ICSP2012 Proceedings
Robust Widenull Anti-jamming Algorithm for High Dynamic GPS Lu Dan, Wu Renbiao, Wang Wenyi Tianjin Key Lab for Advanced Signal Processing, Civil Aviation University of China, Tianjin, China e-mail:
[email protected] around each original interference source in order to generate the widenull beampattern. The Mailloux method need not know interference directions and compute simply. However, the method would alter the noise contribution to the covariance matrix. For a theoretic covariance matrix, the alteration is equal to diagonal loading and be neglected.
Abstract—In this paper, a robust null broadening algorithm is proposed for high dynamic global positioning system (GPS) interference suppression. The new algorithm reconstructs interference subspace by additional discrete sources of interferences, which could broaden the width of nulls and adapt to the rapid change of the interference directions in the high dynamic environment. Also the proposed algorithm would form the robust beam pattern utilizing the diagonal loading technique, which improves the performance of the conventional null broadening beamformer in snapshot deficient. The simulation results show that the feasibility of the proposed algorithm. Keywords-global positioning system; interference suppression; null broadening
I.
high
The null broaden algorithms mentioned above are proposed for moving interference suppression, not GPS anti-jamming. Because the Mailloux method needs not known interferences direction, the algorithm is used to high dynamic GPS interference suppression in this paper. However, in the high dynamic environment, snapshots used to compute sample covariance matrix is very deficient due to rapidly change of interferences directions. In this case, noise contribution to the covariance matrix must be considered. Therefore, an improved robust null broaden algorithm combining the idea of Mallioux method with diagonal loading technique is proposed in this paper. The new algorithm would improve the performance and robustness of the Mallioux algorithm for snapshot deficient in high dynamic environment.
dynamic;
INTRODUCTION
Global Positioning System (GPS) receiver is susceptible to interferences from either intentional or unintentional sources because the signal is 20dB lower than the receiver noise[1]. Therefore the GPS receiver would be unable to lock onto the GPS signals, which results that the performance of GPS navigation and positioning degrades dramatically. Therefore, it is necessary to make some measure to mitigate the interferences.
II.
MALLIOUX ALORTIHM FOR HIGH DYNAMIC GPS ANTIJAMMING
In recent years, the promising solution to the interference problem is use of adaptive antenna arrays[2-6]. Various adaptive anti-jamming algorithms have been proposed, including the power minimization approach[3,4], minimum variance distortionless response beamformer[5] based on signal direction known a prior and blind adaptive beamforming algorithms utilizing the characteristics of satellite signal[6]. However, the nulls formed by the methods mentioned above are very sharp[7]. In high dynamic environment, the direction of interference relative to GPS receiver is changing rapidly with time. The beamformer with narrow nulls could not track the rapidly changing of the interference directions, which could lead to the interferences moving out of the nulls, and could not be suppressed.
A. Data model in high dynamic environment The distance between GPS satellite and the receiver is far more than the one between the interference equipment and the receiver, so which is equivalent to the situation that the GPS receiver is in a stationary state but the interference is moving in a very high speed in high dynamic environment[10]. Therefore, considering an M-element antenna arrays, when the interferences impinge on the array, the received signals can be modeled as x t
An effective method is null broadening[8,9], which makes interference locate in the null all the time during the weight vector application. Gershman has proposed a null broadening algorithm which incorporates derivative constraints on jammer directions[8]. However, derivative constraints could lead to significantly increased computing and could be insensitive for the width of the null control. Mailloux has proposed another null broadening algorithm[9]. In this algorithm, a cluster of equal-strength incoherent sources are artificially distributed
L
Q
l 1
q 1
¦ a Tl sl t ¦ b T q t jq t e t .
Where sl t is the l th l
(1)
1 " L GPS signal, Tl
denotes the direction of arrival of the l th GPS signal and is invariant in the model due to the direction is slowly changing with time, a Tl is corresponding to the steering vector. jq t is the q th q
1" Q
interference signal , T q t
denotes the direction of arrival of the q th interference signal
___________________________________ 978-1-4673-2197-6/12/$31.00 ©2012 IEEE
and is time variant in the model due to the direction is rapidly changing with time, b T q t is corresponding to the steering
In Mallioux approach[9], The m th row and n th column is element of the reconstructed covariance matrix R
vector. e t denotes the receiver noise vector which is modeled as zero-mean white Gaussian processing. GPS signal, interference and noise are not correlated with each other in this paper. For M-element linear antenna array, let xm denotes the m th element location and there is
a Tl
ª j 2OS x1 sin Tl «e ¬
" e
j
2S
O
xm sin Tl
" e
j
2S
O
xM sin Tl
º » ¼
2S Q sin K ' mn V 2 e j O xn xm sinTq p V 2G . R mn q v mn sin ' mn q 1
Where ' mn S xm xn E / O , E B / K 1 and B is the trough width between the outermost nulls. K is the number of artificial jammers. Let
T
(2)
Tmn
and b T q t
T
2S 2S j j x sin T q t x sin T q t º ª j 2OS x1 sin Tq t O m O M e " e " e » (3) « ¼ ¬
B. Mallioux Algorithm for interference suppression GPS signals are far lower than the noise floor, then the covariance matrix of the array receiver signals is dominated by interference covariance matrix and noise covariance matrix. Therefore the m th row and n th column element of the covariance matrix R can be approximated Q
R mn V q2 e
j
2S
O
sin K ' mn sin ' mn
V v2G mn .
