Detection of Spectrum Misuse Behavior in Satellite-Terrestrial Spectrum Sensing Based on Multi-Hypothesis Tests

As the increasing of both users’ demand and the number of communication satellites, spectrum resource is becoming increasingly scarce. Cognitive radio (CR) can effectively alleviate the scarcity of spectrum resource. By using CR technology in the satellite and terrestrial integrated network, the satellite can share the spectrum resource with the terrestrial communication equipment to improve the spectrum utilization. However, due to the satellite communication’s inherent openness and the increasing intelligent level of devices, it makes the satellite communication vulnerable to spectrum misuse, which seriously threatens the reliability and efficiency of the satellite communication system. This paper studies the detection of spectrum misuse in the satellite-terrestrial spectrum sharing system. First, we model the problem as a ternary hypothesis test problem by using generalized multi-hypothesis Neyman-Pearson (GMNP). Then, we utilize a two-step method to solve the complex formula problem in the multiple authorized users scenario. Reasonable decision thresholds are derived by means of the generalized likelihood ratio test (GLRT) and maximum a posterior (MAP) criterion, respectively. Finally, the performance of detection is evaluated and analyzed comprehensively to verify the feasibility of the study.

are similar problems with shortage of spectrum resources, so CR technology can be well used in the satellite communication system. In 2010, Dr. Kandeepan of Italy first proposed the concept of cognitive satellite terrestrial radio (CSTR) [20]. For CSTR systems, it is very important for secondary users to correctly perceive the spectrum of primary users, so secondary users need to have the ability to detect illegal behavior if an illegal user violates the rules of spectrum usage and emotional communication [21], [22]. However, it is difficult to analyze and detect illegal users because of the random appearance of illegal users.

B. RELATED WORK
As for satellite cognitive communication system, reference [23] utilizes hypothesis testing and maximum posteriori to detect NGEO satellite signals which impact GEO system. Reference [12] deduces the protection radius of the earth station by analyzing the interference to the satellite system. Reference [24] suggests that beamforming technology can reduce the interference caused by satellite signals to terrestrial communication systems. In reference [25], a dynamic spectrum access (DSA) decision framework, which can also perceive the spectrum well under interference, is proposed. References [9] proposes a space segment design based on a spectrum-sensing-based cooperative framework, in applying cognitive radio to future broadband satellite communications toward 5G. Reference [17] proposes the concept of weighted cooperative spectrum sensing, which can better cope with the interference to the primary user caused by a secondary user. In fact, the above references only study how to make normal spectral perception under interference, and the interference discussed is predictable. However, illegal frequency users are a priori unknown, the above method cannot be applied.
For the detection of spectrum misuse behavior, reference [26] constructs a joint spectrum sensing and access framework to prevent malicious behavior by rational and irrational malicious users. Reference [27] proposes a malicious user detection framework based on low rank matrix completion. However, their detection method is based on the current frequency state known. In reference [21], a generalized multi-hypothesis Neyman-Pearson (GMNP) is proposed, which can detect illegal behaviors while judging the frequency state of use. However, the above methods are all based on the terrestrial spectrum sharing system. The satellite-terrestrial spectrum sharing system is quite different from the terrestrial spectrum sharing system, the detection method should be designed according to its characteristics.
The main differences between the satellite-terrestrial spectrum sharing system and the terrestrial spectrum sharing system are as follows: • Satellite-terrestrial channel is very different from terrestrial channel, which is mainly reflected in channel gain. Satellite signal is beam signal, and different antenna gain should be considered in different cognitive scenarios. In this paper, the analysis of the system model is added, and the characteristics of the satellite-terrestrial spectrum sharing system are reflected in more detail.
• For traditional terrestrial spectrum sharing system, the location of the primary user is generally fixed and the beam coverage is small (such as cellular communication systems), so the coverage of the primary users usually do not overlap. However, in the satellite-terrestrial spectrum sharing system, the position of NGEO satellite is generally not fixed. A satellite's beam coverage is greater than that of a base station on the ground. The satellite beam coverage area may also vary as the satellite moves, and beams of multiple primary users may overlap. Therefore it is necessary to consider the spectrum misuse behavior detection scenario under the coverage of multiple primary users.

