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ORIGINAL RESEARCH article

Front. Phys., 01 February 2023
Sec. Interdisciplinary Physics
This article is part of the Research Topic Advanced Signal Processing Techniques in Radiation Detection and Imaging View all 13 articles

Direction and distribution sensitivity of sup-DOF interference suppression for GNSS array antenna receiver

Yifan SunYifan SunFeiqiang ChenFeiqiang ChenFeixue Wang
Feixue Wang*Wenxiang LiuWenxiang LiuBaiyu LiBaiyu LiJie SongJie Song
  • College of Electronic Science, National University of Defense Technology, Changsha, China

Introduction: Distributed wideband jamming (interference) is commonly used in navigation countermeasure. Due to the limited volume of GNSS (Global Navigation Satellite System) array antenna receiver, the number of interferences usually exceeds the number of array elements. At present, the anti-jamming capability and mechanism of Global Navigation Satellite System array antenna against distributed sup-DOF (Degree of Freedom) interference have not been fully studied.

Methods and innovation: To solve this problem, this paper analyzes the characteristics of GNSS System array antenna against sup-Degree of Freedom interference by formula derivation and simulation. Firstly, the definition of sup-Degree of Freedom interference of Global Navigation Satellite System array antenna is proposed from the perspective of spatial anti-jamming; Secondly, the directional characteristics of Global Navigation Satellite System array antenna for sup-Degree of Freedom interference suppression are analyzed.

Results: The results show that the performance of sup-Degree of Freedom interference suppression is sensitive to the direction and distribution of interference. On the one hand, the residual interference power varies from interference direction and distribution, while the minimum value of which is zero and the maximum value is the sum of interference power. On the other hand, the suppression performance of UCA (Uniform Circular Array) and central Uniform Circular Array is periodic along azimuth. If the number of elements on the circumference is M (M ≥ 3), the period of the suppression performance is 4π/M/3+(1M+1.

Discussion: The conclusion of this paper show the upper and lower bounds of sup-Degree of Freedom interference suppression performance and the variation rule in azimuth, which can be used in the fields such as interference deployment, anti-jamming performance evaluation and anti-jamming algorithm development.

1 Introduction

GNSS (Global Navigation Satellite System) provides convenient positioning, navigation and timing services for its application terminals [1]. It has played an important role in transportation, marine fishery, geological disaster monitoring and emergency rescue. However, the GNSS signal power received on ground is weak [2], which is 30 dB lower than the thermal noise of the receiver [3]. The GNSS receiver is vulnerable to unintentional or intentional interference (jamming) under the complex electromagnetic environment, resulting in the receiver performance degradation or failure [4].

Distributed interference is a commonly used interference style in navigation countermeasure [5]. In this case, the number of interferences usually exceeds the number of elements of GNSS antenna array, which might make the receiver unavailable for positioning [6]. On the one hand, it is difficult to completely suppress interferences since the orthogonality between the spatial filter coefficients and the interference steering vectors no longer exist [7, 8]. On the other hand, most GNSS array antennas have only 4–7 elements [9, 10]. Due to limited space of navigation facilities, half wavelength of L-band GNSS signal and low cost of interference equipment, it is easier to increase the number of interferences than the number of array elements [11, 12].

There is a lack of definition of sup-degree-of-freedom interference in GNSS anti-jamming research. In array signal processing, it is generally considered that the DOF (Degree of Freedom) of an array with N elements is N-1. In the field such as sparse array [13], virtual array [14], polarized array antenna [15] and synthetic aperture [16], there is only the concept of “interference exceeds array DOF” without clear definition; In the field of DOA (Direction of Arrival) estimation, direction estimation ambiguity is defined and classified [17, 18], which provides reference for the study of interference direction against array DOF; Some researchers propose that N-element GNSS antenna array receiver can at most suppress N-1 interference [19], but this statement is not accurate considering the preconditions for the conclusion are not clearly explained, and the signal types and parameters are not limited; At the same time, most beamforming algorithms and DOA estimation algorithms focus on the precondition that the number of interference is less than N [20, 21]. In order to further study the anti-jamming characteristics of array antenna while the number of interferences is equal or larger than N, a clear and simple definition of sup-degree-of-freedom interference is needed to specify the background. It would be better if the definition focuses on the array anti-jamming module rather than considering the positioning performance of GNSS receivers. Otherwise, parameters related to signal and data processing should be further introduced [22], such as receiver acquisition and tracking threshold and DOP (Dilution of Precision) constrains, which makes the definition complicated.

To solve the above problems, from the perspective of spatial anti-jamming, this paper first analyzes the precondition that N-element array can suppress at most N-1 interferences, and proposes the definition of sup-DOF interference; Secondly, according to theoretical analysis and simulation, the paper proposes that the suppression performance of array antenna is sensitive to direction and distribution of sup-DOF interference. The structure of the paper is as follows: In the second section, the model of array signal reception and anti-jamming is established; In the third section, the definition of sup-DOF interference is proposed and is explained by numerical calculation; In the fourth section, based on theoretical analysis and numerical calculation, it is proposed that the suppression performance has directional sensitivity, distribution sensitivity, as well as azimuthal periodicity; In the fifth section, the conclusion in the fourth section is verified by simulation; The structure block diagram of the article is shown in Figure 1, in which the orange part is the innovation of this article.

