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Abstract

Large onboard antennas with multi-beams are used almost in all the Geostationary Earth Orbit satellite communications systems, which are always disturbed by jams and inferences. However, there are lots of factors restricting interference localization in satellite communications system because the system is not designed for location-oriented. This article makes use of the characteristic that each spot beam of multi-beams antenna has different gain at the position of interference, establishes the localization equations set by using the ground projection model of multi-beams antenna pattern, and gives two methods of solving equations. Thus, it proposes interference localization algorithm using single satellite, based on antenna gain and the distribution of signal strength in multi-beam space. Because of not depending on extra onboard equipments and other facilities, it has no effect on the construction and operation of satellite communications system. From the results of analyses and simulations, the localization precision can satisfy the general practical requirement. The algorithm could be used in checking the interference of the system, decision supporting of anti-jamming, and improving operation and management of the system.

Keywords

Satellite communications multi-beam antenna interference localization antenna gain pattern

Introduction

With the development of the satellite comprehensive application over the world, the orbits of satellites and frequency have become crowded. The number of all kinds of earth stations for satellite communication increased drastically, which leads to the electromagnetic environment of satellite communication deteriorating gradually. Therefore, satellite communication systems are often disturbed by various kinds of radio-frequency interference. Almost all of satellite mobile communication systems based on Geostationary Earth Orbit (GEO) adopted extensible onboard antenna with multi-beam. Because of the big aperture of onboard antenna and the high sensitivity of receiver, it is easier to be disturbed by interference intentionally (jamming) or involuntary for satellite mobile communication system with onboard antenna with multi-beam.¹ In order to operate the system properly and reliably, it is urgent to find out interference quickly and reduce its effect furthermore. However, the precondition is localizing the interference. As communication systems are always not designed for localization and the interference source is unknown a priori, there are lots of limits to achieve the goals mentioned above. In other words, the requirement of localization condition is not easy to be satisfied. Therefore, we need to develop a new method to localize the interference in satellite communication system which has the least requirement for the system and has little dependence on the other equipments.

There has been great interest toward interference localization in the satellite communication system.^2–4 The interference localization methods can be divided into single and multiple satellites localization by the number of satellite used for positioning. The localization principles^3–6 include measuring the time difference location and direction finding location. The former derives time difference of arrival (TDOA) along with phase measurements to localize an unknown interference using two satellites. Direction finding location^6–9 is preferred in single satellite localization, including direction finding by phase interferometer, phase comparison direction finding, amplitude comparison direction finding, time difference direction finding, spatial spectrum estimation, and so on.

Right now, the most ordinary methods to localize the interference in GEO satellite communication system include TDOA and frequency difference of arrival (FDOA), both playing important roles in past decades.^10–14 There should be at least one neighbor satellite with the same frequency, which is the basic condition of employing these two methods. It is easier for GEO satellite fixed communication system, but it is difficult for a mobile communication system. The other methods are based on direction finding for single satellite localization. However, it is difficult to add the special localization equipment of direction finding onboard for the restriction of weight, volume, power and running orbit, leading to much problem to application.

The overlap area may be highly large between neighbor beams when a GEO satellite adopts a multi-beam antenna. Therefore, when the system is disturbed by strong interference, the interference signal will be received by both main jammed beam and other neighbor beams. According to the pattern of antenna, the gain of antenna in the direction of different beams will be changed with each other when interference locates at different position. The strength of interference signal arriving at each beam changes with location of interference, which gives birth to the idea of interference localization for single satellite based on signal strength distribution in multi-beam antenna. This article proposes a novel method of interference localization for single satellite based on antenna gain and the strength distribution, and using the gain of multi-beam antenna at the direction is different from at the location of interference of each spot beam. The method does not need the extra onboard equipment and facility which is specially used for localization only. The benefit is that it will not impact the normal operation and running of satellite communication. What is more, the localization technology can be used to other communication system with onboard multi-beam antenna and electronics observation satellite system.

The remainder of the article is organized as follows. In section “System model,” the system model and the proposed interference localization architecture are described. Section “Interference localization algorithm” of the article discusses algorithm for interference localization. The analyses and simulations results are given in sections “Localization error analysis” and “Simulation results,” respectively. Finally, the conclusions are given in section “Conclusion.”

System model

When the GEO satellite communications system is disturbed by strong interference signal, besides the beam major been jammed, the other common frequency reuse beams also will be jammed by the interference signal. The common frequency reuse beams may be close to each other, but also interval some beams. We will illustrate the principle of localization in the scene described in Figure 1, in which seven beams reuse the frequency.

Figure 1.

The principle of interference localization using single satellite.

The interference radiation source located at P is labeled by red color, and the frequency of interference signal falls into the radio band used by beam A, leading to the result that beam A is jammed. Meanwhile, at least two beams can receive the signal, labeled by beams B, C, and D which use the same frequency band. The location of the beam center is priori known, and the satellite location in the space can be deduced by ephemeris. According to the distribution of multi-beam antenna pattern, the antenna gain at the direction from each beam to the interference radiation source is different from each other. Therefore, the signal strength received by different beams is different. From the equal signal strength of each beam described by four dashed circles in Figure 1, we can get interference location by obtaining the intersection of at least three circles.