C. CONTRIBUTIONS
This paper proposes a detection framework for illegal behavior in the satellite-terrestrial spectrum sharing system. The main contributions are as follows: • We propose the detection of spectrum misuse behavior in satellite-terrestrial spectrum sharing system and model the problem as a multiple hypothesis test problem. Then the hypothesis is reasonably simplified to a ternary hypothesis test problem. Finally, the GLRT is used to solve the problem.
• We analyze the influence of the number of primary users on the detection performance of illegal behaviors by deducing the judgment threshold of illegal behaviors. In addition, the detection of illegal behavior of single and multiple satellites is analyzed and deduced, respectively. Furthermore, the detection method and decision threshold in two scenarios are given.
• We provide detailed simulation under different parameter settings, which validates the feasibility of the detection algorithm and has a high detection probability for illegal behaviors.

D. ORGANIZATION
The rest of this paper is organized as follows. Section II analyzes the system model of the typical spectrum sharing scenario and classifies it into two cases. Then it establishes multiple hypothesis test model when the number of authorized users is one. Section III gives the GMNP criterion and deduces the detection threshold of the illegal behavior when the number of authorized users is one. Section IV establishes multiple hypothesis test model when the number of authorized users exceeds one and deduces the detection threshold of illegal behavior in this scenario. Section V carries on the simulation, Section VI gives the conclusion.

II. SYSTEM MODEL
There are three typical satellite-terrestrial spectrum sharing systems: • Satellite authorization and terrestrial cognition. The satellite network is the authorization network while the terrestrial network is the cognitive network.
• Terrestrial authorization and satellite cognition. The terrestrial network is the authorization network while the satellite network is the cognitive network.
• Satellite authorization and satellite cognition. The satellite network is both the authorization network and the cognitive network. Before proceeding further, we make some assumptions as follows: • The power level sets from the satellite's system are known to cognitive users, since the power level sets are predetermined and could be acquired from ITU (International Telecommunication Union) database.
• The gain patterns of the earth stations and the satellites conform to relevant ITU-R recommendations [28]. The positions of the satellites can be obtained through the electronic fence system [29], which adopts phased array radar to search and track space targets. Its phased array antenna controls the direction of the beam electronically and it can simultaneously search and measure multiple beams of different directions.
• Generally speaking, it is difficult for illegal users to obtain accurate information about the working status of the primary user, and they can only obtain limited information by means of transient time slot perception. So considering an extreme case where an illegal user has no information about the true status of the primary user, illegal users can only attack at random completely, that is to say, the prior probability of the illegal user and the primary user are independent of each other.

