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2Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, P. R. China e-mail: [email protected].
STAP Compensation Technique Based on Homomorphic Filtering in GPS Renbiao Wu1,2, Rulan Xu2, Dan Lu2, Jianpin Yu1 1

Intelligent Information Institute of ATR Lab, Shenzhen University, Shenzhen 518060, P. R. China Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, P. R. China e-mail: [email protected]

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Abstract—Space time adaptive processing (STAP) can be used to remove hostile jammers interfering for Global Positioning System (GPS) receivers, especially in the presence of jammer multipath. However, the STAP filter can bring in distortion to the GPS signal because it does not have a uniform frequency response across the operating band, which would dramatically reduce the accuracy of the users’ position. Based on the homomorphic filtering a new method is proposed to compensate the distortion caused by the STAP filter. Finally, simulation results show the proposed method is effective.

of the user's position with the characteristics of simple realization and not so much calculation. This paper will be structured as follows. In Section 2 we give a uniform data model and describe the structure characteristic of STAP technique. In section 3 we explain the reason of the distortion caused by STAP and propose a novel approach based on homomorphic filtering. In section 4 simulation results verify the effect of the proposed algorithm.

2. STAP TECHNIQUE STAP has been proposed for mitigating multipath jamming environments. Consider a single jamming emitter as seen by the array, if there is no dispersion[10], the signal seen at different elements in the array will be completely correlated, differing only by a gain and phase. Because of this, they can be linearly combined to self-cancel: thus nulling the jammer. Now we assume that there is some dispersion in the array. The signal seen at the different antenna elements will no longer be completely correlated at any instance in time, and no linear combination of the antenna outputs can completely cancel the interference when using a spatial-only beamforming algorithm.

1. INTRODUCTION In recent years GPS has gained increased attention for numerous applications in both commercial and military operations. However, the jamming threat is serious because of the physical design of the GPS system. When the antijamming array is mounted on an aircraft, jammer multipath reflections from distant portions of the airframe will cause the signals at different antennas to become only partially correlated. Under these circumstances, a spatial only adaptive array will require multiple degrees-of-freedom to null out a single jamming source which makes adaptive nulling of interference signals more difficult. Then STAP [1] is proposed for mitigating multipath jamming environments.

STAP technique for GPS interference mitigation is to extend the space adaptive processing to space-time adaptive processing which would has a lot of space and time domain degrees of freedom to eliminate the jammer and jammer multipath effectively. The architecture of STAP is shown in figure1. M elements equi-spaced linear array with P taps at each antenna is used.

However the architecture of STAP does not have a uniform frequency response across the operating band. It is possible that STAP will introduce a distortion to the desired GPS signal, which will broaden the peak or shift the peak position of cross correlation function (CCF) between the array output and a desired signal. The impact to CCF would potentially bias the GPS positioning solution[2]. So we want to equalize the impact of STAP after resisting the interference. The existing weight constraint method[3] equals the different satellites signal distortion to the same, which brings complexity to the derivation of different STAP algotithms by adding orthogonal constraint; least square inverse filter method [5] will bring time delay after compensation and has large amount calculation. In this paper, a new method based on homomorphic filtering is proposed to compensate satellite signal distortion introduced by STAP, which achieve accurate measurement 978-1-4244-5128-9/10/$26.00 ©2010 IEEE

We assume the array is in the far field and thus that it receives a plane wave. Assume that the first antenna of the linear array is the reference antenna. The incidence direction of emission target signals is T  [S / 2, S / 2 ] ˈ then the steering vector of the equi-spaced linear array can be denoted as: a

841

2S ª  j 2S d sin T j ( M 1) d sin T º ,..., e O «1, e O » «¬ »¼

T

(1)

1

x11

T

x12

T

xM 2

T w11

w12

M

xM 1

T wM 1

T

wM 2

T xMP

x1P w1P

wMP

ě

y (t ) Figure 1 - STAP technique for GPS interference mitigation The incidence signals of the array include GPS signal and K jammers. The space-time data model[6] can be written as: x n