The flow of interference localization is as follows, as shown in Figure 2:

Set up the model of multi-beam antenna pattern. According to the relative position of beam center and interference source and the antenna gain pattern, get the relationship between the location and antenna gain.

Select location beams. After the main interference beam is confirmed, the three most strong strength signal beams are selected according to interference signal strength and relative position. In some satellite communication systems, wide beam and zone beam can be selected as spot beam used for localization.

Signal strength measurement. Get the signal strength in the beams selected in step (2). In order to get the accurate strength of very weak signal, the measurement can be assisted by feature of signal in the major jammed beam whose signal is strong.

Establish the localization equations set. Combined with the propagation loss, signal strength curve of the interference can be obtained, by the function $L_{i} = F (r_{i}), i = 1, 2, 3$ , where $L_{i}$ is the attenuation relative to the center of the beam, $r_{i}$ is the distance from the point of the equivalent signal strength curve to the beam center, and $F (\cdot)$ denotes the relationship between $r_{i}$ and $L_{i}$ , which can be calculated by the antenna gain model.

Solve the localization equations set. Jointly consider the constrain condition of interference coordinate and solve the localization equations set to get the interference location.

Figure 2.

The flow chart of interference localization in using single satellite with onboard multi-beam antenna.

Interference localization algorithm

The overlap area between each other beams may be large when satellite adopts a multi-beam antenna. Therefore, when the communication system is disturbed by strong interference, whose strength is large and affect the normal operation of the system, the interference signal will be received by both main jammed beam and other neighbor beams. After the main interference beam is confirmed, the three most strong strength signal beams are selected according to interference signal strength and relative position. A novel method for locating interference is proposed utilizing the difference of the received interference power level between different spot beams. The interference localization algorithm includes establishing and solving of localization equations set described as follows.

Localization equations set

When choosing the spot beam for localization, the principle of maximizing the signal strength is adopted. Three beams with the same frequency are chosen which are nearest to the source of interference. The centers of the three beams construct an equilateral triangle, called localization triangle, shown in Figure 3 marked by dotted lines. Therefore, as long as the position of the interference is in the coverage of the satellite, it will be in one of localization triangle. Without loss of generality, the changes of positions of the interference are limited in a localization triangle in the following study. When using seven color frequency reuse, the side of the triangle is $\sqrt{21} R_{b}$ , and $R_{b}$ is the radius of the beam.

Figure 3.

The localization triangle marked by dotted lines is equilateral triangle.

The model of interference localization is shown in Figure 1. The radius of the earth is known as $R$ . The coordinate vectors of the satellite and the source of interference are represented by $S = [x_{0}, y_{0}, z_{0}]$ and $P = [x, y, z]$ , respectively, where P is unknown. The center coordinate vectors of the three spot beams are $B_{i} = [x_{i}, y_{i}, z_{i}], (i = 1, 2, 3)$ . The connecting vector between the satellite and interference source is $\vec{SP}$ , and the one between the satellite and each spot beam center is $\vec{S B_{i}}$ . The $θ_{i} (i = 1, 2, 3)$ demonstrates the angle between the center of the spot beam and the connecting line of the source of interference and the satellite. From the law of cosines, we can get

$θ_{i} = \arccos \frac{\vec{S B_{i}} {\vec{SP}}^{T}}{| \vec{S B_{i}} | | \vec{SP} |} (i = 1, 2, 3)$ (1)

where $θ_{i}$ is the function of the coordinate of the source of interference P. If we use $G (θ_{i}) (i = 1, 2, 3)$ describing the gain in the direction of interference for onboard receiving antenna of each spot beam, the function of $G (\cdot)$ is function of P. In the remainder of the article, a specific antenna pattern is used as an example to analyze the feasibility and effectiveness of the localization method. Although different antenna types and antenna parameters have different function of $G (\cdot)$ , the localization method and the error analysis method are the same. Without the loss of generality, we adopt the model function for the gain of onboard antenna in Zhong et al.,¹⁵ as function (2)

$G (θ) = 10 \underset{10}{\log} (G_{0} \cdot {(\frac{J_{1} (k \sin θ)}{2 k \sin θ} + 36 \frac{J_{3} (k \sin θ)}{{(k \sin θ)}^{2}})}^{2})$ (2)

where $J_{1} (\cdot)$ and $J_{3} (\cdot)$ are the first-order and third-order Bessel functions, respectively, $k = 2.07123 / \sin (θ_{3 dB})$ , and $θ_{3 dB}$ is the angle of 3 dB attenuation in the gain of the beam relative to the center of the beam. It can be calculated by $θ_{3 dB} = 70 λ / D$ , where D is the diameter of the antenna and $λ$ is the wave length of the signal. The gain of satellite antenna in the center point is $G_{0}$ $(θ = 0)$ , as function (3)

