A hybrid beamforming approach to anti jamming for multi channel GNSS receivers

We consider the problem of anti-jamming in GNSS navigation. In this work, we introduce the novel concept of a hybrid beamforming approach to antijamming for multi-channel GNSS receivers.

Công nghệ thông tin A HYBRID BEAMFORMING APPROACH TO ANTI-JAMMING FOR MULTI-CHANNEL GNSS RECEIVERS Nguyen Huu Trung1*, Nguyen Minh Duc1, Thai Trung Kien2 Abstract: We consider the problem of anti-jamming in GNSS navigation In this work, we introduce the novel concept of a hybrid beamforming approach to antijamming for multi-channel GNSS receivers In this method, three separate techniques are integrated: a) the system suppresses jams by nulling based on eigendecomposition of correlation matrices; b) the direction of desired satellite is estimated by phase difference based on antenna geometry and carrier-phase measurements; and c) an adaptive beamformer is used to optimize desired signal while minimizing interferences The characteristics of the proposed system model is demonstrated using Monte-Carlo simulations Keywords: GNSS, Anti-jamming, Interference mitigation, Hybrid beamforming, Multi-channel GNSS receivers INTRODUCTION The Global Navigation Satellite System (GNSS) includes the Global Positioning System (GPS) operated by the United States, the Global Orbiting Navigation Satellite System (GLONASS) and various other systems such as COMPASS operated by China and GALILEO operated by Euro in future With GNSS, multi-constellation signals will be available to increase the performance of receivers operating with some or all of these systems [1] Global Navigation Satellite Systems (GNSS) plays an important role in many aspects of life, from military and defense to transportation, rescue, surveying maps, marine navigation, aviation… with high demanding requirements for security and integrity [2] GNSS receivers have to operate at very low signal levels and in the presence of RF interference including jams, multipath and noise For GPS system, the satellites have an orbit altitude of 20200Km from earth GPS L1 signals are transmitted with a power of 44.8 W at 1575.43MHz and GPS satellite antenna gain is 12dBi Assume receiver antenna gain is 4dBi, the power level received by user located near the Earth surface is -125dBm using free space loss model while background inband noise (2.046MHz) is -110dBm However, spread spectrum processing gain is 43dB (10log1.023Mbs/50bps ≈ 43dB), so the signals are recovered at power level of -110dBm - 43dB = -153dBm In fact, the received power strength requirement will be several dB above the theoretical level [3] With this very low signal levels, it is easy for GNSS receivers to be subjected to unintended and intended jams In the case of jams, the signals are unable to be synchronized Especially in military applications, GNSS receivers have to operate in the ruggedenvironment and mitigate interferences in order to provide reliable navigation solutions [4] The effect of multipath on GNSS receiver performance has been widely analyzed, whereas many anti-jamming and multipath mitigation algorithms have been proposed [5-10] In order to mitigate interferences, a single-antenna receiver can make use of time and frequency diversity, such asadaptive space-time equalization techniques [11] A remarkable approach is based onthe use of antenna 112 N H Trung, N M Duc, T T Kien, “A hybrid beamforming approach … GNSS receivers.” Nghiên cứu khoa học công nghệ arrays that can benefit from spatial-domain processing and thus mitigate the effects of multipath Their capability is usually applied to the signal tracking operation of GNSS receivers, and there exists an extensive bibliography on this topic [12] Recently, multi-antenna techniques have been presented in [13] for interference mitigation A digital beamformer works as a spatial filter to mitigate the interferences and acquire the signal using the beamformer output [14] The beamformer design relies on a priori knowledge of a reference waveform or the spatial signature of the signal using DOA estimation [15] Antenna arrays and beamforming algorithms may estimate direction-of-arrivals (DOAs) [16] But, in a GNSS receiver, due to the extremely low receiving power of the satellite signals, it is difficult to estimate the DOA before signal correlation, and DOA-based beamforming techniques usually