Since the array place nulls in the direction of the interferences, we can easily find that K

K

As n  ¦ B j i j n  v n

w H ¦ B ji j n | 0

(2)

Denoting

x( n)

as

the

space-time

snapshot And then the array output can be written as:

T

, where A I Pu P … a , a is steering vector of the desired GPS signal, … is denoted as the Kronecker product. The satellite signal can be written as the P taps of the array received signal: x(n) [ x11 (n). x.M 1 (n) x12 (n). x.M 2 (n). x.1P ( n). x.MP ( n)]

s n

ª¬ s n s n  1 ... s n  P  1 º¼

T

y n | w H As n  w H v n

(3)

h n

additive white Gaussian noise vector.The STAP works by choosing the adaptive weights w with different algorithms to preserve the desired GPS signal while simultaneously minimizing all interference.

> w11,.., wM 1, w12 ,.., wM 2 ,.., w1P ,.., wMP @T

y n

(4)

ª º s n « » « s n  1 » « » ¬ª h 0 h 1 h 2 ... h P  1 ¼º « s n  2 »  v n ... « » « s n  P  1 » ¬ ¼ h 0 s n  h 1 s n  1  ...  h P  1 s n  P  1  v n

The array output signal is different from the desired GPS signal s (n) , which is actually the convolution of the array response to GPS signal with noise:

w H x n K

(8)

(9)

STAP can suppress interference effectively, but it will also bring some distortion to GPS signal, which must be taken into consideration. The array output signal after STAP is:

w H As n  w H ¦ Bi j n  w H v n

¬ª h 0 h 1 h 2 ... h P  1 ¼º

We will find that

3. HOMOMORPHIC FILTERING COMPENSATION ALGORITHM

y n

(7)

Let h(n) w H A , v(n) w H v(n) , where h(n) is 1 u P vector, corresponds to response coefficients of array different time delay

B j and i j have the same structure as A and s(t ) , v (n) is an

w

(6)

j 1

j 1

y n

(5)

i 1

842

h n s n  v n

(10)

Then STAP will effect a distortion to GPS signal and affect the CCF of the received signal with the known signal, which will bias the GPS position solution.

With the interference mitigation algorithm and the direction of satellite signals, we can obtain the STAP response h(n) . Then the distortion caused by STAP can be compensated. From equation (15), additive distortion can be subtracted,

If the GPS satellite emits a signal with a Fourier transform S ( jZ ) , then the signal output after passing through the STAP adaptive filter is H ( jZ ) S ( jZ ) , then the CCF [8] is r l

1 2S

log S1 jZ

(11)

S1 jZ

S

Where P( jZ ) is the power spectrum of GPS signal, so that in the absence of H ( jZ ) , it can introduce both a broadening of the correlation peak and a shift of that peak from the correct value. And the basic idea of acquisition in GPS is to measure the beginning of the C/A code very accurately, which can be carried out by measuring the CCF of array output signal and the C/A code. So it will cause large error to the high-precision GPS if the CCF is not precise. Therefore, we need to compensate the distortion made by STAP.

noise, S ( jZ )

and

the

desired

satellite

signal

is 

S1 ( jZ )  V1 ( jZ ) . With inverse Fourier transformˈ

4. SIMULATIONS In this section, we will present some simulation results to evaluate the STAP performance of interference suppression and the effect of the proposed novel compensation algorithm. In the simulations, a uniform linear array with no mutual coupling is used. Consider the case of an M 4 element linear array with P 3 taps at each antenna. The array element spacing is half-wavelength. The intermediate frequency is 1.25MHz and sampling frequency is 5MHz. The SNR is 20dB and JNR is 20dB . The jammers involve a 1.25MHz single frequency interference and dispersion multipath jammers simulated by the method metioned in[3],[11]. Five satellites incident to the array and the direction of arrival (DOA) of different signals are listed in Table 1 respectively. The initial position of user receiver is located in a certain Latitude Longitude Height (LLH) . Firstly, the performance of interference suppression and acquisition is compared between space-only and STAP power minimization algorithm. Secondly, the array output signal of STAP MMSE[8] algorithm and the output signal of homomorphic filtering are correlated to the local C/A code respectively to compare the normalized CCF. Thirdly, the positioning root mean square error (RMSE) comparison is carried out by using the STAP MMSE algorithm and three compensation approaches.