FIGURE 1
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FIGURE 1. Article structure block diagram.

For the convenience of reading, the commonly used symbols in this paper are shown in Table 1. In the text, symbols in italics represent variables, and non-italics in bold represent vectors or matrices.

TABLE 1
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TABLE 1. Common symbols in this article.

2 Signal reception and anti-jamming model of antenna array

Suppose that the navigation signal and interference signal are received by an N-element array antenna. Denote navigation signal, signal power and steering vector as st, ps, and as respectively. Denote the interference, power and steering vector as jt, pj, and aj respectively. Denote the noise power as pn, while t represents time. The steering vector is N × 1 dimensional column vector. Let the number of navigation signals and interference be I and K respectively, the received signal of N-element array is

xt=i=1Ipsisitasi+k=1Kpjkjktajk+pnnNt(1)

where i represents the ith navigation signal, k represents the kth interference signal, and nNt represents the thermal noise of the N-dimensional array. The steering vector is composed by the signal phase difference between each array element and the reference element. In order to simplify the analysis, the non-ideal factors that cause the steering vector mismatch is ignord, and the narrowband signal model is adopted. The steering vector has the following form, where τ1,τ2,τN1 is the time delay between the received signal of each array element and the reference element, and fc is the carrier frequency.

a=1expj2πfcτ1expj2πfcτN1T(2)

Denote the normalized spatial filter coefficient as w¯.it is an N × 1 dimensional column vector with the modulus of 1.

w¯=w1w2wNT(3)
w¯=1(4)

Then the output signal of spatial anti-jamming processing is

yt=w¯Hxt(5)

If interference signals are independent of each other, the covariance matrix of interference signals is

Rjj=k=1KpjkajkajkH(6)

3 Definition of sup-DOF interference and numerical analysis

In array signal processing, it is generally considered that the DOF of an N-element array is N-1. However, the statement that N-element array can suppress at most N-1 interferences is not accurate. In this section, from the perspective of spatial anti-jamming, the preconditions for the above statement are analyzed, and the definition of sup-DOF interference is proposed.

3.1 Definition of sup-DOF interference

According to Eqs 1, 5, the residual interference signal is

yjt=w¯Hkjktajk(7)

There are two ways to interpret principle of spatial interference suppression. One is to figure out w¯ satisfies w¯0 and the following equation set

w¯Haj1j1t=0w¯Haj2j2t=0w¯HajKjKt=0(8)

The constrains make sure that the antenna gain of direction described by ajk is 0, thus suppress the interference coming from this direction. The other way is to solve w¯ that makes the residual interference power zero:

pjres=Ek=1Kw¯Hajkjk(t))k=1Kw¯Hajkjkt)*=0(9)

where pjres is the residual interference power.

Denote the interference steering vector matrix as

Aj=aj1aj2ajK(10)

According to Eq. 6, the covariance matrix of the interference signal is non-negative. As a result, Eq. 8 is equivalent to the following form:

w¯HAj=0r(11)

where 0r is a row vector, Aj is an and matrix of dimension N×K. Denote the rank of Aj as rankAj, the maximum value of rankAj is N. Analyze the solution set of Eq. 8. If rankAjN-1, w¯ satisfies Eq. 8 and w¯0, thus the interferences can be completely suppressed. If rankAj = N, Eq. 8 is satisfied only if w¯=0, thus the navigation signal cannot be retained while suppressing interference, and anti-interference processing becomes invalid. It can be seen that for mutually independent interferences, the anti-jamming ability of the N-element array antenna depends not only on the number of interferences, but also on the spatial correlation of the interference signal steering vector. The precondition for the N-element array to suppress at most N-1 interference is that the steering vectors of the interference signal are linearly uncorrelated. The precondition for N-element array to suppress more than N-1 interferences is that the rank of the interference steering vector matrix is not greater than N-1.

Under the condition that the interferences are independent of each other, the definition of array sup-DOF interference is proposed according to spatial anti-jamming principle.

Definition 1: Assume interferences are independent of each other, and their steering vectors can be represented by a finite number of column vectors. When the number of interferences simultaneously received by the antenna array is greater than or equal to the number of array elements N, and at least N steering vectors are linearly uncorrelated, it is defined that the number of interferences surpass the array degree of freedom.

This definition can be described as follows:

rankAj<NinterferencesdonotsurpassthearraydegreeoffreedomrankAj=Ninterferencessurpassthearraydegreeoffreedom(12)

The above definition is only applicable to the case where interference signals are independent of each other. If the interference signal has correlation, or the steering vector of the signal cannot be represented by a limited number of column vectors, the interference suppression principle can be analyzed through the residual interference power of anti-jamming processing.