$G_{0} = \frac{π^{2} D^{2} η}{λ^{2}}$ (3)

where $η$ is the efficiency of the antenna. As a matter of fact, it is difficult to describe sometimes the antenna pattern of multiple beams onboard, using precise functions of the analytic relationship. However, a discrete numerical model of antenna pattern of the beams (including side beam) can be constructed according to the result of measurement and calibration in antenna pattern, which is also suitable in the algorithm of localization in this article. In the localization scenario, when the interference signal reaches the receiver of the onboard multi-beam antenna from the radiation source, the propagation path of the interference signal is exactly the same. So, the propagation losses and weather conditions are the same in the localization model. When interference signal arrives at the antenna onboard, the propagation loss in free space is

$L_{f} = 10^{*} \log_{10} {(\frac{4 π \cdot d}{λ})}^{2}$ (4)

where d is the propagation distance of signal $| \vec{SP} | = \sqrt{{(x - x_{0})}^{2} + {(y - y_{0})}^{2} + {(z - z_{0})}^{2}}$ , $λ$ is the wave length of signal, and $L_{f}$ is the function of the coordinate of the source of interference P, too. The loss caused by weather and atmosphere is $L_{a}$ . So, the total loss of propagation is $L = L_{f} + L_{a}$ .

If the radiation strength of the interference is $Q$ , and the signal strength of interference received by each spot beam is $A_{i} (i = 1, 2, 3)$ , then the localization equations set is

${\begin{matrix} Q - L + G (θ_{1}) = A_{1} \\ Q - L + G (θ_{2}) = A_{2} \\ Q - L + G (θ_{3}) = A_{3} \\ P P^{T} = R^{2} \end{matrix}$ (5)

In equations set (5), if only we measure the interference signal strength in three spot beams, there are four unknown quantities, x, y, z, and Q, four equations could be constructed, and the position of the interference P will be calculated by solving the equations set.

Furthermore, the first three equations in equation (5) all include parameters Q and L. If we make the three equations be pairwise subtracted from each other, eliminates the unknown signal transmitting strength and loss, and be expressed as the function of coordinate of interference, then the equations set is changed as in equation (6)

${\begin{matrix} G (θ_{2}) - G (θ_{1}) = F_{1} (x, y, z) = A_{2} - A_{1} = Δ_{1} \\ G (θ_{3}) - G (θ_{2}) = F_{2} (x, y, z) = A_{3} - A_{2} = Δ_{2} \\ \sqrt{P P^{T}} = F_{3} (x, y, z) = R \end{matrix}$ (6)

From equation (6), we can achieve our localization method by measuring the relative deviation value of the interference signal strength in two spot beams. The degree of difficulty for measurement is reduced compared with using absolute signal strength in equation (5). It can be seen from the latter analysis that the requirement of the signal strength measurement accuracy is not too high, and the existing measurement methods can almost meet the requirements. When the interference signal is too weak or difficult to be detected, the impact on the system is limited. Moreover, the measurement accuracy of the signal strength in the other beams can be improved with the assistance of the signal parameters in the main jammed beam, in which the signal parameter could be estimated more accurate.

Solution to the localization equations set

The localization equations set (6) is nonlinear, which is complicated to solve. This article gives two methods of solving, from the aspects of the complexity of algorithm, the accuracy of localization and the practicability.

Newton iteration based on least square method

We first adopt Newton iteration based on least quare method (NILSM)¹⁶ in this article, which combines the advantages of both well estimation property in least quare method and fast convergency in Newton iteration.

Newton iteration method¹⁶ is a frequently used method in solving nonlinear system of equations. Each step of Newton iteration method is as follows: first, each equation is linearized at the estimated value of a root, and then the linearized equations set should be solved; finally, the estimated value of the root could be updated according to the result above. The linearization of Newton method needs differential operation, which requires the functions to be derivable. Thus, this method is suitable for the scene when the function of satellite antenna pattern function is derivable.

Suppose the value of $P (x, y, z)$ is $P_{m - 1} (x_{m - 1}, y_{m - 1}, z_{m - 1})$ at the time of $m - 1$ and is $P_{m} (x_{m}, y_{m}, z_{m})$ at the time of $m$ , when iteration. Each equation in equation (6) can be linearized by solving the differential at the estimated value. Then, it can be denoted by matrix as follows

$H \cdot Δ_{p} = b$ (7)

Equations set (6) can be solved by the least square method

$Δ_{p} = (H^{T} H)^{- 1} H^{T} b$ (8)

The updated solution is as follows

$P_{m} = P_{m - 1} + Δ_{p}$ (9)

where $P_{m}$ can be regarded as the new initial iteration value, and the operation above could be continued until the accuracy of localization is satisfied. Then, the estimated values of the position of the interference can be inferred.

