operations. In particular, the responses of space-time processing system under the influence of radiation sources signals can be received with the help of procedures of direction finding in adaptive antenna-arrays, spaced at a distance of a great number of wave lengths, and their self-focusing during correlation identification.

2.

PRINCIPLE OF CREATION OF THE OPTIMAL SPACE-TIME FILTER ON THE BASIS OF SPATIALLY SEPARATED AAAS Consider the posibility of increasing of efficiency of emitting targets space filtering by using of Keypon adaptive algorithm (minimization of external noise dispersion) separately in each antenna array of bistatic system and subject to correlation selection of the detected sources in the adaptive inter-aperture channel. The analysis will be carried out for the space-coherent system model (Fig. 1), which comprises the M^ -elemental AAA in the central position and the M^ -elemental AAA in the additional position put on the base 6. Using of the AAA in the monostatic radar systems allows to localize the signal source in relation to it by the direction of radiation. Increasing of signal-to-noise ratio in the monostatic AAA due to the adaptation of the vector of weighting factors is typical at that. In this case AAA amplification factor hence the amplitude of signal source field) is a little bit reduced in relation to the direcfion of the maximal intensity of radiation of the desired signal. At the same time the nought in the jamming directions (Fig. 2) are fomied. If there is not one but some sources of radiation in the beam of the generated adaptive direction-finding

Focusing of the SpatiaUy Separated Adaptive Antenna Arrays on Multiple Radiation Sources ...

characteristic, their mutual moving in the plane, which is perpendicular to the boresight line, results in the fast fluctuations of the weight factors vector (noise of adaptation). In AAA of gradient type quality of adaptation depends on depth of a correlation feedback. However in this case the process of getting into the steady state is delayed and AAA space selection capabilities are reduced to the potential ones of not adaptive antenna array. It is possible to reduce the adaptation time and to raise quality of space selection by means of creation of information redundancy [5, 7] If signal processing is realized in two adaptive arrays spaced by some base 6 , there is a possibility to locate the source radiation not only by its direction but also by difference of time of its arrival to the receiving points by means of focusing the second array on the source of the desired signal subject to the desired signal source parallax. Presence of the additional information about radiation sources, particularly, about their selection by time of delay of the envelopes of the cross-correlation fimctions (CCF) of signals in AAs, which are spaced at a distance of a great number of wave lengths, allows to make additional selection of the chosen source and to make the amplitude-phase distribution (APD) more precise subject to these data. This process can be regarded as a next qualifying phase of localization, during which correction of the wei^t factors of adaptation is made due to the additional freedom degrees of a system and signal-to-noise relation is optimized due to the reduction of influence of the interfering sources of radiation in AAA's beam (Fig. 3). In contrast to the system of space-time processing [6] adaptation of complex factors of amplification in each element of antenna arrays is carried out not by the way of gradient forming of the weighting vector but on the basis of the methods of direct conversion of the space correlation matrix of jamming. Besides, the survey on radiation sources propagation difference is carried out in the between-positions-delay-time matrix correlator (MC). 2.1. THE FIRST STAGE OF THE PROCEDURE OF MEASUREMENT OF RADIATION SOURCES COORDINATES WITHOUT ADAPTATION AA Consider a special case of signal-jamming conditions when there are three stochastic signal sources in the AAAl and AAA2 coverage sector (Fig. 4). Their searching is carried out in accordance with the optimal space processing algorithm without preliminary estimation of their number. As a result the bearing estimations vector (relative AAAl - Oi and relative AAA2 - ^2) is formed. Searching on the generated in both positions bearings by delay time is performed with the purpose of determining of amount of the targets on the bearing line. The reradiating objects such as close located air ones or ground surface at small altitudes of flight can get in the illumination field. In other words there is an abnormal reproduction of sources in quantity to 77 (where rj quantify of objects in the illumination field) when multipath signal distribution. In this case these are objects 01 and 02. Moreover, as it is known [5], observation of radiation sources on the spatially separated apertures by the