need a calibrated array [17].However, if we make use of plural of correlators that work by carrier-phase measurements then we can estimate the direction of desired satellite by phase difference based on antenna geometry and carrier-phase measurements In this paper, weproposeahybrid beamforming approach to anti-jamming for multi-channel GNSS receivers design The approach suppresses jams by nulling based-on subspace orthogonal projectionand maximizesthe gain of useful GNSS signals by minimum variance distortionless response beamformer We overcome the problem of the DOA estimation by using differential carrier-phase measurements The rest of the paper is organized as follows In section 2, the necessary background is given about notations and GNSS signal structure and carrier-phase measurement for satellite direction estimation In section3, the proposed system is shown with beamforming scheme Section provides simulation results and characteristics of the proposed system Finally, the conclusions of this paper are for concluding remarks, and suggestions for further researches PRELIMINARIES 2.1 Notations Throughout the paper, following conventions are used Bold capital letters are denoted for matrices, while low-case bolt letters are for vectors.ℝ{ } stands for real, { } for either expectation or average value of { } The super scripts ρ(.), (.)T, (.)H, (.)+, Re{.} stand for rank, transpose, conjugate transpose, pseudo-inverse, real part of (.) Non-negative (positive) definite matrix ≽ is a symmetric one having only non-negative (positive) eigenvalues 2.2 GNSS Signal model The term GNSS means interoperability and compatibility between different satellite navigation systems: Modernized GPS + GLONASS + GALILEO GNSS makes use of CDMA, BOC modulation (binary offset carrier) and QPSK Transmitted signal for the normalized complex envelope (i.e base-band version) ( ) ( ) of a RF signal ( ) ( ) includes inphase component and quadrature ( ) ( ) component ( ), ( ) of kth satellite respectively [18] We have: Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san CNTT, 12 - 2017 113 Công nghệ thông tin ( ) ( )= ( ) ( ( ) ( )− ( ) ( ( )) ( ) ( )) ( ) (1) ( ) ( ): Average signal power of satellite k where: Re{}: Real part of the signal; ( ) at frequency of ; : Phase of the signal transmitted from satellite k ( ) ( ) and pilot ( ) ( ) are modulated by DS-CDMA The components data and BOC (in order to minimize interference to existing available GPS signals) typical of modernized global positioning system [19] The BOC modulation is a signal subcarrier modulation where the signal is multiplied by a rectangular subcarrier (sine or cosine phased) Both the subcarrier frequency fs and the pseudorandom noise sequence (PRN code) with chip rate fc are an integer multiple of the reference frequency f0=1,023MHz BOC signal is denoted as BOC( , ) or BOC( , ) where the integer represents the ratio between fs and reference frequency f0 and the integer represents the ratio between code rate and f0 ( ) ( )=∑ ( ) [/ ] ( ) , ( ) ( )=∑ ( ) , ( ) , : ( ( ) sin (2 ( ) ( ) ( − )) ( − ) ) (2) (3) where: [.] is integer part, , , Brimary spreading sequences for data (D) and pilot (P), ( ): Navigation message, : Carrier frequency, Tc: Chip rate and ( ): Unit pulse 2.3 GNSS carrier-phase measurement Since the carrier-phase measurement have higher resolution than that of the code phase measurement, it can be used for the attitude determination of vehicles with multiple antennas confuguration In this case, we use carrier-phase measurements to estimate phase shifts relative to the reference antenna for steering vector in beamforming stage In general, the carrier-phase measurement model is as follows [20]: () = () + − () () − + () + () + , − ( ) + () + (4) where the subscript k indicates the k-th receiver (for master, denote b, for slave, () denote r), the superscript i indicates the i-th satellite, λ is the wavelength, is the true range between the receiver antenna k and GNSS satellite i, c is the speed of () light, is receiver clock error from GNSS time, is the satellite clock () () offset from GNSS time, is ionospheric delay error, is tropospheric () () delay error, is delay error due to satellite ephemerides error, is the carrier-phase integer ambiguity and represents other errors such as multipath, ( ) inter channel receiver biases, thermal noise ( ), = ( ) ( ) are , = phases at initial time The phase‐range is expressed as: Φ () =λ () (5) 114 N H Trung, N M Duc, T T Kien, “A hybrid beamforming approach … GNSS receivers.” Nghiên cứu khoa học công nghệ The single and double carrier-phase difference is modeled as shown in Figure 2: () ∆ () = ( ) − () (6) () ( ) ∇∆ =∆ −∆ (7) Given that the common clock is used in the system, the satellite clock error () can be removed by a single difference as: ∆ () = ∆ () +∆ () () −∆ +∆ () +∆ () +∆ () +∆ (8) () Where: ∆ ( ) = − (9) Due to the short baselines used, common mode error terms () () () ∆ ,∆ ,∆ between satellites and receivers can be eliminated or greatly reduced, we have: ∆ () = ∆ () +∆ +∆ () +∆ (10) Satellite i direction ∆ () = () ∆ ∆ + ∆ ̅( ) () θ Master ⃗Slave Figure Vector Diagram for phase shift calculation PROPOSED HYBRID BEAMFORMING SCHEME FOR GNSS RECEIVERS 3.1 System model The RF Front-end consist of a low-noise amplifier (LNA), a ceramic filter (BPF), a mixer to form the IF frequency signal, an IF filter and an ADC It converts the RF signal at the output of each antenna elements to a digitally sampled signal There are mixer modules in the array working with the same common clock which is synchronized to local oscillator Consider the array with K elements The sample from ADC of each element of the array is multiplied with a weight ∗ where the superscript * represents the complex conjugate The weighted signals are added together from K elements to form the output signal: Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san CNTT, 12 - 2017 115 Công nghệ thông tin Where The ∗ =∑ = (11) represents the weighted vector of length , is received vector signal: = [ ∗ , ∗ , … , ∗ ] = [ ]∗ (12) Figure Block diagram of proposed system × observation vector of the array at time t is given by: ( )=∑ ( ) , ( )+ ( ) +∑ (13) ( ) is GNSS signal of interest (SOI) arriving from pth satellite Where (p=1 P) is a steering vector, ( ) and ( ) are broad band interference and noise vector, respectively Number of satellites in view is P and number of jammers is Q Steering vector is expressed as: , = ( )/ ( )/ … ( ) ( )/ (14) 3.2 Nulling based-on subspace orthogonal projection For simplicity, we omit the parameter of time in the equations The covariance matrix of input signals is: } = { (15) The covariance matrix can be divided into GNSS signal, interference and noise component as: = + + (16) GNSS signal level is far below noise floor, so that: ≈ + (17) Where is the average noise power The eigen-decomposition of the covariance matrix is given by: = (18) Where the columns of = , and = diag( , , … , , , ,… , ,…, ,… are the K eigenvectors of ) contains the corresponding K 116 N H Trung, N M Duc, T T Kien, “A hybrid beamforming approach … GNSS receivers.” Nghiên cứu khoa học công nghệ eigenvalues , ,…, are the Q largest eigenvalues corresponding to Q jammers Denote interference subspace as = span{ , , … }, then its orthogonal complementary space is: = − ( ) (19) And the subspace orthogonal projection matrix is: = ( ) (20) New steering vector for interference nulling is: , = , (21) 3.3 Beamforming based-on carrier-phase measurements Beamforming is an important technique in array processing in order to optimize desired signal while minimizing interferences Statistically optimal beamforming techniques include maximization of SNR, Minimum Mean Squared Error (MMSE), Linearly Constrained Minimum Variance (LCMV), minimum variance distortionless response (MVDR) are widely applied [21], [22] Design of the beamformer under statistically optimal method requires statistical properties of the source and the statistical characteristics of the channel In this case, after interference nulling by subspace orthogonal projection, the output power of the beamformer is minimized andthe response according to direction of arrival of the desired signalis fixed in order to preserve desired signal while minimizing the impact of undesired components including noise and remaining interference We have the output response of signal source with direction of arrival and frequency is determined by ( , ) Linear ( , ) = , where c is a constant to ensure constraint for the weighs satisfy that all signals with frequency come from direction of arrival are passed with response c Minimization of output due to interference is equivalent to minimizing the output power (minimum