A. SATELLITE AUTHORIZATION AND TERRESTRIAL COGNITION
The satellite authorization and terrestrial cognition refers to the scenario that the satellite network is the authorization network while the terrestrial network is the cognitive network. There are two scenarios, satellite uplink sharing and satellite downlink sharing. For the scenario of satellite uplink sharing, the wide area of satellite coverage and the communication elevation angle of the earth station increase the difficulty of spectrum sensing of cognitive users, and it is difficult for terrestrial cognitive users to effectively use the satellite spectrum resources. Therefore, the existing satellite authorization and terrestrial cognition network is mainly applied in the terrestrial cognitive user how to use CR technology to share the channel resources of satellite downlink. Fig. 1 shows a typical scenario of satellite authorization and terrestrial cognition wireless network based on satellite downlink sharing. In this scenario, the ground cognitive user finds and utilizes the idle satellite spectrum resources through spectrum sensing. In traditional CR systems, detecting whether the authorized primary user works on a given frequency band is a binary hypothesis test problem [23]. In this scenario, for this binary hypothesis test problem, the authorized user is a communication satellite, and the cognitive base station needs to perceive whether the signal of the primary user exists or not. Therefore, there are two assumptions here. One is that H S 0 is not used by the authorized user, i.e. the channel is idle; the other is that H S 1 is used by the authorized user, i.e. the channel is busy.
where y s (t) is the received signal of the sensor at sampling time t; s(t) is the primary user's satellite transmission symbol, which follows a circularly symmetric complex gaussian (CSCG) distribution with zero mean and unit variance; P s is the transmitted power of a satellite, and in a single time slot is constant; h s for the channel gain, concrete expression as follows, φ for channel phase; n(t) for additive white gaussian noise, the variance for σ 2 n .
where G ner,max represents the maximum gain of the receiving antenna of the sensor, G gst (θ 1 ) represents the gain of the satellite transmitted antenna with the direction angle of θ 1 , c represents the speed of light, f represents the central frequency of the spectrum band, and d S→B represents the distance between the satellite and the sensor. Propagation factors A g and A c represent gaseous absorption and cloud or fog attenuation, respectively, [30]. Gaseous absorption A g (dB) due to the two constituent gases(oxygen and water vapor) is given by [31].
where A ω and A ω are the specific attenuations due to dry air and water vapour, respectively, and N (f ) is the imaginary part of the frequency-dependent complex refractivity that the specific expression refers to [31]. On the other hand, the specific attenuation within a cloud or fog A c (dB) can be written as [32] A c = K l M , (4) VOLUME 8, 2020 where K l = 0.89 f / (1 + η) is the specific attenuation coefficient with η = 2 + / , and are the real part and imaginary part of the dielectric permittivity of water, M represents liquid water density in the cloud or fog.
Vectors − → AB and − → AC can be expressed by the following equation: where h sat represents the altitude of the satellite from the ground, R e is the radius of the earth, SE represents the angle between the satellite to the center of the earth and the satellite earth station to the center of the earth, EB represents the angle between the satellite to the center of the earth and the terrestrial cognitive base station to the center of the earth. Therefore, θ 1 and d S→B can be expressed as: In addition, φ is channel phase, which is independent of the sensing mode that mainly involving signal energy proposed in this paper, n(t) is additive white gaussian noise (AWGN) with mean value of zero and variance of σ 2 n . Therefore, h s follows a cyclic symmetric complex gaussian (CSCG) distribution, and can be expressed as: h s ∼ CN (0, h s P s + σ 2 n ). In the perceptive slot with time length T, the sensor earth station obtains N sampling values, expressed as y = (y 0 , y 1 , · · · , y N ).
For unauthorized users, channels may be used at a time slot. This state can also be described by the following binary hypothesis test: where H I 0 refers to the assumption that there is no illegal user in the current time slot, H I 1 refers to the assumption that the illegal user runs with transmitted power of P x , h x represents the channel gain, since the prior information of the illegal user is unknown, the transmitted power P x and channel gain h x are also unknown. s(t) refers to the signal emitted by the illegal user at time t. Therefore, y I (t) also follows a CSCG distribution and can be expressed as: In practical use, the coverage of multiple satellites using the same frequency band may overlap due to the wide range of satellite coverage and the increase in the number of satellites. The cognition terrestrial user may receive signals from multiple authorized satellites at the same time, as shown in Fig. 2. For this scenario, the detection method of illegal users is different from the previous scenario, which increases the difficulty of detection. The following sections will analyze the problem of illegal user detection in this scenario in detail.