Fig. 1. Model of space-time processmg system operation Signs) s«iree

Inlcrfcrcncc source

t V

Fig. 2. Adaptation by direction in AAA The position dirwlion characterislic AAAl ■fter second adaptalion pliase TTie position difeclion characteristic AAAl IIKT first adaptation phase

AAAl

AA*2

Fig. 3. Adaptation in the spatially separated AAAs by the time of delay of the envelopes of the signal cross-correlation fimctions (CCF) methods of a passive location is accompanied with the false targets due to mutual crossing measuring beams. Generally number of false crossings is equal to n^ - n, where n - true number of radiation sources. Thus information about signal and jamming conditions contains n'^ - n + rj false points, which as well as true ones can be considered by the space processing system as the coordinate measurement objects. Using of matrix correlators in the structure of model allows on the one hand to reduce the time of reception of the information about between apertures complex envelope delay to the minimum in contrast to the procedure of consecutive focussing on the jamming sources in correlation interferometer [6]. And on the other hand fliere is possibilify to realize a rule of an identification of the radiating targets in accordance with

International Conference on Antenna Theory and Techniques, 9-12 September, 2003, Sevastopol, Ukraine

43

Yu. N. Sedyshev, P. Yu. Sedyshev, S. N. Rodenko

when processing in the identification device, cause of the hyperbolas corresponding to their positions are not generated. Thus the correspondence of the numbers of targets received on tfie spatially separated apertures AAAI and AAA2 is determined (Fig. 6) 11 xl, ^ Xr lilt is possible to associate each pair \\xl 4^

AAAI

AAA2

Fig. 4. A principle of space selection in the two AAAs system

XT

I

with the amplitude-phase distribution (APD) Xi(a,)and ^2(^7) on the first and the second AAA apertures. Thus AAAs are focused on the chosen i -th source of radiation (i = Ln). 2.2.

THE SHCOND STAGE OF RADIAI

li)N

AAA

ADAPTATION

SOURCES COORDINATES

MEASUREMENT PROCEDURE

y-^j '''Matrix of AND elements

Matrix adder

F^. 5. The scheme of the feature identification device Matri't cwrclalor 1

x; Taryd* echo ktx,-|K-x;,,xi-x;...xj-xi,|

corrcfaior 7

Fig. 6. The scheme of getting the information for focusing the main beams of AAA pair in the directions on the radiation source i-th interference source

As well as in the works [6,9] the quality of system functioning can be estimated having concretized the spacetime characteristics of signals on apertures of the spatially separated positions. At that consider the bistatic location system as the unique phased array (PA) comprised of the sub-arrays AAAI and AAA2 spaced at a distance of a great number of wave lengths. In this PA the coordinates of radiation sources are estimated in relation to the centre of the base-correlation system (BCS) in the plane of the base line - target (Fig. 7). Suppose there are N radiating sources in the BCS's coverage zone. Consider radiation of every i-th (i = l,n) source to be stochastic signal. Considering the amplitude variety of a signal from the i -th source to be small within the range of aperture of one array, express the APD vectors in the following form: - for the first antenna-array exp jk- -rfisin(7ij k

(I)

-mil, mil,

where|miiI = \mi-2\, mn -f mi^ -f 1 = M^ number; - for the second antenna array Fig. 7. Representation of BCS as PA with the broken aperture the principle that the sum of signals delays between the positions in the direction of one source is always equal to2rt. This is based on the exchanging of the current estimations of delays of signals from the targets (X'randx?) between the positions. The device of the identification functioning in accordance with the given rule (Fig. 5) will not form a pair from the measurements II Xr, i(7i,ft)] N{t) = 2=1

Ni{t-t,.2i)X2hi,Pi)X xexp[#2(7nA)]

N,{t-t,u)Xi{ji,Pi)x xexp[i^(7„A)] N2oit)

+E t=i Ni{t-t,u -At,)X2{J^,p^)X

xexp[#2(7uft)] where i^ii, t^2i - delay time of a signal at its propagation from the external i-th source up to the cenfre of the first and second antenna-array accordingly; Nio (i), ^^20 W- correlation matrix of internal noise of the first and second antenna-array accordingly; ^H = h2i ~ ''zli •