From equation (7)

$Δ_{p} = P_{m} - P_{m - 1} = [\begin{matrix} x_{m} \\ y_{m} \\ z_{m} \end{matrix}] - [\begin{matrix} x_{m - 1} \\ y_{m - 1} \\ z_{m - 1} \end{matrix}]$ (10)

$b = [\begin{matrix} A_{1} - F_{1} (x_{m - 1}, y_{m - 1}, z_{m - 1}) \\ A_{2} - F_{2} (x_{m - 1}, y_{m - 1}, z_{m - 1}) \\ A_{3} - F_{3} (x_{m - 1}, y_{m - 1}, z_{m - 1}) \end{matrix}]$ (11)

$H = [\begin{matrix} \frac{\partial F_{1}}{\partial x} & \frac{\partial F_{1}}{\partial y} & \frac{\partial F_{1}}{\partial z} \\ \frac{\partial F_{2}}{\partial x} & \frac{\partial F_{2}}{\partial y} & \frac{\partial F_{2}}{\partial z} \\ \frac{\partial F_{3}}{\partial x} & \frac{\partial F_{3}}{\partial y} & \frac{\partial F_{3}}{\partial z} \end{matrix}]$ (12)

From equation (12)

$\frac{\partial F_{1}}{\partial \nabla} = \frac{\partial G}{\partial θ_{2}} \cdot \frac{\partial θ_{2}}{\partial \nabla} - \frac{\partial G}{\partial θ_{1}} \cdot \frac{\partial θ_{1}}{\partial \nabla} (\nabla = x, y, z)$ (13)

$\frac{\partial F_{2}}{\partial \nabla} = \frac{\partial G}{\partial θ_{3}} \cdot \frac{\partial θ_{3}}{\partial \nabla} - \frac{\partial G}{\partial θ_{2}} \cdot \frac{\partial θ_{2}}{\partial \nabla} (\nabla = x, y, z)$ (14)

where

$\begin{array}{l} \frac{\partial θ_{i}}{\partial \nabla} = \frac{- [\frac{\nabla_{i} - \nabla_{0}}{| \vec{S B_{i}} | | \vec{S P} |} - \frac{\vec{S B_{i}} {\vec{S P}}^{T} (\nabla - \nabla_{0})}{| \vec{S B_{i}} | {| \vec{S P} |}^{3}}]}{\sqrt{1 - {(\frac{\vec{S B_{i}} {\vec{S P}}^{T}}{| \vec{S B_{i}} | | \vec{S P} |})}^{2}}} \\ (i = 1, 2, 3; \nabla = x, y, z) \end{array}$ (15)

Suppose $G_{U} (θ) = ((J_{1} (k \sin θ) / (2 k \sin θ)) + 36 (J_{3} (k \sin θ) / (k \sin θ)^{2}))^{2}$ , then equation (2) can be rewritten as $G (θ) = 10 \log_{10} (G_{0} \cdot G_{U} (θ))$ . Furthermore, we can get

$\frac{\partial G}{\partial θ} = \frac{10}{\ln 10 \cdot G_{0} \cdot G_{U}} \frac{\partial G_{U}}{\partial θ}$ (16)

where

$\begin{array}{l} \frac{\partial G_{U}}{\partial θ_{i}} = - 2 [\frac{J_{1} (k \sin θ_{i})}{2 k \sin θ_{i}} + 36 \frac{J_{3} (k \sin θ_{i})}{{(k \sin θ_{i})}^{3}}] \\ \cdot [\frac{J_{2} (k \sin θ_{i})}{2 \tan θ_{i}} + 36 \frac{J_{4} (k \sin θ_{i})}{k^{2} \sin^{2} θ_{i} \tan θ_{i}}] (i = 1, 2, 3) \end{array}$

Maximum likelihood iterative search method

Ordinarily, the approximate iteration method for solving nonlinear equations involves dealing with the derivatives of the function. However, when the function of antenna pattern onboard is not derivable, or when there is no function to denote the antenna pattern, this kind of method cannot be used. A direct way is blindly searching the coordinate in the interfered area which can minimize the error of the objective function. Thereby, the true position of interference will be approached. In this section, according to the idea mentioned above, the maximum likelihood iterative search method (MLISM) is designed for solving the localization equations.

If the estimated position coordinate of interference is $\hat{P}$ , and the mean square error, defined as $\sum_{i = 1}^{2} {(Δ_{i} - F_{i} (\hat{P}))}^{2}$ , is the objective function, then the maximum likelihood search equation is as follows

${\hat{P}}_{ML} = \underset{| | \hat{P} | | = R, \hat{P} \in Z}{\arg} [min \sum_{i = 1}^{2} {(Δ_{i} - F_{i} (\hat{P}))}^{2}]$ (17)

In the equation, the subscript ML denotes the solution of maximum likelihood search, Z means restricted searching range, which is the set of positions lied in the localization triangle, and arg shows the value range of the variable. The definite meaning of equation (17) is when the position vector $\hat{P}$ satisfies the constraint condition of the sphere of the earth, all the locations within the range will be searched, and the vector $\hat{P}$ which can minimize objective function $\sum_{i = 1}^{2} {(Δ_{i} - F_{i} (\hat{P}))}^{2}$ will be the solution of the localization equations.