B. TERRESTRIAL AUTHORIZATION AND SATELLITE COGNITION
Terrestrial authorization and satellite cognition network refers to the scenario that the terrestrial network is the authorization network while the satellite network is the cognitive network. Since it will cause serious interference to the terrestrial primary user when the satellite downlink as the cognitive user link, this scenario mainly focuses on the satellite uplink. The typical application of terrestrial authorization and satellite cognition network is shown in Fig. 3.
The cognitive earth station finds and utilizes the free terrestrial spectrum resources through spectrum sensing. The signal y B (t) received by the cognitive earth station can be modeled as: where H B 0 indicates that the authorized user is not working, H B 1 represents the authorized user working. P B represents the transmitted power of the base station, h B represents channel gain, n(t) represents noise. s 2 (t) is the signal transmitted by the ground station and obeys the gaussian distribution of zero mean value and unit variance. P B and h B are known a priori, so y B (t) also obeys the gaussian process, y B (t) ∼ N (0, h B P B + σ 2 n ). Again, unauthorized illegal user modeling is the same as before. It follows the same gaussian. Since the beams of the ground communication network do not generally overlap, it makes no sense for this scenario to discuss multiple authorized users. In this scenario, it is a spectrum misuse detection problem with single authorized user.

C. SATELLITE AUTHORIZATION AND SATELLITE COGNITION
According to the difference of satellite orbital altitude, the cognitive wireless network constructed by the dual-satellite system is generally composed of the high-orbit satellite network as the authorized user and the low-orbit satellite network as the cognitive user. A typical satellite authorization and satellite cognition network is shown in Fig. 4. In this scenario, the signal y ss (t) received by the earth station of cognitive satellite can be modeled as: where P ss represents the transmitted power of the authorized satellite, h ss represents channel gain, s 3 (t) is a symbol for satellite transmission, following a CSCG distribution. G ner (θ 3 ) represents the receiver antenna gain of the cognitive earth station, G gst (θ 2 ) represents the transmitted antenna gain of the cognitive earth station, d S→E represents the distance between the authorized satellite and the cognitive earth station. Vectors − → AB, − → AD and − → CD can be expressed as follows: where h sat1 is the altitude of satellite 1, h sat2 is the altitude of satellite 2, EE represents the angle between the earth station of satellite 1 and the earth station of satellite 2 to the center of the earth, SS represents the angle between satellite 1 to the center of the earth and satellite 2 to the center of the earth. Therefore, θ 2 , θ 3 and d S→E can be expressed as: Similarly, y ss (t) also follows a CSCG distribution and can be expressed as: y ss (t) ∼ CN (0, h ss P ss + σ 2 n ). Again, unauthorized illegal user modeling is the same as before. It follows the same gaussian distribution.
In the same way, the cognition earth station may receive signals from multiple authorized satellites at the same time, as shown in Fig. 5. For this scenario, the detection method of illegal users is different from the previous scenario, which increases the difficulty of detection. The following sections will analyze the problem of illegal user detection in this scenario in detail.
Through the analysis of the previous three scenarios, we can divide the detection of spectrum misuse behavior in the satellite-terrestrial spectrum sharing system into two situations: • When the number of authorized users is one, the detection of spectrum misuse behavior.
• When the number of authorized users exceeds one, the detection of spectrum misuse behavior. Next, in the case of satellite authorized ground cognition, the detection methods of illegal frequency behavior in the above two cases will be analyzed. The detection methods in the other two scenarios are basically the same, so the analysis is not repeated. VOLUME 8, 2020