According to Eq. 7, the residual interference power is

pjres=Eyjtyj*t(13)

The above equation can be written as

pjres=w¯HRjjw¯(14)

For GNSS array antenna receiver, the goal of anti-jamming process is to make the residual interference power zero, while retaining the navigation signal as much as possible. That is to say, the solution w¯ satisfies w¯0 while

pjres=w¯HRjjw¯=0(15)

According to Rayleigh entropy theorem, the value range of interference residual power is

λminpjresλmax(16)

where λmin and λmax is the minimum and maximum eigenvalues of Rjj. The equal signs are taken when w¯ equals to the corresponding eigenvector. In this case, the residual interference power of the anti-jamming process is optimized.

If the minimum eigenvalue of Rjj is zero, the residual power is 0, and the interference signal is completely suppressed, which means that Rjj is not a full-rank matrix; If the minimum eigenvalue of Rjj is not zero, the interference signal cannot be completely suppressed, at this moment Rjj is a full-rank matrix.

Thus, the sup-DOF interference can be defined by the rank of the interference covariance matrix:

Definition 2: Assume the number of interferences received by the array antenna simultaneously is K (K ≥ 1). If the rank of the covariance matrix of the received interference equals to the number of antenna elements N, the interferences surpass the array degree of freedom. The above interferences are collectively referred to as array sup-degree of freedom interference, or sup-DOF interference for short.

Denote the rank of the interference covariance matrix as rankRjj. This definition can be described as

rankRjj<NinterferencesdonotsurpassthearraydegreeoffreedomrankRjj=Ninterferencessurpassthearraydegreeoffreedom(17)

According to Formula. 17, rankRjj is only related to the second-order statistical characteristics of the received array interference. The scope of application of this definition has no limit on the signal correlation and spatial correlation of the interference signals.

Based on the above analysis, the following conclusion can be drawn:

Conclusion 1: If the number of interferences is greater than or equal to the number of array elements (K ≥ N), the interference may not surpass the array degree of freedom. The existence of a specific direction allows the antenna array to completely suppress K interferences.

The specific incident directions in conclusion 1 can be divided into two categories. The first category is that the incident directions of interference are different, while their steering vectors are equal or conjugate; The second category is that the K steering vectors corresponding to different incident angles are correlated with each other and can be represented by N-1 vectors. Among them, it is difficult to find the incident direction of the second category by enumerating. Finding the incident direction of the second category has become an open problem in the field of differential geometry, which will not be further researched in this paper.

3.2 Numerical calculation and analysis

Taking the central UCA (Uniform Circular Array) of four elements as an example, the typical interference incidence direction is taken to further explain conclusion 1. The coordinates of array element are given in Eq. 18. Assume that the power of interferences in Figure 2 is 1, and the interference signals are independent of each other. In Figure 2A, the incident direction of interference 1 is θj1,φj1. Interference 1 and interference 2 are symmetrical about the xOy plane, interference 1 and interference 3 are opposite in the incident direction, interference 1 and interference 4 have the same incident elevation angle, and the azimuth difference is π. The incident directions of interference 2, 3, and 4 are θj1,φj1, θj1,φj1 and θj1,φj1+π respectively.

FIGURE 2
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FIGURE 2. The position of the central UCA with four elements in the Oxyz coordinate system. (A) Three-dimensional figure; (B) xOy Planar 2D figure.

In the Oxyz coordinate system, the array element coordinate matrix is as follows. The first, second and third columns of the matrix are respectively the x, y, and z coordinates of the array element, and λc is the signal carrier wavelength.

Pele=00012λc0012λc32λc012λc32λc0(18)

The interference steering vector is

ajk=expj2πfcPelerjkc(19)

where c is the speed of light, rjk is the unit direction vector, and fc is the carrier frequency.

rjk=cosθjkcosφjkcosθjksinφjksinθjk(20)

Simplify the steering vector expression, it can be written as

ajk=expjπcosθjk0cosφjksinφjkπ6sinφjk+π6(21)

The steering vectors of interference 1–4 is

aj1=aj2=1expjπcosθj1cosφj1expjπcosθj1sinφj1π6expjπcosθj1sinφj1+π6,aj3=aj4=1expjπcosθj1cosφj1expjπcosθj1sinφj1π6expjπcosθj1sinφj1+π6(22)

It can be seen that the steering vectors of interference 1 and 2 are conjugate with interference 3 and 4. According to Eq. 6, the covariance matrix of each interference can be obtained as follows

Rj1=Rj2=Rj3=Rj4(23)

Denote the steering vector matrix and covariance matrix of the above four interferences as Aj_1 and Rjj_1 respectively. Considering the sup-DOF interference definition 1 and definition 2, it can be obtained that

rankAj_1=1(24)
rankRjj_1=1(25)

Therefore, the above four interferences are equivalent to one interference. Interference 1–4 belong to the incident direction of first category interference mentioned at the end of Section 3.1.