(6)
denotes the m th row and n th column element of the augment matrix T without angle dependence. According to (5), could be the off-diagonal term of the new covariance matrix R rewritten as T :R . R mn mn mn
(7)
For the diagonal terms, there is T q R K 1 V 2 . R mn mn mn v
xn xm sin T q t
(5)
(4)
(8)
Compared (7) with (8), It is not possible to exactly by multiplying the duplicate the new covariance matrix R measured matrix coefficients by the augment matrix T , namely
q 1
Where V q2 is the power of the q th interference signal,
V v2 is the power of noise and G mn is the Kronecker G function. Obviously, the covariance matrix in (4) is time variant. So the conventional adaptive beamforming technique with narrow nulls will fail to suppress the interferences. The Mailloux algorithm assumes that there are a cluster of equal-strength incoherent sources around each original interference source as shown in Fig.1. According to the above hypothesis, the covariance matrix could be reconstructed to broaden the nulls in order to compensate the time-varying covariance matrix obtained by the array received signals.
R : T. R
(9)
are exactly reproduced and the But the off-diagonal of R diagonal terms have the noise contribution multiplied by K , which is equal to diagonal loading. Therefore new covariance in (9) could be used to replace the covariance matrix matrix R R to broaden the nulls. In practice, the covariance matrix R is unknown and has to be replaced by its sample estimate P ˆ 1 R x p x H p . P p 1
B
(10)
However, in the high dynamic environment, the snapshot number P is very deficient and the estimated error covariance matrix R is equal to color noise. So noise contribute would reduce the performance of Mailloux method by not accurate diagonal loading value in (9).
Tq
III. Figure 1. The distributed figure of artificial interference in Mallioux approach
ROBUST WIDENULL ALOGRITHM
In snapshot deficient case, making eigenvalue decomposition
for R , we can obtain the following form M
Q
M
m 1
m 1
m Q 1
¦ Om em emH | ¦ Om em emH V v2 ¦
e m e mH .
(11)
Where Om m 1," Q is the Q largest eigenvalues and Q is the number of interferences. V v2 is the rest M Q
Fig.2 is a comparison of the beam pattern formed by the Mallioux method and the new method via 50 Monte-Carlo simulations. Fig.2(a) is the result of the Mallioux method and Fig.2(b) is the result of the new method. Compared Fig.2(a) with Fig.2(b), it can be noted that beampatterns obtained via the new method are more robust and nulls are wider.
eigenvalues, which are assumed to be equal. e m m 1," Q denotes the eigenvector corresponding with the mth eigenvalue, which span the interference subspace. Set interference subspace is U J span{e1 ," , eQ } , then the reconstructive interference covariance matrix by the augment matrix T can be become as
U J U HJ : T.
(12) Gain,dB,
R
Clearly, (12) is not include the noise term, so the influence of noise term will be reduce when (12) is used to beamforming in high dynamic. However, array performance loses will occur without noise terms and in the presence of mismatch between the presumed and actual steering vector of the GPS signal. Therefore, based on subspace theory and robust Capon beamforming idea in [11,12], the constrained optimization problem for the l th GPS signal is proposed as followed min a Tl R a T l H
a
0
-10
-20
-20
-30
-30
-40
-40
-50
-50
-60
-60
-70
-70 -80
-60
-40
-20
0
20
40
60
80
-80
-60
-40
-20
0
20
DOA,degree,
DOA,degree
(a)
(b)
40
60
80
Figure 2. The comparison of beam pattern obtained via two method (a) Mallioux method (b) The new method
The curves of input SNR versus output SINR calculated by different method are shown in Fig.3. From Fig.3, it can be seen that the array performance obtained by the conventional beamforming algorithm is greatly deteriorative due to sharp nulls ˈ but the new method works better than the Mallioux method when snapshot number is deficient due to rapidly changing interference direction with time in high dynamic environment
1
subject to || a T l as T l ||2 H .
0
-10
Gain,dB
R
IV. SIMULATION RESULT In the simulation experiment, consider a ULA composed of 16 elements. One GPS signals impinge on the array from 0q . The DOAs of two original interferences are 50q and 40q respectively and rapidly change with time. The noise is white Gaussian with SNR 18dB and INR 20dB . K 40, B S / 40 and the snapshot number is 10.