D. MULTIPLE HYPOTHESIS TEST MODEL WHEN THE NUMBER OF AUTHORIZED USERS IS ONE
In the scenarios of satellite authorizing terrestrial to recognize only a single satellite is considered as the primary user, the perception of the system by the sensor can be modeled as the following quaternion hypothesis test problem: where S 0 means the channel is idle, S 1 means the channel is used by the primary user satellite, S 2 means the channel is occupied by illegal users, and S 3 means that both the primary user and illegal users exist. y(t) represents the observation of the earth station for sensing at time t. From the signal received by the earth station for sensing, it can be seen that both primary users and illegal users are subject to CSCG distribution and are independent of each other. By analyzing the characteristics of each hypothesis, it can be found that the difference of each hypothesis lies in the variance of observed values. Because when S 1 is assumed to be true, σ 2 = P s + σ 2 n ; when S 2 is assumed to be true, σ 2 = P x + σ 2 n ; when S 3 is assumed to be true, Because the illegal user's received power p x is unknown, the above model can be reformulated as: where σ 2 0 = σ 2 n , σ 2 1 = P s + σ 2 n . In S 3 , σ 2 > σ 2 1 when both primary and illegal users are received. When S 2 is true, because the parameter space is continuous and the probability of σ 2 = σ 2 1 is 0, it can be distinguished from the assumption S 1 .
Back to the purpose of this paper, that is to detect whether the channel is occupied, and if it is occupied, determine whether there are illegal users. Among them, the first part is to find out the spectrum opportunity, while the second part is to protect the scarce spectrum resources from being occupied illegally. In order to implement part one, S 0 needs to be distinguished from other states, and in order to implement part two, S 0 and S 1 need to be distinguished from other states. Therefore, detection targets can be achieved even if S 2 and S 3 are not distinguished. Therefore, the original quaternion hypothesis test problem can be transformed into a ternary hypothesis test problem, that is, S 2 and S 3 can be merged into one term. Then the original problem can be modeled as the following ternary hypothesis test [21]: where H 0 and H 1 represent channel idle and channel occupied by the main user satellite respectively, while the mixed assumption H 2 indicates the existence of illegal users.
To facilitate subsequent statements, σ 2 is used to represent the unknown variance of the observed data in H 2 .

III. DETECTION OF ILLEGAL BEHAVIOR A. GENERALIZED MULTI-HYPOTHESIS NEYMAN-PEARSON CRITERION
In the ternary hypothesis model, not only the prior probability is unknown, but also it is difficult to obtain the prior distribution of the unknown variance of the observed data. At the same time, there are conflicts among various state detection probabilities. To solve these problems, we can use the following generalized multi-hypothesis Neyman-Pearson (GMNP) criterion [21]: Under the constraint of the detection probability of H 0 and H 1 (Pr(H 0 |H 0 ) and Pr(H 1 |H 1 )), the detection probability of illegal behavior is maximized, i.e.
where Pr(H i |H j ) represents the probability that judgment H i is true when the true state is H j , i, j ∈ {0, 1}, both α and β are the performance constraint parameters, 0.5 <α< 1, 0.5 <β< 1. In addition, R i represents the decision domain of H i , including: where p(y; H j ) is the probability density function of observation sequence y when j is true. Next, Lemma 1 is obtained by equivalent simplification of optimization criterion (23). Lemma 1: The optimization criterion in (23) can be equivalent to: [21] 50404 VOLUME 8, 2020 Remark 2: In the Neyman-Pearson criterion of traditional binary hypothesis test, the detection probability of the alternative hypothesis is maximized under false alarm constraint of the original hypothesis [33]. Therefore, the GMNP criterion proposed in Eq. 23  First, in order to maximize the detection probability of illegal behaviors under given constraints, the Lagrange factor method is used to construct the objective function [21]: Then: According to the above equation, if F is to be maximized, λ 0 p (y; H 0 )−p (y; H 2 ) > 0 when the decision domain is R 0 ; when the decision domain is R 1 , λ 1 p (y; H 1 ) − p (y; H 2 ) > 0, where λ 0 and λ 1 are determined by the constraint in the equation. Therefore, in order to maximize Pr (H 2 |H 2 ), judge H 2 as true if and only if: In order to realize the GMNP criterion, two likelihood ratios p(y;H 2 ) p(y;H 0 ) and p(y;H 2 ) p(y;H 1 ) need to be considered in the detection process. By comparing the two equations, it can be seen that in the constraint, Pr (H 0 |H 0 ) is related to the former likelihood ratio, while Pr (H 1 |H 1 ) is related to the latter likelihood ratio. Therefore, the optimization problem can be decomposed into two sub-problems: [21] and max R 1 ,R 2 Pr (H 2 |H 2 ) , The final solution is the intersection of two subproblems. Therefore, the original problem can be simplified by decomposition. Moreover, the two problems have significant physical significance, which can be called detection subproblems and identification subproblems.

