A Novel GPS Antijamming Receiver Based on Noncircularity

Abstract

We present a novel antijamming global position system (GPS) receiver that relies on the unique noncircularity of the GPS signal. Since the GPS signal is BPSK modulated, it is a noncircular signal while most interferers are circular signals. The proposed receiver utilizes this circular difference property to excise interferences while preserve the GPS signals. It is shown that the proposed receiver is a blind receiver that does not require any angular information of the satellites.

1. Introduction

A Global Positioning System (GPS) is a satellite-based all-weather navigation system and having a large success in civil application because its low prize receives its global coverage and its precision in navigation solutions [1]. However, the intentional interferences and the increasing pollution of electromagnetic environment may make GPS receivers lose their efficacy. Interference suppression in GPS can be conducted in the time, space, or frequency domain, or in a domain of joint variables [2–4]. It is noted, however, that the existing GPS interference cancellation techniques do not fully utilize the noncircularity property of the GPS signal.

Recently, more and more researches concerned on using the non-circularity of signals to improve the performance of signal processing such as root-MUSIC algorithm [5] and enhanced unitary ESPRIT [6] for non-circular sources DOA estimation, enhanced blind estimator [7], and nondata aided carrier frequency offset estimation in non-circular transmissions through frequency-selective channels [8]. GPS signal is BPSK modulated and is a non-circular signal while most interferers are circular signals.

This letter proposes a new interference suppression technique using, in effect, the circular difference between the GPS signal and the interferes. Unlike previous contributions in this area, the proposed GPS receiver exploits the unique non-circular property to suppress a large class of narrowband and broadband interferers. Furthermore, neither the knowledge of the satellite locations nor the synchronization between the satellite and the receiver is required to perform interference cancellation.

Section 2 describes the system model and introduces the proposed method. Section 3 will present simulation results and show that the proposed method can achieve a similar performance as an MMSE receiver. Section 4 will make conclusions.

2. Proposed Antijamming Receiver

A block diagram of the GPS receiver with an M-element spatial uniform linear array(ULA) is depicted in Figure 1.

Figure 1 

Block diagram of the GPS receiver.

The waveforms impinging on the array are those of the GPS signals, interferences, and noise. After down-conversion and sampled by chip-rate, the discrete-time format of the received signal vector form can be presented as

𝐱(𝑛)=𝐚_𝑠⋅𝑠(𝑛)+^𝑃_𝑝=1𝐚_𝑗𝑝⋅𝑗_𝑝(𝑛)+𝐯(𝑛), (1) where 𝑠(𝑛) is signal component, as steer vector of the signal; 𝐚_𝑠=[1,𝑒^{𝑗2𝜋(𝑑/𝑐)sin𝜃𝑠},…,𝑒^{𝑗2𝜋(𝑑/𝑐)(𝑀−1)sin𝜃𝑠}]^𝑇𝜃_𝑠 is steer vector of the GPS signal, where 𝑑 is the sensor spacing, 𝑐 is the propagation speed of the waveform, and 𝜃_𝑠 represents the arriving angle of the signal; 𝑗_𝑝(𝑛) is the waveform of the 𝑝th interfere; 𝐚_𝑗𝑝=[1,𝑒^{𝑗2𝜋(𝑑/𝑐)sin𝜃𝑗𝑝},…,𝑒^{𝑗2𝜋(𝑑/𝑐)(𝑀−1)sin𝜃𝑗𝑝}]^𝑇 is steer vector of the 𝑝th interferer, where 𝜃_𝑗𝑝 represents the arriving angle of the 𝑝th interferer; 𝐯(𝑛) is additive white Gaussian noise sample vector with variance 𝜎_𝑣.

Equation (1) can be rewritten as

𝐱(𝑛)=𝐬(𝑛)+𝐣(𝑛)+𝐯(𝑛) (2) where 𝐬(𝑛) denote the signal vector and 𝐣(𝑛)=∑^𝑃_𝑝=1𝐚_𝑗𝑝⋅𝑗_𝑝(𝑛) is the compound interference vector.