(13)
Where as T l is presumed steering vector and H is user parameter. To simplify the notation, in what follows, we sometimes omit the argument Tl of a T l and
-15
as T l .According to [11], the solution of (13) is
-20 -25 -30
a s (I N R ) 1 a s . (14)
Output SINR,dB
a
| zm |2 H , which can Where N is the solution of ¦ 2 m 1 (1 N/ m ) be calculate by Newton's method. zm is the m th element of M
z
wl as H (R +
1
N
N
-60 -65 -25
1
N
.
-24
-23
-22
-21 -20 Input SNR,dB
-19
-18
-17
-16
Figure 3. Curve of input SNR versus output SINR obtained by different method
In order to further illuminate the performance of the proposed anti-jamming algorithm for high dynamic GPS, the simulation experiment on the receiver’s acquisition capability is made. In the experiment, the acquisition is achieved by cross-correlating the received signal with the locally generated C/A code. When the receiver acquires the satellite, there is a maximum correlation. The comparison of acquisition
I ) 1 a s
I ) 1 R (R +
New method Conventional method Mailloux method
-45
-55
eigenvalue of R . According to (14) and literature [11], the weight vector of the proposed algorithm for the l th GPS signal is (R +
-40
-50
V H a s and V is the eigenvector of R . / m is the m th
1
-35
(15)
I ) 1 a s
performance obtained by the different method is shown as Fig.4. From Fig.4 it can be note that conventional beamforming algorithm is failed to acquire the GPS signal, and the Mallioux algorithm can acquire the GPS signal but the acquisition performance is poor. However, the new method exhibits much better acquisition performance because it can form robust widenull beam pattern in the high dynamic environment.
beamforming. The new method exhibits the wider nulls than the Mallioux method and more robust than the one. Simulation results have demonstrated the performance of the new method. ACKNOWLEDGMENT This work was supported by National Natural Science Foundation of China 61172112 and 61179064 and Fundamental Research Funds for the Central Universities ZXH2009A003.
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Normalized cross-correlation
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REFERENCES
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[1]
E. D. Kaplan, C. J. Hegarty, Understanding GPS principles and applications, Second Edition, ARTECH HOUSE, INC,2006. [2] D. Moelker, T. Van der Pol, and Y.Bar-Ness, “Adaptive antenna arrays for interference cancellation in GPS and GLONASS receivers,” in Proceeding of the IEEE 1996 Position Location and Navigation Symposium, pp. 191-198, 1996. [3] A. Gecan, M. Zoltowski, “Power minimization technique for GPS null steering antennas,” Institute of Navigation Conference, Palm Springs, CA, pp.13-15, September 1995. [4] R. L. Fante, J. J. Vaccaro, “Wideband cancellation of interference in a GPS receiver array,” IEEE Transaction on Aerospace and Electronic Systems,Vol 36, No.3,pp.549-564, 2000. [5] M. Johan, “Robust Navigation with GPS/INS and Adaptive Beamforming,” FOI-R—0848—SE, Swedish Defence Research Agency System Technology Division, pp.31-38, April 2003. [6] W. Sun, M .G. Amin, “A self-coherence anti-jamming GPS receiver,” IEEE Transactions on Signal Processing, Vol. 53, No.10, pp. 3910-3915, 2005. [7] R. A. Monzingo, T. W. Miller, Introduction to adaptive arrays, New York, Wiley,1980. [8] A. B. Gershman, G. V. Serebryakov and J. F. Bohme, “Constrained Hung-Turner adaptive beam-forming algorithm with additional robustness to wideband and moving jammers,” IEEE Trans. Antennas Propagate, Vol. 44, No. 3, pp.361-366, 1996. [9] R. J. Mailloux, “Covariance matrix augmentation to produce adaptive array pattern troughs,” Electronics Letters, Vol. 31, No.10, pp. 771–772, 1995. [10] R. B. Wu, C. Li, and D. Lu, “Power minimization with derivative constraints for high dynamic GPS interference suppression,” Sci China Inf Sci, Vol 55, pp. 857-866, 2012. [11] J. Li, P. Stoica, and Z .Wang, “On robust Capon beamforming and diagonal loading,” IEEE Transactions on Signal processing, vol 51, pp. 1702-1715, 2003. [12] G.S.Liao,H.Q.Liu, and J.Li, “A subspace-based rpbust adaptive Capon beamforming,” Proc.Proess in Electromagnetics Research Symposiun, PIERS 2006,Cambridge,Cambridge,USA,pp.374-379 ,2006.
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(c) Figure 4. Comparison of acquisition performance obtained by the different method (a) the conventional beamforming method (b) the Mallioux method (c) the new method
V.
CONCLUSION
In this paper, null broaden technique is used to high dynamic GPS anti-jamming. According to characteristic of snapshot deficient in the high dynamic environment, a robust widenull algorithm is proposed base on the interference subspace reconstruction and the idea of robust capon