1) DETECT SUBPROBLEMS (IDLE OR BUSY)
The subproblems in the equation can be regarded as derived from the following binary hypothesis: Let σ 2 x represent the variance of observation data under H 2 . The above problem is unilateral detection because σ 2 x > σ 2 0 . Therefore, under the limit of Pr(H 0 |H 0 ) = α, in order to maximize Pr(H 2 |H 2 ), likelihood ratio detection is used: Since the observed data y plays a role in the form of N −1 n=0 y 2 n , the test statistic is Y = N −1 n=0 y 2 n , and since σ 2 x > σ 2 0 , L 0 (y) increases with the increase of Y . So L 0 (y) < λ 0 is the same thing as Y < η 0 and η 0 is the threshold for λ 0 . At the same time, since H 0 is true, Y = N −1 n=0 y 2 n ∼ χ 2 N (σ 2 0 ), so, where (·) and (·, ·) are Gamma functions and upper bound incomplete Gamma functions respectively. Therefore, the judgment threshold is where, −1 (·, ·) is the inverse function of the incomplete Gamma function. Therefore, the detection subproblem is determined as VOLUME 8, 2020

2) RECOGNITION SUBPROBLEM (LEGAL OR ILLEGAL)
The second subproblem in the equation is believed to come from the following binary hypothesis test problem: Different from detecting subproblems, identification subproblem bilateral detection means that unknown parameters σ 2 x may be higher or lower than σ 2 1 . Then, the GLRT is used to solve the bilateral problem.
First, the maximum likelihood estimation of unknown varianceσ 2 x is calculated: The generalized likelihood ratio can be obtained Since the test statistic is Y = N −1 n=0 y 2 n and L 1 (y) are functions of Y, it can be obtained that: So at Y > N σ 2 1 , L 1 (y) increases as Y increases; otherwise it decreases as Y increases. Therefore, the minimum value of likelihood ratio 1 can be obtained at N σ 2 1 . Therefore, L 1 (y) < λ 1 is equivalent to η 1 < Y < η 2 , where η 1 and η 2 are the two solutions of L 1 (y) = λ 1 : At the same time, when only the primary user exists, that is, if H 1 is true, Y = N −1 n=0 y 2 n ∼ χ 2 N σ 2 1 . Therefore, in order to satisfy the constraint Therefore, the thresholds η 1 and η 2 can be solved. Therefore, this part is determined as Considering the complexity of GLRT, it is difficult to get a closed form of the threshold. However, the asymptotic properties of the proposed method can be obtained by proper approximation.
First, when N is sufficiently large and the channel is occupied by the primary user, 2 ln L 1 (y) ∼ χ 2 1 . Let z 2 = 2 ln L 1 , where z obeys the gaussian distribution of zero mean and unit variance, namely, z ∼ N (0, 1), and the approximate test probability can be obtained where Q(·) is the complementary cumulative distribution function of the standard normal distribution. Therefore, the asymptotic solution of λ 1 is In addition, based on equation (41), the approximate solutions of the thresholds η 1 and η 2 can be obtained.
Although the complexity of calculating threshold is reduced, the closed solution of threshold cannot be obtained, which hinders the evaluation of detection performance. However, asymptotic detection performance can be obtained by following propositions.
When N is sufficiently large and H 2 is true, then z 2 = 2 ln L 1 (y) ∼ χ 2 1 (θ ), where χ 2 1 (θ ) is the non-central chisquare distribution, and the non-central parameter is θ = σ 2 2 − σ 2 1 2 I σ 2 1 , σ 2 1 is Fisher information. Specifically, Fisher information is calculated as follows: (46) So the gradual probability that H 2 is misjudged as H 1 is where, based on Eq. 45), 2 ln Based on the Eq. 35 and Eq. 43, the decision domain can be obtained: where in order to meet the detection performance constraint, R 2 is the intersection of the decision domain of H 2 in two subproblems.
In Eq. 48, there is a problem that η 0 may be less than η 1 . This means that there may be overlaps between decision domains R 0 and R 1 , making the performance constraints in the target function unsatisfied. Specifically, when constraint parameters α and β satisfy certain conditions, the overlap of decision fields occurs, for which theorem 1 is given.
Theorem 1: If GLRT is adopted, the decision domain R 0 and R 1 will overlap if and only if constraint parameters α and β meet the following conditions: [21] β > N 2 , Proof: see appendix for detailed proof process. When the conditions in theorem 1 are not satisfied and there is no mutual influence between the two subproblems, the asymptotic value of the probability of the illegal behavior being correctly detected can be deduced: On the other hand, when overlap occurs, the decision domain needs to be adjusted to avoid overlap and meet performance constraints. Returning to Eq. 27, it can be seen that, in order to meet the performance constraint, the thresholds λ 0 and λ 1 need to be improved. However, since the overlap has occurred, an increase of λ 0 would lead to an increase of η 0 , thus increasing the overlap area, while η 1 and η 2 in Eq. 48 would increase with the increase of λ 1 , and the decision domain R 1 would further expand. Therefore, when the condition in theorem 1 is satisfied, η 1 needs to be the same as η 0 in order for the constraint condition Pr (H 0 |H 0 ) = α to be satisfied, and in order for the other constraint condition to be satisfied The threshold for η 2 Finally, in the case of overlap, the decision domain is: where, η 0 is calculated from Eq. 34, and η 0 = η 1 , and η 2 is obtained from Eq. 52.