Denote the incident directions of four interference in Figure 2B are θj1,φj1, θj5,φj5, θj6,φj6 and θj7,φj7 respectively. The four incident directions have the following relationship:

θj1=θj5=θj6=θj7(26)
φj1=φj7,φj5=φj6(27)

Denote the steering vector matrix and covariance matrix of the above four interferences as Aj_2 and Rjj_2 respectively,

Aj_2=aj1aj5aj6aj7(28)
Rjj_2=k=1,5,6,7ajkajkH(29)

It can be simplified that

Rjj_2=100r14010r2400110000(30)

Among which

r14=ejπcosθj13cosφj5+3sinφj5×1+ejπcosθj13cosφj13cosφj5+3sinφj13sinφj5e23jπcosθj1sinφj5+ejπcosθj13cosφj13cosφj53sinφj13sinφj5e2jπcosφj1cosθj1e2jπcosφj5cosθj1(31)
r24=e2jπcosθj1cosφj1+cosφj5×e2jπcosθj1sinφj1π6e2jπcosθj1sinφj5π6+e2jπcosθj1sinφj1+π6e2jπcosθj1sinφj5+π6e2jπcosφj1cosθj1+e2jπcosφj5cosθj1(32)

It can be seen that the rank of the covariance matrix is 3, and the rank of the guidance vector matrix is 3.

rankAj_2=3(33)
rankRjj_2=3(34)

Interference 1, 5, 6, and interference 7 exist in the subspace of dimension-3, so they are equivalent to three interferences. They belong to the incident direction of the second category interference mentioned at the end of Section 3.1.

4 Direction and distribution sensitivity of sup-DOF interference suppression

If the number of interferences is greater than or equal to the number of array elements (KN), the spatial anti-jamming algorithm may not completely suppress them. According to Eq. 16, the residual interference power obtained by optimal spatial filter is the minimum eigenvalue of the interference covariance matrix, so the minimum eigenvalue of the interference signal covariance matrix can be used to characterize the anti-jamming performance. Denote the minimum eigenvalue as the optimal residual interference power, this section will specifically analyze the interference suppression performance on the condition of KN.

4.1 Influence of interference direction and distribution of super-DOF interference on residual interference power

According to the typical interference deployment scenarios in navigation countermeasure, the sup-DOF interferences are usually gathered in certain space angles while their exact locations are uncertain [23]. To study the rules of anti-sup-DOF-jamming performance, it is simpler to depict a cluster of jammers by their DOA boundaries rather than concentrate on specific jammer configurations. As a result, the interference deployment is described in the following parameters. Suppose the interference number is K, the incident angle in elevation is θj1,θj2,θjK, the azimuth angle is φj1,φj2,φjk, and the power is pj1=pj2==pjK. In order to describe the positions of the K interferences, the central incident direction of the interferences is defined as θj0,φj0, and the interference distribution is simplified as ΔΘj,Δϑj. Their expressions are as follows.

θj0=k=1KθjkKφj0=k=1KφjkK(35)
ΔΘ=maxθjiθjli,l1,K,ijΔϑ=maxφjiφjli,l1,K,ij(36)

See Figure 3 for the schematic diagram of the central incident direction and distribution.

FIGURE 3
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FIGURE 3. Schematic diagram of central incident direction and interference distribution.

In order to simplify the analysis of the central incident direction and distribution, the interference signals are assumed to be independent and the intersection angle of adjacent interferences are equal. A two-element antenna is taken as an example for analysis.

The positions of the two elements are shown in Figure 4. Its coordinates are as follows, where λc is the signal carrier wavelength.

P2ele=x1y1z1x2y2z2=0000012λc(37)

FIGURE 4
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FIGURE 4. Position of two-element array in Oxyz coordinate system.

The steering vector of the kth interference is

ajk=expjπ0sinθjk(38)

The characteristic polynomial of the interference covariance matrix is a one-variable quadratic equation about the eigenvalue λ:

fλ=λIRjj=λ2M11+M22λ+Rjj(39)

Wherein, M11 and M22 are the algebraic cofactors of two diagonal elements respectively, representing the determinant of the matrix.

Take two independent interferences as an example. Set the interference power as pj1 and pj2 respectively. The interference central direction and distribution are set as follows.

θj0=θj1+θj22(40)
ΔΘj=θj2θj1(41)

Combine Eq. 6 and Eq. 39, let fλ=0, then the optimal residual interference power is obtained:

pjresθj0,ΔΘj=pj1+pj2pj1+pj2)22pj1pj2(1cos(πsin(θj0ΔΘj2)πsin(θj0+ΔΘj2)(42)

The partial derivative of the above equation is obtained from θj0 and ΔΘj:

pjresθj0,ΔΘjΔΘj=pj1pj2πsinπsinθj0ΔΘj2πsinθj0+ΔΘj2cosθj0ΔΘj2+cosθj0+ΔΘj22pj1+pj2)22pj1pj2(1cos(πsin(θj0ΔΘj2)πsin(θj0+ΔΘj2)pjresθj0,ΔΘjθj0=pj1pj2πsinπsinθj0ΔΘj2πsinθj0+ΔΘj2cosθj0ΔΘj2cosθj0+ΔΘj2pj1+pj2)22pj1pj2(1cos(πsin(θj0ΔΘj2)πsin(θj0+ΔΘj2)(43)

It can be seen from the observation that it is difficult to simplify pjresθj0,ΔΘj to the multiplication of two one-variable functions. The central incident direction is closely coupled with the interference distribution.