Assume noises of each external radiation sources to be uncorrelated in time and to have uniform power specfral density in the receiver Iln 's band of frequencies. Then [10] according to Hinchin-Winner formula: fi{t,s) = NAit - s), where N.i - spectral density of capacity of i -source of interferences, 7r[i — s]

Provided that the signal spectrum width is more than receiver B's bandwidth, it is possible to replace "quasiwhite" noise with delta-correlated process. Then,, by virtue of mutual independence of internal noises in the various reception channels, noises of internal and external sources, proportionality of correlation matrix elements to delta-ftinction(5(f - s), - expression for the correlation mafrix a of a mix of AAA's internal noises and signals of external sources looks like:

xexpl j(4^2ili^Pi) - \

(Circular

We have realized two compact horns with low cross polarization and high aperture efficiency, respectively, by introducing the waving taper (1), (2). The desigii method is based on the optimization technique combined with the mode-matching approach. This paper presents such horns with high performance.

2.

Dfj I (Circular g, aperture)

DESIGN METHOD

The design parameters are defined by inner diameters A, (n = 1, 2, ..., N) and the axial length L. The continuous configuration of the taper is given by interpolating these dimensions through the third-order spline function (see Fig. 1). The scattering matrix for the whole structure of the interpolated taper is obtained by the cascade connection of step discontinuities and uniform waveguides with different diameters as shown hi Fig. 2. The generalized scattering matrices for each discontinuity can be easily computed by the cylindrical mode-matching approach. To avoid much calculation time for step discontinuities in the optimization procedure, the dimension of the step discontinuities are kept same for all steps, and hence an arbitrary taper configuration is represented only by changing the lengths A/^ of uniform waveguide between the adjacent step discontinuities as shown in Fig. 3. The objective function in the optimization procedure is defined by using the higher-order mode coefficients for high aperture efficiency or the radiation pattern by the dominant hybrid mode in corrugated horns, corresponding to the purpose of the horn use. Finally, the minimum length L is determined by iteration of the above procedure.

Fig. 1. Basic geometry of proposed horn antennas A Ai

Bi A2

ai=l-aA/,

B2

'■^f&^dfi;^J Si

!

I

!

I

T TT nB2' Bi'A2'

A' A]'

Bi B ^ TE,i fS=S5TM|, TEi2

S2

SI Ai'

t

i

tf? Bi' B'

Fig. 2. Equivalent ckcuit represented by the generalized scattering matrices

-H h-M

(ZA,i.] 37 dB; cross polarization < -30 dB (averaged over the beam); 1" side lobe '.' ii=-yv

Where E(g,i/„), H(g,z/„), 3'{q,v„), 3'"(q,u„) are, respectively, the complex amplitudes of intensities of electric and magnetic fields and of surface densities of electric and magnetic currents at frequency I'u - "Jo>,W() -I- mi„wi -I-... -I- Tnx,iU)j^:

Fig. 1.

3.

mfa, =0,±1,±2,...

NONLINEAR BOUNDARY CONDITIONS IN SPACE-TIME AND SPACE-FREQUENCY DOMAINS

As the device under analysis is nonlinear, so in general case, it is impossible to manipulate with complex amplitudes, that is NBC must be formulated in the spacetime domain. Correspondingly, we suppose that instantaneous values of tangential components of electric E{q.t) and magnetic U{q,t) fields intensity vectors on surface Z are linked by the following relation n, xE(9,i) = Z[n, xH(9,0], or in the equivalent form 3"'{q,t) = -Z[3"{q,t)].

("*)

Sn =1, if u„ = 0 and 6„ = 1/2, if i/„ ^ 0. Under these conditions, (1) and (2) are transformed to the following form T

n, xE(9,i/„) = lim ;/Zx X

E*i(n, xH(9,i',))e^"'' e-^""' dt t = -Af

Vi/„,

d)

(5) T-oc

n = -N,N,

or

(2)

3'"(g,K) = T

Here n, is the external normal to the surface at point lim

g; J'" (