In the searching process mentioned above, as the number of times of searching increases, the time spent increases, and also the complexity. In order to reduce the computation and improve the efficiency, we adopt hierarchical search method in this article. When getting the solution in some accuracy, we regard the point as a center, reduce the radius, and then, search once more, and so on. The accuracy of the solution will be improved step by step, until the result is less than the required threshold. At this time, the result will be the final estimated location of the interference. The algorithm of step-by-step decline is adopted for the sake of reducing the false probability when narrowing the range, meanwhile, for taking account of searching efficiency.

Localization error analysis

There are many possible sources of error, such as ephemeris, antenna pointing, antenna pattern, and so on. In order to highlight the focus of the article, these errors are eventually translated into signal strength errors. The article only analyzes the influence of signal strength measurement error on localization results. Conducting differential operation for the three equations in equation (6) at the target point $(x, y, z)$ , we can get

$\begin{matrix} \begin{matrix} d Δ_{1} = & \frac{\partial F_{1}}{\partial x} d x + \frac{\partial F_{1}}{\partial y} d y + \frac{\partial F_{1}}{\partial z} d z \\ + \frac{\partial F_{1}}{\partial x_{0}} d x_{0} + \frac{\partial F_{1}}{\partial y_{0}} d y_{0} + \frac{\partial F_{1}}{\partial z_{0}} d z_{0} \\ + \frac{\partial F_{1}}{\partial x_{1}} d x_{1} + \frac{\partial F_{1}}{\partial y_{1}} d y_{1} + \frac{\partial F_{1}}{\partial z_{1}} d z_{1} \\ + \frac{\partial F_{1}}{\partial x_{2}} d x_{2} + \frac{\partial F_{1}}{\partial y_{2}} d y_{2} + \frac{\partial F_{1}}{\partial z_{2}} d z_{2} \end{matrix} \end{matrix}$ (18)

$\begin{array}{l} d Δ_{2} = \frac{\partial F_{2}}{\partial x} d x + \frac{\partial F_{2}}{\partial y} d y + \frac{\partial F_{2}}{\partial z} d z + \frac{\partial F_{2}}{\partial x_{0}} d x_{0} \\ + \frac{\partial F_{2}}{\partial y_{0}} d y_{0} + \frac{\partial F_{2}}{\partial z_{0}} d z_{0} + \frac{\partial F_{2}}{\partial x_{2}} d x_{2} + \frac{\partial F_{2}}{\partial y_{2}} d y_{2} \\ + \frac{\partial F_{2}}{\partial z_{2}} d z_{2} + \frac{\partial F_{2}}{\partial x_{3}} d x_{3} + \frac{\partial F_{2}}{\partial y_{3}} d y_{3} + \frac{\partial F_{2}}{\partial z_{3}} d z_{3} \end{array}$ (19)

$d R = \frac{\partial F_{3}}{\partial x} d x + \frac{\partial F_{3}}{\partial y} d y + \frac{\partial F_{3}}{\partial z} d z$ (20)

where

$\frac{\partial F_{1}}{\partial \nabla_{0}} = \frac{\partial G}{\partial θ_{2}} \cdot \frac{\partial θ_{2}}{\partial \nabla_{0}} - \frac{\partial G}{\partial θ_{1}} \cdot \frac{\partial θ_{1}}{\partial \nabla_{0}} (\nabla_{0} = x_{0}, y_{0}, z_{0})$ (22)

$\frac{\partial F_{1}}{\partial \nabla_{1}} = - \frac{\partial G}{\partial θ_{1}} \cdot \frac{\partial θ_{1}}{\partial \nabla_{1}} (\nabla_{1} = x_{1}, y_{1}, z_{1})$ (23)

$\frac{\partial F_{1}}{\partial \nabla_{2}} = \frac{\partial G}{\partial θ_{2}} \cdot \frac{\partial θ_{2}}{\partial \nabla_{2}} (\nabla_{2} = x_{2}, y_{2}, z_{2})$ (24)

$\frac{\partial F_{2}}{\partial \nabla_{0}} = \frac{\partial G}{\partial θ_{3}} \cdot \frac{\partial θ_{3}}{\partial \nabla_{0}} - \frac{\partial G}{\partial θ_{2}} \cdot \frac{\partial θ_{2}}{\partial \nabla_{0}} (\nabla_{0} = x_{0}, y_{0}, z_{0})$ (26)

$\frac{\partial F_{2}}{\partial \nabla_{2}} = - \frac{\partial G}{\partial θ_{2}} \cdot \frac{\partial θ_{2}}{\partial \nabla_{2}} (\nabla_{2} = x_{2}, y_{2}, z_{2})$ (27)