output power): }, s.t ( , )= = arg {| | } = arg min{ (22) Using the method of Lagrange multipliers, find min[ ( ; λ)], where: ( ; λ) = {| | } + λ( − )= = + λ( − ) (23) +λ (24) = (25) Solution of the equation [23]: = −λ is the constraint steering vector toward the target satellite after interference nulling The steering vector is set as phase shifts relative to the reference antenna, which are represented by: , = = [1 (∆ … , ∆ , ) Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san CNTT, 12 - 2017 ] (26) (27) 117 Công nghệ thông tin Where ∆ , is the phase difference based on antenna geometry and the direction of desired satellite ∆ , is the phase resulting from the difference of cabling and RF chain including down converter and digitization among elements of antenna array ∆ , can be calculated via carrier-phase measurement model by (10) The covariance matrix is estimated by computing sample covariance matrix with assuming the sample mean is zero as follows: ⋯ , , ⋱ ⋮ [ ]= ⋮ (28) ⋯ , , , = ∑ [ ] [ ] (29) Where N is the number of samples to compute the covariance matrix The overline sign denotes complex conjugation In practice, uncorrelated noise component ensures is invertible If c = the beamformer is called minimum variance distortionless response, MVDR, beamformer The MVDR beamformer does not require the knowledge of the direction of the interferences for weight vector calculation, it requires only the direction of the SOI [19] The minimization process minimizes the total noise including interference and uncorrelated noise NUMERICAL RESULTS The performance of the system is performed by means of the Monte-Carlo simulation The simulation estimates the influence of some parameters on the performance of the system These parameters include ISR (Interference to Signal Ratio); SNR (Signal to Noise Ratio), array configuration (UCA); Number of antennas (M); Sampling rate fs; Difference DOA between transmited signal and interference Δθ Monte-Carlo simulation algorithm includes sequence steps: generation of transmit signal, interference and AWGN by parameters of SNR, INR and DOAs; Reception of signal by steering vector a(t), interference and AWGN at sensors; Beamforming weights calculation by processing signal samples; Compare output signal to source signal and evaluate NRMSE by Monte-Carlo method The number of samples to compute the covariance matrix N = 103 Transmitted signal is determined narrow band sine wave signal Signal is transmitted continuously through the training sequence and the amplitude of the signal can vary or change in order to get the desired SNR at each antenna The transmit signal is of the form: ( ⁄ ) ( )=√ (30) Where fc is carrier frequency, fs is sampling frequency, is phase of signal, = [1: ] with N is simulated number of samples Interference can be narrow band with the same frequency as signal or broadband interference as: ( )=√ (0,1) (31) 118 N H Trung, N M Duc, T T Kien, “A hybrid beamforming approach … GNSS receivers.” Nghiên cứu khoa học công nghệ AWGN ( ) = (0,1) has normal standard deviation appears at every antennas The system performance in simulation is Normalized Root Mean Square Error, NRMSE, the final value is the average value of all Q values after each simulation: ∑ = mean ( ) ( ) ( ) ( (32) ( )) Besides the simulation of system performance according to SNR we also simulate the performance of the system according to the interference The severity of the interference is determined by INR (interference to noise ratio): (33) [ ] = [ ]+ [ ] [ ] (interference to signal ratio) is in dB In the simulations, Uniform Circular Arrayhave been used, array element spacing is 1/2 signal wavelength; GPS navigation symbols are in the BPSK symbols transmitted at 50 b/s; The C/A-code is a Gold code with a chip rate of 1.023 Mcps (or code period P = 1023) and repeats every millisecond; Carrier frequency Fc = 1.57542GHz; its DOA θs=30o, Jammers used in the simulations are generated as broadband binary signals with DOAs = -20, 30, 55 degrees MonteCarlo experiment number is 200 -5 Normalized Array Gain [dB] -10 -15 -20 -25 -30 -35 -40 -45 -50 -90 -70 -50 -30 -10 10 Angle [degree] 30 50 70 90 (a) -5 Normalized Array Gain [dB] -10 -15 -20 -25 -30 -35 -40 -45 -90 -70 -50 -30 -10 10 Angle [degree] 30 50 70 90 (b) Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san CNTT, 12 - 2017 119 Công nghệ thông