(13)

where V ( jZ ) H ( jZ )V1 ( jZ ) ˈlet S1 ( jZ ) S ( jZ )  V1 ( jZ ) ˈ H jZ S1 jZ

(17)

The process of the homomorphic filtering algorthm is sample and it overcome the algorithm complexity compared with weight constraint method, and it only involves FFT, logarithmic and exponentiation processing, which will not cause large calculation compared with inverse filter method.

For deriving convenience, we rearrange the equation to be:

Y jZ



we can obtain the signal s (n)  v1 (n) , where  v1 (n) is additive noise.

Considering the principle of homomorphic filtering algorithm, it’s obvious that the distortion in GPS caused by STAP can be compensated by homomorphic filtering. From equation (10), the effect of STAP to GPS signal can be regarded as a convolution distortion. So the homomorphic filtering approach can be used to restore the desired satellite signal. Firstly, Fourier transform is made to equation (10), converting the convolution distortion to multiplicative distortion (12) Y jZ H jZ S jZ  V jZ H jZ S jZ  V1 jZ



exp log Y jZ  log H jZ

 S1 ( jZ ) is the frequency response of satellite signal with

Homomorphic filtering algorithm[9] is a nonlinear processing method which is based on homomorphic mapping, it is always used as a deconvolution technique. In the process of homomorphic filtering, firstly the convolution distortion is converted to multiplicative distortion by fast Fourier transform (FFT), and logarithmic transformation is used to the spectrum of distortion signal, making multiplicative distortion into additive distortion which can be subtracted. Then we can restore the original desired signal.

Y jZ

(16)

After exponential operation to equation (16)

S

³ P jZ H jZ exp  jZl dZ

log Y jZ  log H jZ

(14)

We obtain the position result of user receiver by a Danmark GPS software receiver[12] which is able to perform acquisition, code and carrier tracking, navigation bit extraction, navigation data decoding, pseudorange estimation, and position computations.

Utilizing logarithmic transform to make multiplicative distortion into additive distortion log Y jZ log H jZ S1 jZ log H jZ  log S1 jZ (15)

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TABLE 1 DOA of different signals incident to the array DOA Code Type q q 20 ,5 ,15q , 25q ,35q Satellite 1, 14,22,25&6 Single frequency interference 70q 50q ,80q Wideband interference and dispersion multipath

0 -5 -10

Gain

-15 -20 -25 -30 -35 -40 -45 -100

-80

-60

-40

-20

0 degree

20

40

60

80

100



 Figure 2 - Antenna beam pattern via spatial-only algorithm

Figure 3 - Antenna beam pattern via STAP algorithm

From figure 2 and figure 3, it’s clear to find that the STAP interference mitigation algorithm can form deep nulls in the directions of all jammers. But the space-only interference mitigation algorithm cannot null deep enough in the direction of dispersion multipath jammers, which cause the

interference can not be completely inhibited. As a result, the acquencision performance shown in figure 4 is influenced. Just three satellite signals were acquired, which can’t calculate the user’s position. However, figure 5 shows that the STAP algorithm can acquire all the satellite signals. Acquisition results

Acquisition results 5

9

4.5

8

4

7

Acquisition Metric

Acquisition Metric

3.5 3 2.5 2 1.5

6 5 4 3 2

1 0.5 0

Not acquired signals Acquired signals

1

Not acquired signals Acquired signals 0

5 10 15 20 25 PRN number (no bar - SV is not in the acquisition list)

0

30

0

5 10 15 20 25 PRN number (no bar - SV is not in the acquisition list)