$\frac{\partial F_{2}}{\partial \nabla_{3}} = \frac{\partial G}{\partial θ_{3}} \cdot \frac{\partial θ_{3}}{\partial \nabla_{3}} (\nabla_{3} = x_{3}, y_{3}, z_{3})$ (28)

where

$\begin{array}{l} \frac{\partial θ_{i}}{\partial \nabla_{0}} = \frac{- {\frac{[2 \nabla_{0} - \nabla - \nabla_{i}]}{| \vec{S B_{i}} | | \vec{S P} |} - \frac{\vec{S B_{i}} {\vec{S P}}^{T} (\nabla_{0} - \nabla_{i})}{{| \vec{S B_{i}} |}^{3} | \vec{S P} |} - \frac{\vec{S B_{i}} {\vec{S P}}^{T} (\nabla_{0} - \nabla)}{| \vec{S B_{i}} | {| \vec{S P} |}^{3}}}}{\sqrt{1 - {(\frac{\vec{S B_{i}} {\vec{S P}}^{T}}{| \vec{S B_{i}} | | \vec{S P} |})}^{2}}} \\ (i = 1, 2, 3; \nabla = x, y, z) \end{array}$ (30)

Rewriting the functions (18)–(20) in the form of matrix

$\begin{matrix} \begin{matrix} [\begin{matrix} d Δ_{1} \\ d Δ_{2} \\ d R \end{matrix}] & = [\begin{matrix} \frac{\partial F_{1}}{\partial x} & \frac{\partial F_{1}}{\partial y} & \frac{\partial F_{1}}{\partial z} \\ \frac{\partial F_{2}}{\partial x} & \frac{\partial F_{2}}{\partial y} & \frac{\partial F_{2}}{\partial z} \\ \frac{\partial F_{3}}{\partial x} & \frac{\partial F_{3}}{\partial y} & \frac{\partial F_{3}}{\partial z} \end{matrix}] \cdot [\begin{matrix} d x \\ d y \\ d z \end{matrix}] \\ + [\begin{matrix} \frac{\partial F_{1}}{\partial x_{0}} & \frac{\partial F_{1}}{\partial y_{0}} & \frac{\partial F_{1}}{\partial z_{0}} \\ \frac{\partial F_{2}}{\partial x_{0}} & \frac{\partial F_{2}}{\partial y_{0}} & \frac{\partial F_{2}}{\partial z_{0}} \\ 0 & 0 & 0 \end{matrix}] \cdot [\begin{matrix} d x_{0} \\ d y_{0} \\ d z_{0} \end{matrix}] \\ + [\begin{matrix} \frac{\partial F_{1}}{\partial x_{1}} & \frac{\partial F_{1}}{\partial y_{1}} & \frac{\partial F_{1}}{\partial z_{1}} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}] \cdot [\begin{matrix} d x_{1} \\ d y_{1} \\ d z_{1} \end{matrix}] \\ + [\begin{matrix} \frac{\partial F_{1}}{\partial x_{2}} & \frac{\partial F_{1}}{\partial y_{2}} & \frac{\partial F_{1}}{\partial z_{2}} \\ \frac{\partial F_{2}}{\partial x_{2}} & \frac{\partial F_{2}}{\partial y_{2}} & \frac{\partial F_{2}}{\partial z_{2}} \\ 0 & 0 & 0 \end{matrix}] \cdot [\begin{matrix} d x_{2} \\ d y_{2} \\ d z_{2} \end{matrix}] \\ + [\begin{matrix} 0 & 0 & 0 \\ \frac{\partial F_{2}}{\partial x_{3}} & \frac{\partial F_{2}}{\partial y_{3}} & \frac{\partial F_{2}}{\partial z_{3}} \\ 0 & 0 & 0 \end{matrix}] \cdot [\begin{matrix} d x_{3} \\ d y_{3} \\ d z_{3} \end{matrix}] \end{matrix} \end{matrix}$ (32)

In the function above, $d Δ_{i} (i = 1, 2)$ is the measurement error of the relative signal strength, and $dR$ is the radius error of the earth, which is also called target elevation error. According to the corresponding relationship, equation (32) can be simplified as follows

$\begin{array}{l} d V = C \cdot d X + C_{0} \cdot d X_{0} + C_{1} \cdot d X_{1} \\ + C_{2} \cdot d X_{2} + C_{3} \cdot d X_{3} \end{array}$ (33)

where $d V$ is the observation error vector, $d X$ is the localization error vector, and both $C$ and $C_{i} (i = 0, 1, 2, 3)$ are coefficient matrixes

$\begin{array}{l} d X = C^{- 1} \cdot (d V - C_{0} \cdot d X_{0} - C_{1} \cdot d X_{1} \\ - C_{2} \cdot d X_{2} - C_{3} \cdot d X_{3}) \end{array}$ (34)

The covariance matrix of error vectors is shown in equation (35), where function $E (\cdot)$ denotes calculating mathematical expectation