If ΔΘj0, let

pjresθj0,ΔΘjθj0=0(44)

The central interference incident direction that minimizes the optimal residual interference power can be solved

θj0=π2+πl,l=0,±1,±2,(45)

Combine Eq. 45 with Eq. 42, it can be obtained that

pjres(θj0,ΔΘj)=0(46)

Extending to K interferences (K ≥ 2), the optimal residual interference power is

pjres=k=1Kpjkk=1Kpjk22u=1,v=1uvKpjupjv1cosπsinθjuπsinθjv(47)

According to the above formula, if cosπsinθjuπsinθjv=1, the optimal residual interference power reaches the minimum value pjres,min=0; If cosπsinθjuπsinθjv=1, the optimal residual interference power reaches the maximum value pjres,max. The maximum value pjres,maxk=1Kpjk, and the condition for the equality is K = 2 (see Appendix A for detailed proof). Therefore, if the total power of interference is fixed, two interferences can achieve better jamming effect than multiple interferences for two-elements array.

Interference cancellation ratio (ICR) is defined as the ratio of input interference power to residual interference power:

ICR=k=1Kpjkpjres(48)

Setting two typical interference configurations in Table 2, the above analysis results are numerically illustrated.

TABLE 2
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TABLE 2. Typical interference configuration.

Assume that the interference power is equal, and the intersection angle of adjacent interferences are equal. The variation of optimal residual interference power and ICR against the interference distribution and central incident direction is shown in Figure 5. Figures 5(A−D) show the numerical calculation results of configuration ① and configuration ② respectively.

FIGURE 5
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FIGURE 5. The variation of residual interference power and ICR against the interference distribution and central incident direction (two-element array). (A) Optimal residual interference power of config ①; (B) ICR of config ①; (C) Optimal residual interference power of config ②; (D) ICR of config ②.

As shown in Figure 5A, for a two-element array, if the interference number K = 2, the maximum optimal residual interference power is 1 and the minimum value is 0. If the incident direction of interference is symmetrical about ±90°, sinθjk=sinπθjk=sinθjk , thus two interferences are equivalent to one interference. Note that in Figure 5B, ICR should be positive infinity at extreme points, where ΔΘ=0 and θj0=±90. It can be seen from Figures 5C, D that when the interference number K ≥ 2, the maximum value of optimal residual power is less than 1, which shows that it is less effective than two interferences, the analysis at Eq. 47 is verified. For the case of multi element array, if the interference number KN and the total interference power is 1, it can be either concluded that the maximum optimal residual interference power is 1 (see Appendix B for detailed proof), and the minimum is 0 (according to Section 3.2).

Conclusion 2: If the number of interferences is greater than or equal to the number of array elements, the interference residual power obtained by optimal spatial filter is closely related to the central incident direction and the interference distribution. The maximum value of the interference residual power is the sum of the interference power, and the minimum value is 0. In particular, when the element number N = 2, the maximum residual power equals to the sum of interference power only if the number of interference K = 2.

The above conclusion show that for the evaluation of sup-DOF anti-jamming capability, the interference incidence direction will cause huge differences in the evaluation results, and the anti-interference capability needs to be evaluated separately for different interference deployment scenarios; For the deployment of jammer, if the number and power of several jammer are determined, the interference efficiency can be improved by reasonably setting the incident direction of interference.

4.2 Azimuth periodicity of interference suppression performance

It can be seen from Section 4.1 that in order to study the rule of sup-DOF interference suppression performance, it is necessary to test multiple groups of interference incidence directions, and the workload of simulation calculation is huge. Eq. 45 shows that the interference suppression performance of the linear array takes π as the cycle. If the anti-jamming performance of the plane array also has periodicity, it can reduce the repetitive calculation and improve the simulation efficiency. In order to solve this problem, this section analyzes the periodicity of the interference suppression performance of the navigation receiver array antenna, assuming that the interference signals are independent of each other.

First, take the central UCA of four elements as an example. The element positions are shown in the orange circle in Figure 6, and the coordinates are shown in the Eq. 18.

FIGURE 6
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FIGURE 6. Position of array element in Oxyz coordinate system.

Suppose the incident direction in elevation remains constant, if the azimuth angle φjk deviates 2π3+2π3mm=0,±1,±2, from the initial direction, then the deviated steering vector is

ajk=expjπλccosθk0sinφjk+π6cosφjksinφjkπ6(49)

ajk equals to line exchange of ajk, which can be written as follows:

ajk=Ts1ajk(50)

where Ts1 is the matrix representing row exchange.