Under the assumption that GPS signals, interference, and noise are independent, the normal covariance matrix 𝐑_𝑥𝑥 of the received signal becomes

𝐑_𝑥𝑥=𝐸{𝐱(𝑛)⋅𝐱^𝐻(𝑛)}=𝐑_𝑠+𝐑_𝑗+𝐑_𝑣, (3) where 𝐸{⋅} represents the statistical expectation. (⋅)^𝐻 denotes complex conjugate transpose, and 𝐑_𝑠, 𝐑_𝑗, and 𝐑_𝑣 are covariance matrices of the GPS signals, the interference, and the noise, which are defined, respectively, as

𝐑_𝑠≜𝐸{𝐬(𝑛)⋅𝐬^𝐻𝐑(𝑛)},_𝑗≜𝐸{𝐣(𝑛)⋅𝐣^𝐻𝐑(𝑛)},_𝑣≜𝐸{𝐯(𝑛)⋅𝐯^𝐻(𝑛)}=𝜎_𝑣𝐈, (4) where 𝐈 is an 𝑀×𝑀 identity matrix.

GPS signal is BPSK modulated and is a non-circular signal while jammers and noise are circular signals. The elliptic covariance matrix 𝐑^′_𝑥𝑥 of the received signal should be

𝐑^′_𝑥𝑥=𝐸{𝐱(𝑛)⋅𝐱^𝑇(𝑛)}=𝐑^′_𝑠+𝐑^′_𝑗+𝐑^′_𝑣, (5) where (⋅)^𝑇denotes transpose, and 𝐑^′_𝑠,𝐑^′_𝑗, and 𝐑^′_𝑣 are elliptic covariance matrices of the GPS signals, the interference, and the noise. To circular signals, its elliptic covariance matrix is equal to zero, which means that

𝐑^′_𝑗≜𝐸{𝐣(𝑛)⋅𝐣^𝑇𝐑(𝑛)}=0,_′𝑣≜𝐸{𝐯(𝑛)⋅𝐯^𝑇(𝑛)}=0. (6) Then, (5) becomes

𝐑^′_𝑥𝑥=𝐑^′_𝑠. (7) Let 𝐰 be the 𝑀×1 weight vector. Then, the output of the beamformer is given by

𝐳(𝑛)=𝐰^𝐻⋅𝐱(𝑛), (8) and the beamformer 𝐰 is obtained by Maximizing the Non-circular to Circular Ratio (MNCR) of the output signal:

𝐰_MNCR=argmax_𝐰𝐸{|𝐳(𝑛)⋅𝐳^𝑇(𝑛)|}𝐸{|𝐳(𝑛)⋅𝐳^𝐻(𝑛)|}=argmax_𝐰|𝐰^𝐻𝐑^′_𝑥𝑥𝐰^∗|𝐰^𝐻𝐑_𝑥𝑥𝐰, (9) where (⋅)^∗denotes complex conjugate.

Since the denominator of (9) is positive real, we notice that (9) is unchanged when 𝐰 undergoes an arbitrary phase rotation. Then, by choosing a 𝐰 such that Im{𝐰^𝐻𝐑^(𝑇)_𝑥𝑥𝐰^∗}=0 where Im{⋅} denotes the imagine part, then, (9) can be rewritten as

𝐰_MNCR=argmax_𝐰Re{𝐰^𝐻𝐑^′_𝑥𝑥𝐰^∗}Re{𝐰^𝐻𝐑_𝑥𝑥𝐰}. (10) Let

𝐰_𝑟=Re{𝐰};𝐰_𝑖𝐑=Im{𝐰},_𝑟𝑥𝑥=Re{𝐑_𝑥𝑥};𝐑_𝑖𝑥𝑥=Im{𝐑_𝑥𝑥𝐑},_{′𝑥𝑥𝑟}=Re{𝐑^′_𝑥𝑥};𝐑^′_𝑥𝑥𝑖=Im{𝐑^′_𝑥𝑥}, (11) where Re{⋅} denotes the real part. Then we can expand (10) in complex form as

𝐰_MNCR=argmax_{(𝐰𝑟,𝐰𝑖)}Re{(𝐰^𝑇_𝑟−𝑗𝐰^𝑇_𝑖)(𝐑^′_𝑥𝑥𝑟+𝑗𝐑^′_𝑥𝑥𝑖)(𝐰_𝑟−𝑗𝐰_𝑖)}Re{(𝐰^𝑇_𝑟−𝑗𝐰^𝑇_𝑖)(𝐑_𝑥𝑥𝑟+𝑗𝐑_𝑥𝑥𝑖)(𝐰_𝑟+𝑗𝐰_𝑖)}=argmax_𝐰𝐰^𝑇(𝐑^′_𝑥𝑥)𝐰𝐰^𝑇(𝐑_𝑥𝑥)𝐰, (12) where 𝑗=√−1 and 𝐰=[^𝐰𝑟_𝐰𝑖] is a 2𝑀×1 real vector; 𝐑^′_𝑥𝑥=[^{𝐑′𝑥𝑥𝑟𝐑′𝑥𝑥𝑖}_{𝐑′𝑥𝑥𝑖−𝐑′𝑥𝑥𝑟}] and 𝐑_𝑥𝑥=[^{𝐑𝑥𝑥𝑟−𝐑𝑥𝑥𝑖}_{𝐑𝑥𝑥𝑖𝐑𝑥𝑥𝑟}] are both 2𝑀×2𝑀 real matrices.