$\begin{matrix} P_{d X} & = E (d X \cdot d X^{T}) \\ = E [C^{- 1} \cdot (d V - C_{0} \cdot d X_{0} - C_{1} \cdot d X_{1} - C_{2} \cdot d X_{2} - C_{3} \cdot X_{3}) \\ \cdot (d V^{T} - d X_{0}^{T} \cdot C_{0}^{T} - d X_{1}^{T} \cdot C_{1}^{T} - d X_{2}^{T} \cdot C_{2}^{T} - {dX}_{3}^{T} \cdot C_{3}^{T}) \cdot C^{- T}] \end{matrix}$ (35)

When the errors of each beam center position are independent, and the errors in both satellite position and relative signal strength are also independent, the localization errors caused by all the error factors can be denoted as in equation (36)

$\begin{matrix} P_{d X} = C^{- 1} {E [d V d V^{T}]} C^{- T} \\ + C^{- 1} C_{0} {E [d X_{0} d X_{0}^{T}]} C_{0}^{T} C^{- T} \\ + C^{- 1} C_{1} {E [d X_{1} d X_{1}^{T}]} C_{1}^{T} C^{- T} \\ + C^{- 1} C_{2} {E [d X_{2} d X_{2}^{T}]} C_{2}^{T} C^{- T} \\ + C^{- 1} C_{3} {E [d X_{3} d X_{3}^{T}]} C_{3}^{T} C^{- T} \end{matrix}$ (36)

Suppose the signal strength measurement errors of the three beams all obey normal distribution, the mean value is 0, and the standard deviation is $σ$ . They are independent and the errors of other terms are 0

$E (d V d V^{T}) = [\begin{matrix} σ^{2} & 0 & 0 \\ 0 & σ^{2} & 0 \\ 0 & 0 & 0 \end{matrix}]$ (37)

$P_{d X} |_{A} = C^{- 1} \cdot E (d V d V^{T}) \cdot C^{- T}$ (38)

The geometric dilution of precision(GDOP) of localization error is as follows (GDOP describes three-dimensional geometric distribution)

$GDO P_{A} = \sqrt{tr (P_{d X} |_{A})}$ (39)

where $tr (\cdot)$ denotes the trace of the matrix, which means the sum of diagonal element.

Simulation results

The orbit altitude of GEO satellite is 35,786 km and the radius of Earth is about 6371 km. When using the antenna pattern function in equation (2), the two methods introduced in the article are both appropriate. If the diameter D of the satellite antenna is 12.5 m, and the carrier frequency of signal is 2 GHz, which means the wave length λ is 0.15 m, then the half-power beam width is $θ_{3 dB} = 70 \cdot λ / D = 0.84$ degree. The radius of the spot beam is 262 km, and then the side length of the localization triangle is $\sqrt{21} R_{b}$ , almost 1200 km. Without loss of generality, suppose the longitudes and latitudes of the vertex of localization triangle are(–5.339205, 0), (0, 9.358678), and (5.339205, 0), respectively. The point under satellite is located at (0, 0), and the unit is degree, totally.

Under the scene above, the localization simulation is carried out with the interference source at (0, 3.11646). First, when MLISM is adopted, the relationship between the result of localization error and the searching times is shown in Figure 4.

Figure 4.

The relationship between the result of localization error and the searching times.

From Figure 4, the localization error decreases drastically along with the increase of searching times. When searching times are 10,000, 20,000, 50,000, and 100,000, the accuracy of localization, respectively, reaches under 13, 6, 4, and 2 km, which still has rising space as searching times increase. Although the searching efficiency maybe not satisfy some applications, it reveals the convergence of the algorithm and feasibility of the localization method in this article. In satellite communications system, the diameters of spot beams are always hundreds of or thousands of kilometers, while the users distribute sparsely. Combined with the prior information of user distributions in the system, the localization accuracy within 50 km can almost solve most of practical issues. Meanwhile, the searching method is appropriate for the situation that localization equations are not derivable. In the ideal case that there is no error in signal strength measurement, the localization is carried out by the way of NILSM, with the three interference sources at (–2.669, 1.558), (0, 3.11646), and (5.339205, 0), where the searching stop thresholds are, respectively $10^{- 3}$ , $10^{- 4}$ , and $10^{- 6}$ . Simulations were performed for three independent interference sources, respectively. The simulation results of the localization errors are shown in Table 1 in the unit of km, and the numbers in circles next to the errors denote the searching times.

Table 1.

Localization errors by means of NILSM.

	$10^{- 3}$	$10^{- 4}$	$10^{- 6}$
(–2.669, 1.558)	1.36e−10, ④	1.36e−10, ④	1.33e−10, ⑤
(0, 3.11646)	1.24e−11, ④	1.24e−11, ④	1.24e−11, ⑤
(5.339205, 0)	4.46e−10, ⑤	4.46e−10, ⑤	2.22e−10, ⑥

NILSM: Newton iteration based on least quare method.