Ts1=1000000101000010(51)

Ts1 is a unitary matrix, Ts1H=Ts11.

If all K interferences deviate 2π3+2π3mm=0,±1,±2, relative to the original incident direction, denote Rjj as the deviated interference covariance matrix. The relationship between Rjj and the original covariance matrix Rjj is as follows.

Rjj=Ts1RjjTs11(52)

Matrix Rjj is similar to Rjj, so they have same (smallest) eigenvalues. As a result, this deviation of incident direction in azimuth does not change the optimal residual interference power and ICR.

In the same way, it can be proved that if the azimuth incidence angle deviates π3+2π3mm=0,±1,±2,, the deviated steering vector ajk is equal to the line exchange of the conjugate of ajk.

ajk=Ts2ajk*(53)
Ts2=1000001000010100(54)

Similarly, if all K interferences are deviated π3+2π3mm=0,±1,±2, relative to the original incident direction, denote Rjj as the deviated interference covariance matrix. The relationship between Rjj and the original covariance matrix Rjj is as follows.

Rjj=Ts2RjjTTs21(55)

Matrix Rjj is similar to RjjT. What’s more, the transpose transformation does not change the eigenvalues of the matrix, so the minimum eigenvalue of Rjj is the same as that of Rjj. This deviation of incident direction in azimuth does not change the optimal residual interference power or ICR.

To sum up, if the relative relationship between the interference incident directions is certain, the residual interference power of four-element central UCA is periodic in the azimuth direction. The period is π3.

Secondly, take the five-element central UCA as an example for analysis. The array element position is shown in the golden circle in Figure 6, where one element is located at the origin, and the other four elements are uniformly distributed on the circumference with half-wavelength radius. By the same derivation method, when the incident direction of the interference is deviated π2+2π3mm=0,±1,±2, from the original incident direction, the new interference covariance matrix is similar to the original covariance matrix, and the minimum eigenvalue is the same, so the period of the interference residual power is π2.

Take the interference number K = 10 as an example to illustrate the numerical calculation of the above analysis results. See Figure 6 for array antenna geometry. Interference parameters are given in Table 3.

TABLE 3
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TABLE 3. Jamming configuration.

The relative incident direction of the interference remains constant, and the central incident direction of the azimuth is changed. The changes of residual interference power and ICR against central incident direction in azimuth are shown in Figure 7. Wherein, Figures 7A, B show that the anti-jamming performance period of four-element central UCA is π3, while Figures 7C, D show that the anti-jamming performance period of five-element central UCA is π2, which verifies the above analysis.

FIGURE 7
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FIGURE 7. Azimuth periodicity of interference suppression performance. (A) Optimal residual interference power of four-element central UCA; (B) ICR of four-element central UCA; (C) Optimal residual interference power of five-element central UCA; (D) ICR of five-element central UCA.

In spite that the central UCA is taken as an example for theoretical derivation and numerical calculation, it can be seen from the derivation process that the azimuth period of the suppression performance is only related to the number of elements uniformly distributed on the circumference, and whether or not to deploy elements at the center of the circle has no effect on the period size. If the number of elements uniformly distributed on the circumference is M (M ≥ 3), it can be further generalized that the interference suppression performance period of UCA or central UCA is.

πMMisodd2πMMiseven(56)

Correspondingly, the anti-jamming performance repeats for Np cycles when the interference azimuth changes from 0 to 2 π towards the central incident direction.

Np=2MMisoddMMiseven(57)

The following conclusion can be further generalized:

Conclusion 3: Assume the interference signals are independent of each other. For an UCA with half-wavelength radius circumference, if the number of elements uniformly distributed on the circumference is M (M ≥ 3), and the number of elements at the center of the circle is 0 or 1, then the interference suppression performance period in azimuth is 4πM3+(1M+1. Correspondingly, interference suppression performance repeats 3+(1M+1)M/2 cycles when the central incident direction turns over the range of 2π in azimuth. If the number of elements at the center of the circle is 1, the above conclusion is also applicable to M = 1,2, and the array is degenerated into an ULA (Uniform Linear Array).

This conclusion can be applied to the evaluation of antenna array anti-jamming performance. In the study of the relationship between antenna array anti-jamming performance and azimuth incidence angle, it can reduce the repetitive test or simulation calculation, and improve the evaluation efficiency by 3+(1M+1)M/2 times.

5 Simulation results and analysis

This section verifies conclusion 2 and conclusion 3 through simulation of signal flow.

5.1 Simulation verification of conclusion 2

5.1.1 Simulation scenario 1

Firstly, the theoretical analysis in Section 4.1 is verified by simulation. The simulation parameters are shown in Table 4. Wherein, 1268.52 MHz is the central frequency point of the Beidou navigation system B3I signal. To facilitate comparison with the numerical calculation in Section 4, the total interference power is set as 1 W, and the intersection angles of adjacent interference incident directions are equal.