30

Figure 4 - Acquisition performance of spatial-only algorithm Figure 5- Acquisition performance of STAP algorithm

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In figure 6, we examine the normalized CCF curves of 1th satellite. The output signal of STAP and the output signal of homomorphic filtering are correlated to the local C/A code respectively. The STAP algothem introduces a distortion to the desired GPS signal, which broadens the peak or shifts the peak position of CCF as fork dashed line shown in figure 6. After using the homomorphic filtering, it’s obvious to find that the main lobe of the CCF can be narrowed and the peak position of the CCF can be corrected to zero. The positioning RMSE results of STAP anti-jammer algorithm and three compensation approaches are listed in Table 2. It illustrates that the STAP positioning RMSE can be diminished by three compensation approaches, and the homomorphic filter approach also presents superiority of the positioning accuracy.

Figure 6-Comparison of CCF corresponding to homomorphic filter (“-o”) and uncompensated STAP (“-.h”)

TABLE 2 Comparison of the positioning error results RMSE(m) 2-D 18.91 STAP 13.19 Inverse Filter 3.82 Weight Constraint 2.29 Homomorphic Filter

5. CONCLUSIONS [5] Li Shuangxun, Cheng Zhu, Huang Fukan, “A Compensating Approach for Signal Distortion Introduced by STAP”, Communication Technology, pp. 1-4, Nov. 2730, 2006. [6] D. Lu, Q. Feng, R. B. Wu, “Survey on Interference Mitigation via Adaptive Array Processing in GPS” in Electromagnetics Research Symposium 2006. Cambridge, USA, pp.722-727, March.26-29, 2006. [7] Li Ping, “Adaptive Interference Suppression Techniques Based on Cyclostationarity in GPS”, Master's thesis, Tianjin: Civil Aviation University of China, 2008. [8] R. L. Fante, J. J. Vaccaro, “Wideband Cancellation of Interference in A GPS Receiver Array”, IEEE. Trans. AES, Vol .36(2), pp. 549-564, April 2000. [9] S. V. Vaseghi, “Advanced Digital Signal Processing and Noise Reduction (Third Edition)”, Wliey, 2006. [10] Gary F. Hatke, Tri T. Phuong, Check F. Lee and Robert G. Callahan, “Multipath Effects on F-15 and F-16 Multi-Channel GPS Antenna Performance”, Thirty-Third Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, CA,pp. 922-926, Oct. 1999. [11] Gary F. Hatke, Ali F. Yegulalp “A Novel Technique for Simulating Space-Time Array Data”, Signals, Systems and Computers, 2000. Conference Record of the ThirtyFourth Asilomar Conference on Vol.1, pp. 542 – 546, 29 Oct.-1 Nov. 2000 [12] Kai Borre, Dennis M. Akos,“A software-defined gps and galileo receiver” Boston. Basel .Berlin, 2006.

The necessary of using STAP to remove jammer and jammer multipath is validated. A novel approach based on homomorphic filtering is proposed to compensate the distortion caused by STAP. Positioning results verify the effect of proposed algorithm.

6. ACKNOWLEDGEMENTS This work is supported in part by the 863 High Tech Project of China under grant 2006AA12Z321 and by the Open Foundation of ATR Lab of Shenzhen University.

REFERENCES [1] Sun Xiaoxu, Huang Fukan, “GPS Receiver SpaceTime Joint Anti-Jamming Technique”, Communication Transaction, Vol.4 (9) , Sep.2003. [2] James Bao-Yen Tsui, “Fundamentals of Global Positioning System Receivers”, Wiley, 2005. [3] G. F. Hatke, “Adaptive Array Processing for Wideband Nulling in GPS System”. Signals, Systems & Computers, 1998. Conference Record of the Thirty-Second Asilomar Conference, Vol .2, pp. 1332-1336, Nov. 1998. [4] Myrick, W. J. Goldstein, M. Zoltowski, “Low Complexity Anti-Jam Space-Time Processing for GPS”, Proceedings of 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing, Vol.4, pp. 13321336, 2001.

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