tin -5 Normalized Array Gain [dB] -10 -15 -20 -25 -30 -35 -40 -45 -90 -70 -50 -30 -10 10 Angle [degree] 30 50 70 90 (c) -2 Normalized Array Gain [dB] -4 -6 -8 -10 -12 -14 -16 -18 -90 -70 -50 -30 -10 10 Angle [degree] 30 50 70 90 (d) Figure Beamforming patterns (a): Subspace orthogonal projection, DOA interferences = -20, 20 degrees; (b): MVDR algorithm, DOA signal = 30 degree, DOA interference = -20; (c) LCMV algorithm, DOA signal = 30 degrees, DOA interference = -20 degrees; and (d) Frost algorithm, DOA signal = 30 degrees, DOA interference = 20 degrees We first consider the problem of placing nulls in the directions of interferences Jj(t) while preserving GPS signals si(t) If the relative received power of interference and desired signal at an antenna is taken as a reference, then the rejection of interference with respect to desired signal, is the change in relative power after the null has been placed To evaluate the rejection of the proposed system, we setup the simulation with two jammers at DOA of ±20 degrees, and a signal at DOA of 30 degrees and SNR = 0dB The rejection level is approximately35dB However, this is ideal case In practice, RF component, array mismatch,… decrease the performance 120 N H Trung, N M Duc, T T Kien, “A hybrid beamforming approach … GNSS receivers.” Nghiên cứu khoa học công nghệ Beamforming patterns of beamforming algorithms under the influence of interference sources are shown in Figure4 Figure (a) illustrated the nulling of two jammers;(b), (c), (d) show the responses of the various beamformers including MVDR, LCMV and Frost We see that the response at the incident angle of interference is suppressed Normalized root mean square error (NRMSE) Orthogonal Subspace Projection -40 -35 -30 -25 -20 -15 -10 -5 SNR [dB] (a) MVDR LCMV FrostBeamformer Orthogonal Subspace Projection only Proposed Hybrid Method Normalized root mean square error (NRMSE) 4.5 3.5 2.5 1.5 -30 -28 -26 -24 -22 -20 -18 SNR [dB] -16 -14 -12 -10 (b) Figure NRMSE according to SNR (a) Othorgonal Subspace projection only and (b) NRMSE of the proposed system and various beamformers The simulation results are presented in Figure4 (a,b) according to SNR range Figure (a) shows NRMSE of othorgonal subspace projection method only and (b) shows NRMSE of the proposed system and various beamformers in performance comparing In the figure, we see that the proposed system yields significant result That is, the system is more robust to jamming and multipath due to hybrid beamforming method CONCLUSIONS This paper presented a hybrid beamforming approach to anti-jamming for multichannel GNSS receivers for optimum performance We first apply nulling Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san CNTT, 12 - 2017 121 Công nghệ thông tin technique to suppress jams by nulling based on eigen-decomposition of correlation matrices for acquisition of satellites The direction of desired satellite is estimated by phase difference based on antenna geometry and carrier-phase measurements The direction information is fed into an adaptive beamformer to optimize desired signal while minimizing interferences However, the receiver clock error still exists on single-difference To eliminate the receiver clock error, the second or doubledifference has to be formed by subtraction of the first difference for one satellite to another Ongoing work is being done in the application of nonlinear optimum state filter for carrier-phase tracking and optimum combination of multiple beamforming for multiple satellites Acknowledgements: The authors would like to thank the Ministry of Science and Technology that has supported under the project number NDT.03.ITA/15 REFERENCES [1] P Benevides; G Nico; J Catalão; P M A Miranda, “Analysis of Galileo and GPS Integration for GNSS Tomography”, IEEE Transactions on Geoscience and Remote Sensing, Volume: 55, Issue: 4, Pages: 1936 – 1943, 2017 [2] Christopher Schirmer; Alexander Rügamer; Wim A Th Kotterman; Markus H Landmann; Giovanni Del Galdo, “Evaluation of array antenna systems for GNSS applications using wave-field synthesis in an OTA laboratory”, 2017 11th European Conference on Antennas and Propagation (EUCAP), Pages: 3370 – 3374, 2017 [3] Balaei, A T., Dempster, A G., and Lo Presti, L “Characterization of the effects of CW and pulse CW interference on the GPS signal quality” IEEE Transactions on Aerospace