TABLE 4
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TABLE 4. Settings of simulation parameter.

The variation of anti-jamming performance against central incident direction interference distribution is shown in Figure 8. Four central incident directions are selected for display, and the abscissas are the interference distribution in azimuth and elevation respectively. The simulation verifies conclusion 2. At the same time, it can be concluded from the figure that the anti-jamming performance may be improved by changing the central incident direction and interference distribution of the interference, but the change rule needs to be further studied through statistical data.

FIGURE 8
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FIGURE 8. Anti-jamming performance against central incident direction and interference distribution. (A) ICR at central incident angle of [10°, 0°]; (B) ICR at central incident angle [10°,30°]; (C) ICR at central incident angle [70°,0°]; (D) ICR at central incident angle [70°,30°]; (E) Optimal residual interference power at central incident angle of [10°,0°]; (F) Optimal residual interference power at central incident angle [10°,30°]; (G) Optimal residual interference power at central incident angle [70°,0°]; (H) Optimal residual interference power at central incident angle [70°,30°].

5.1.2 Simulation scenario 2

Since the number of samples in the above simulation is limited, it fails to verify the statement in conclusion 1 that the maximum of optimal interference residual power is equal to the sum of input interference power. This scenario takes a four-element ULA as an example to illustrate the existence of interference incident directions that accord with the statement.

Assume that the element spacing of the four-element ULA is half wavelength, and the interference number is K. The rest of simulation conditions are the same as those in Table 4. Let the power of interferences be 1/K, and the interference distribution ΔΘj=π. The distribution meets Eq. 58 while the central incident direction θj0=0.

θjk=arcsinαjkk=1,2,,K(58)

where

αjk=2kK1k=1,2,,K(59)

Keep the interference distribution unchanged, take the approximate value K = 105 for simulation, and the relationship between the optimal residual interference power and the interference central incident direction is shown in Figure 9. It can be seen that the residual interference power approaches 1 at θj0=0. It can be computed that the optimal residual interference power equals to 1 at θj0=0 while K. The statement in conclusion 2 that the optimum of residual interference power equals to the sum of input interference power is verified.

FIGURE 9
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FIGURE 9. Relationship between the optimal residual interference power and the interference central incident direction.

5.1.3 Simulation scenario 3

This scenario simulates and verifies the theoretical analysis in Section 4.2. The initial incident direction parameters and simulation parameters of interference are the same as Tables 3, 4. The simulation results are shown in Figure 10. The horizontal axis in the figure indicates that the azimuth of the interference has varied by 2π from the initial central incident direction, and the vertical axis is the optimal residual interference power of anti-interference processing. The array used in the simulation is a central UCA. According to conclusion 3, when the number of array elements is N = 2, 4, 6, 8, and the number of elements uniformly distributed on the circumference is M = 1, 3, 5, 7, the azimuth period of the interference suppression performance is π, π3, π5, π7, and the interference suppression performance repeats 2M cycles when the central incident direction turns over the range of 2π; When the number of array elements is N = 3, 5, 7, 9, and the number of array elements uniformly distributed on the circumference is M = 2, 4, 6, 8, the azimuth period of the interference suppression performance is π, π2, π3, π4, and the interference suppression performance repeats M cycles when the central incident direction turns over 2π in azimuth; The simulation results in Figure 10 verify conclusion 3.

FIGURE 10
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FIGURE 10. Azimuth periodicity (signal flow simulation) of four-element central UCA (A–H).

6 Conclusion

On the condition the number of interference is equal to or greater than the number of array elements, in order to study the anti-jamming capability and mechanism of GNSS array antenna, the following two tasks are completed in this paper: First, the definition of sup-DOF interference for GNSS array antenna is proposed from the perspective of spatial anti-jamming; Secondly, the directional characteristics of GNSS array antenna for sup-DOF interference suppression are analyzed, while numerical calculation and simulation verification are carried out. The main achievements and conclusions are summarized as follows.

(1) The definition of sup-DOF interference is proposed. Accordingly, if the number of interferences is greater than or equal to the number of array elements (KN), the interference may not surpass the array degree of freedom. The existence of special directions allows the antenna array to completely suppress K interferences.

(2) If the number of interferences is greater than or equal to the number of array elements, the value of the interference residual power obtained by optimal spatial filter is closely related to the central incident direction and the interference distribution. The maximum value of the interference residual power is the sum of the interference power, and the minimum value is 0.

(3) Assume the interference signals are independent of each other. For an UCA with half-wavelength radius circumference, if the number of elements uniformly distributed on the circumference is M (M ≥ 3), and the number of elements at the center of the circle is 0 or 1, then the interference suppression performance is periodic. The interference suppression performance repeats Np cycles when the central incident direction turns over the range of 2π in azimuth.