From Table 1, when adopting NILSM, the precision of the solving method is high, and the impact caused by the method on the total localization errors can be almost omitted. Also, the searching times are very few. Thus, the localization errors are mainly caused by other factors, such as signal strength measurement. When the signal strength measurement error is 0.3 dB and the area is limited in the red dotted rectangle in Figure 3, the localization error distribution is shown in Figure 5, which includes two halves of a localization triangle and a complete one. In Figure 5, the x-coordinate denotes longitude, the y-coordinate denotes latitude, and the contour line shows the localization errors in the unit of km. The maximum error in the localization area is less than 10 km in Figure 5, which satisfies the general requirement of coarse localization. Within the localization triangle, the accuracy near the edge is high, while it is low near the center. The reason lies in that the localization accuracy has relations with the ratio of the antenna gain gradients and signal strength measurement errors. The localization error in the localization triangle when there exists signal strength measurement error of 0.6 and 1.5 dB is shown in the Figures 6 and 7, respectively. The analysis results show that the localization method can be applied as long as the measurement error is not too large. Because the requirement of the signal strength measurement accuracy is not too high, the existing measurement methods can almost meet the requirements. Thus, we can determine the requirement for measurement accuracy according to the gradients of the antenna gain in applications.

Figure 5.

The contour lines of localization errors in the rectangle when there exists signal strength measurement error of 0.3 dB.

Figure 6.

The contour lines of localization errors in the rectangle when there exists signal strength measurement error of 0.6 dB.

Figure 7.

The contour lines of localization errors in the rectangle when there exists signal strength measurement error of 1.5 dB.

Compared with the literature,⁸ the similarity is as follows. In the multi-beam satellite communications system, the uplink interference waves received by the separate beams onboard are frequency multiplexed and are transmitted to the ground. The ground station selects the beam to be used for localization and measures the ratio of the signals strength. The interference source location is estimated using these measured values. There are both simple but effective methods to estimate the location of unknown interference. However, the differences are as follows. Viewed from the perspective of localization principle, in Matsumoto,⁸ the angle measurement based on amplitude comparison among multiple beams is applied to localization system, whose interference is overlapping with the communication signals received by the multi-beam satellite antenna. However, in this article, the location of the interference source is given by measuring the relative ratio of the received signals strength and building and solving the mathematical equation set. In the view of location accuracy, the total location error of Matsumoto⁸ is calculated as $\pm 0.37 \circ$ square and $\pm 0.21 \circ$ square, corresponding to one-sixth to one-tenth of the antenna beam width. In this article, the location error is about 50 km when the signal strength measurement error is up to 1.5 dB. Compared with the beam diameter 524 km, it is about one-tenth of the beam width, and the performance is equivalent with it in Matsumoto.⁸ The location error is about 20 km when the signal strength measurement error is down to 0.6 dB, which outperforms that in Matsumoto.⁸

From the above, the localization method can achieve finding the interference source without the help of other equipments onboard. There are two ways for the equation set solving: one is MLISM and the other is Newton iteration based on least square method. When MLISM is adopted, the accuracy of localization improves along with the increase of searching times. But the method itself may lead into errors, while in fact, the effects to the overall localization errors should be analyzed together with other issues. The advantage of the method is that it is suitable for the condition that the antenna gain function is not derivable or has no closed-form solution. When using Newton iteration based on least square method, the equations have to be derivable. This method can reach high precision in localization, and thus, the influence to the overall localization errors can be omitted. Generally, when using this method, we can get the solution of desired accuracy by only a small number of iterations.

Conclusion

In this article, we propose a single satellite localization method based on onboard multi-beam antenna gain and signal strength distribution, which makes use of the characteristic that each spot beam of multi-beam antenna has different gain at the position of interference and combines the antenna pattern projection model of the multi-beam antenna. The localization method is independent of the characteristics of interference and the environments of the satellite orbit. Because of not depending on extra onboard equipments and other facilities, it achieves single satellite interference localization by the system itself. A single satellite is enough to locate the source. It is suitable for satellite mobile communication systems, such as TT-1 satellite communication system with onboard multi-beam antenna, because the localization condition is easy to be satisfied and the location accuracy could meet the practical application requirements. Therefore, it can provide a method to predict the link degradation caused by the interference or execute the avoidance operation of the antenna beam from the interference source. The method reduces the requirement of external condition and is convenient to be achieved and applied, which has an important realistic meaning in checking the interference of the system, decision supporting of anti-jamming, and improving operation and management. Meanwhile, the technology can also be popularized and applied in the localization of interference source or radiation source in other communication satellites and electronic reconnaissance satellites, using multi-beam antennas onboard, which will have a better application prospect.

Footnotes

Handling Editor: Haibo Zhou

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research,authorship,and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: The Project Sponsored by the National Natural Science Foundation of China (91738201,61401507) and the China Postdoctoral Science Foundation (2017M613403).

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