and Electronic Systems, 45, (Oct 2009), pp 1418—1431, 2009 [4] Wei Sun, and Moeness G Amin, “A Self-Coherence Anti-Jamming GPS Receiver”, IEEE, IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL 53, NO 10, OCTOBER 2005 [5] Kristine M Larson, John J Braun, Eric E Small, Valery U Zavorotny, Ethan D Gutmann, and Andria L Bilich, “GPS Multipath and Its Relation to NearSurface Soil Moisture Content,” IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, VOL 3, NO 1, pp 91-99, March 2010 [6] Youguo Hou, Wei Guo, and Xiaozhang Jin, “Design of an anti-jamming GPS receiver based on orthogonal projection method”, Journal of Systems Engineering and Electronics, Vol 21, No 1, pp.16–19, February 2010 [7] W C Cheuk, M Trinkle & D A Gray, “Null-steering LMS Dual-Polarised Adaptive Antenna Arrays for GPS”, Journal of Global Positioning Systems, Vol 4, No 1-2: 258-267, 2005 [8] Yu-Hsuan Chen, Jyh-Ching Juang, Jiwon Seo,Sherman Lo, Dennis M Akos, David S De Lorenzo and Per Enge, “Design and Implementation of Real- 122 N H Trung, N M Duc, T T Kien, “A hybrid beamforming approach … GNSS receivers.” Nghiên cứu khoa học công nghệ Time Software Radio for Anti-Interference GPS/WAAS Sensors”, Sensors, Vol 12, pp: 13417-13440; doi:10.3390/s121013417, 2012 [9] James Thomas, H Michele, G Joaquim, “Analog and Digital Nulling Techniques for Multi-Element Antennas in GNSS Receivers”, presented at ION GNSS+ 2015, the 28th International Technical Meeting of the Satellite Division of The Institute of Navigation held in Tampa, Fla., Sept 14–18, 2015 [10] Cui Jianhua; Cheng Naiping; He Panfeng, “Adaptive anti jamming algorithm for GNSS software receiver”, IEEE Information Technology, Networking, Electronic and Automation Control Conference, Pages: 962 - 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96, 2015 [19] Wei Liu; Yuan Hu, “Cosine faded harmonics binary offset carrier modulation for next-generation GNSS”, Electronics Letters, Volume: 52, Issue: 1, Pages: 68 - 70, 2016 Tạp chí Nghiên cứu KH&CN quân sự, Số Đặc san CNTT, 12 - 2017 123 Công nghệ thông tin [20] Yingdong Yang, Xuchu Mao and Weifeng Tian, “Rotation Matrix Method Based on Ambiguity Function for GNSS Attitude Determination”, Sensors, Vol.16, 841, 2016; doi:10.3390/s16060841 [21] R Lorenz and S P Boyd, “Robust minimum variance beamforming,” IEEE Trans Signal Processing, vol 53, pp 1684–1696, May 2005 [22] Sergiy A.V., Alex B.G and Zhi-Quan Luo, “Robust Adaptive Beamforming Using Worst-Case Performance Optimization: A Solution to the Signal Mismatch Problem”, IEEE Trans Signal Processing, Vol 51, No 2, 2013 [23] Yonina C.Eldar, “Minimax MSE Estimation of Deterministic Parameters With Noise Covariance Uncertainties”, IEEE Trans Signal Processing, Vol 54, No 1, 2006 TÓM TẮT HƯỚNG TIẾP CẬN ĐỊNH HƯỚNG BÚP SÓNG HỖN HỢP CHỐNG CAN NHIỄU CHO CÁC BỘ THU GNSS ĐA KÊNH Chúng xét trường hợp chống can nhiễu cho hệ thống định vị GNSS Trong nghiên cứu này, đề xuất phương pháp định hướng búp sóng hỗn hợp cho thu GNSS đa kênh chống nhiễu phá Trong phương pháp này, ba kỹ thuật áp dụng: a) Hệ thống nén can nhiễu đặt điểm không dựa phân tích trị riêng ma trận tương quan, b) Hướng sóng tới vệ tinh xác định dựa phép đo pha sóng mang cấu trúc hình học mảng anten c) định hướng búp sóng thích nghi sử dụng để tối ưu hóa tín hiệu hữu ích loại trừ can nhiễu Các kết mô theo phương pháp Monte-Carlo điều kiện khác minh chứng hiệu phương pháp đề xuất Từ khóa: GNSS, Loại trừ can nhiễu, Định hướng búp sóng hỗn hợp, Bộ thu GNSS đa kênh Nhận ngày 16 tháng năm 2017 Hoàn thiện ngày 26 tháng 11 năm 2017 Chấp nhận đăng ngày 28 tháng 11 năm 2017 Địa chỉ: 1Đại học Bách Khoa Hà Nội; Viện CNTT / Viện KHCNQS * Email: trung.nguyenhuu@hust.edu.vn 124 N H Trung, N M Duc, T T Kien, “A hybrid beamforming approach … GNSS receivers.” ... weproposeahybrid beamforming approach to anti- jamming for multi- channel GNSS receivers design The approach suppresses jams by nulling based-on subspace orthogonal projectionand maximizesthe gain of... due to hybrid beamforming method CONCLUSIONS This paper presented a hybrid beamforming approach to anti- jamming for multichannel GNSS receivers for optimum performance We first apply nulling... Giovanni Del Galdo, “Evaluation of array antenna systems for GNSS applications using wave-field synthesis in an OTA laboratory”, 2017 11th European Conference on Antennas and Propagation (EUCAP),