Np=2MMisoddMMiseven(60)

7 Discussion

The conclusion of this paper gives the upper and lower bounds of the sup-DOF interference suppression capability for typical GNSS antenna arrays, and derives the azimuthal periodic rule of the sup-DOF interference suppression capability. The former item has guiding significance for the jammer DOA deployment. The latter one can be useful in interference suppression performance evaluation, which improves the evaluation efficiency by 3+(1M+1)M/2 times. The conclusions are drawn basically on narrowband array signal model, while it can be proved that they are also tenable for wideband model. The future work includes studying the relationship between interference suppression capability and power, number, direction and distribution of jammers, and to conclude detailed quantitative results.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

Author contributions

YS performed the theoretical study, conducted the simulations, and wrote the manuscript; FC provided the methodology and revised the manuscript; FW provided conceptualizations and research suggestions; WL and BL helped with programming and revised the manuscript; JS helped in figures and correction. All authors have read and agreed to the published version of the manuscript.

Funding

This research was supported in part by the Natural Science Foundation of China (NSFC), grants No. 62003354.

Acknowledgments

The authors would like to thank the editors and reviewers for their efforts in supporting the publication of this paper.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Appendix A

It proves that in Eq. 47, the equality of pjres,maxk=1Kpjk is obtained at K = 2. Eq. 47 is given below.

pjres=k=1Kpjkk=1Kpjk22u=1,v=1uvKpjupjv1cosπsinθjuπsinθjv

Proof

If the total interference power is fixed, k=1Kpjk=pjtot is a fixed value. To maximize the interference residual power, the root term of the above equation needs to take the minimum value. This problem is equivalent to the following optimization problem:

minp1,p2k=1Kpjk2+u=1,v=1uvKpjupjvejπsinθjujπsinθjv+u=1,v=1uvKpjupjvejπsinθjvjπsinθjus.t.k=1Kpjk=pjtot(A1)

Denote

p=pj1pj2pjk(A2)
H=1ejπsinθj1jπsinθj2ejπsinθj1jπsinθjvejπsinθj2jπsinθj11ejπsinθj2jπsinθjvejπsinθjujπsinθj1ejπsinθjujπsinθj21(A3)

Let θj1<θj2<...<θjk, then the above problem is a convex optimization problem of quadratic form, namely

minppHHps.t.pHb=pjtot(A4)

among them b=11T.

It can be solved that when 0°<θj1,θj2,,θjk<90° or 90°<θj1,θj2,,θjk<0°, the optimal value of interference power deployment is

popt=pjtot/200pjtot/2(A5)

herein

k=1Kpjk2+u=1,v=1uvKpjupjvejπsinθjujπsinθjv+u=1,v=1uvKpjupjvejπsinθjvjπsinθju=0(A6)

It can be proved that when the range of θj1,θj2,,θjk is θjkθj190°, the optimal value of interference power deployment has the following form

pju=pjv=ptot2pjk|ku,v=0(A7)

This is equivalent to that the maximum value of interference residual power is obtained when the interference number K = 2.

Appendix B

It proves that in Section 4.1, if the number of array elements N > 2, The optimal residual interference power is the sum of input interference power.

Proof

Denote CN as N-dimensional complex vector space. Assume the total power of interference is 1, the steering vectors of K (K ≥ N) interferences is aj1aj2ajK respectively (not linearly correlated), the power of which is pj1pj2pjK, and the initial phase is γj1γj2γjK. The element space constructed by interferences is

Se=pj1expjγj1aj1pj2expjγj2aj2pjKexpjγjKajK(B1)

Denote the complex number

βjk=pjkexpjγjkk=1,2,,K(B2)

Since K ≥ N and the steering vectors are not linearly correlated, there are βjk (k=1,2,,K) that confirms

Se=Spanaj1aj2ajN=Spanaj1aj2ajK(B3)

Considering

Spanaj1aj2ajK=a|a=β1aj1+β2aj2++βKajK=CN(B4)

It is evident that there is βjk (k=1,2,,K) that confirms

Se=IN(B5)

where IN is the unit matrix. Herein, the minimum eigenvalue is 1, so the maximum of optimal residual interference power is 1. In other words, the optimal residual interference power is the sum of input interference power.

Keywords: anti-jamming, GNSS array antenna, array DOF, direction sensitivity, distribution sensitivity

Citation: Sun Y, Chen F, Wang F, Liu W, Li B and Song J (2023) Direction and distribution sensitivity of sup-DOF interference suppression for GNSS array antenna receiver. Front. Phys. 10:1095109. doi: 10.3389/fphy.2022.1095109

Received: 10 November 2022; Accepted: 28 December 2022;
Published: 01 February 2023.

Edited by:

Jian Dong, Central South University, China

Reviewed by:

Yayun Cheng, Harbin Institute of Technology, China
Du Baoqiang, Hunan Normal University, China

Copyright © 2023 Sun, Chen, Wang, Liu, Li and Song. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Feixue Wang, Znh3YW5nQG51ZHQuZWR1LmNu

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.