Thus, the optimum 𝐰 is the eigenvector corresponding to the maximum eigenvalue of the following generalized eigenvalue problem:

𝐑^′_𝑥𝑥⋅𝐰=𝜆_max⋅𝐑_𝑥𝑥⋅𝐰, (13) where 𝜆_max denotes the maximum eigenvalue, which is also the maximum NCR.

In practice, 𝐑^′_𝑥𝑥 and 𝐑_𝑥𝑥 are replaced by their sample estimate:

𝐑^′_𝑥𝑥≜1𝐾𝐗_𝐾⋅𝐗^𝑇_𝐾,𝐑_𝑥𝑥≜1𝐾𝐗_𝐾⋅𝐗^𝐻_𝐾, (14) where 𝑀×𝐾 data matrix is 𝐗_𝐾≜[^{𝐱(𝑛),…,𝐱(𝑛−(𝐾−1))}].

The updating algorithm may now be outlined as follows:

(1)estimating 𝐑^′_𝑥𝑥 and 𝐑_𝑥𝑥 using data sample matrix 𝐗_𝐾;(2)forming matrices 𝐑^′_𝑥𝑥 and 𝐑_𝑥𝑥 using real and imagine part of 𝐑^′_𝑥𝑥 and 𝐑_𝑥𝑥;(3)determining an estimate 𝐰 according to (13);(4)finally, the estimate of 𝐰_MNCR can be written as

𝐰_MNCR=𝐰_𝑟+𝑗𝐰_𝑖. (15)

3. Simulations

A 7-element uniform linear array with half-wavelength spacing is used in the simulation. GPS navigation symbols are in the BPSK format and spread by the Gold code with processing gain 𝑃 = 1023. At the receiver, chip-rat sampling is performed, and 𝐾 = 800 samples are collected. Jammers used in the simulations are generated as broadband binary signals having the same rates as the C/A-codes but with a different structure than that of the C/A signals.

In the first simulation, to examine the performance of the proposed GPS receiver in presence of multiple strong jammers, we compare the proposed GPS receiver with the MMSE receiver in [4]. In the simulation, the satellite is located at −20^∘, and the three jammers are at 10^∘, 40^∘, and 60^∘. It is noted that the proposed receiver need no prior information of the satellite location while the MMSE receiver does need the knowledge in order to compute the power of the GPS signal. Figure 2 shows that both receivers are capable of suppressing strong jammer while preserving the GPS signal, with the proposed receiver having the advantage of being a blind receiver.

Figure 2 

Comparison of Antenna gains between the self-coherence receiver and the MMSE receiver with SNR = −20 dB, JSR = 60 dB.

In the second simulation, consider the same condition of the first simulation; we examine the effectiveness of the proposed GPS receiver. In GPS, the receiver is ultimately evaluated based on its ability to provide accurate pseudorange measurements. This is achieved by establishing synchronization between the receiver and the satellite, which is decided based on the cross-correlation between the beamformer outputs and a locally generated Gold sequence [1]. The normalized cross-correlations before and after the jammer removal in Figure 3, show that the proposed GPS receiver can effectively cancel directional jammers and achieve synchronization even when the JSR is as high as 60 dB (Figure 3(b)). Without interference suppression, however, synchronization fails, as is evident in Figure 3(a).

(a)

(b)

Figure 3 

(a) Normalized cross-correlation before jammer removal. (b) Normalized cross-correlation after jammer removal.

4. Conclusion

We have presented a new GPS receiver that is based on the non-circularity of the GPS signals. Due to the different circularity between the GPS signal and interferers, we present new adaptive beamforming criteria—the Maximum Non-circular to Circular Ratio (MNCR) criteria. Utilizing this criterion, an anti-jamming GPS receiver is constructed to mitigate a wide class of interferers. Simulations have shown that the proposed receiver is capable of suppressing strong jammers while preserving GPS signals, without requiring any knowledge of the angular information of the satellites.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 60772146, the National High Technology Research and Development Program of China (863 Program) under Grant 2008AA12Z306, and in part by Science Foundation of Ministry of Education of China under Grant 109139.