IV. ILLEGAL BEHAVIOR DETECTION WHEN THE NUMBER OF AUTHORIZED USERS EXCEEDS ONE A. MULTIPLE HYPOTHESIS TEST MODEL WHEN THE NUMBER OF AUTHORIZED USERS EXCEEDS ONE
In the case that multiple satellites are the primary users and there are j satellites, the signals received by the sensing earth station for each satellite can also be modeled as a binary hypothesis test problem: where y j s (t) to perceive earth station in the sampling time t to receive the j-th satellite signal; s j (t) is the j-th primary user satellite transmission symbol, it follows a CSCG distribution, with zero mean and unit variance; P j s for transmitted power of the j-th satellite, and in a single time slot is constant; h j s for the j-th satellite channel gain, concrete expression as follows; φ j for the j-th satellite channel phase; n(t) for AWGN, the variance of σ 2 n .
where G ner,max represents the maximum gain of the receiving antenna of the sensing earth station; G j gst (θ j ) represents the gain of the j-th satellite transmitted antenna with the direction angle of θ; c represents the speed of light; f represents the central frequency of the spectrum band; d j gs→ne represents the distance between the satellite and the sensing earth station. Propagation factors A g and A c represent gaseous absorption and cloud or fog attenuation, respectively.
The problem is modeled by taking into account the existence of all satellites and illegal users and linear combination. Considering the probability of all possible scenarios, the multiple hypothesis test model is as follows: H 0 : Only noise exists, H 1 : Only satellite1 and noise exist, H 2 : Only satellite2 and noise exist, · · · · · · H j : Only satellitej and noise exist, H j+1 : Satellite1,satellite2 and noise exist, H j+2 : Satellite1,satellite3 and noise exist, · · · · · · H k+1 : Only illegal users and noise exist, H k+2 : Satellite1,illegal users and noise exist, H k+3 : Satellite1,satellite2,illegal users and noise exist, · · · · · · H M : All signal exsit.
From the above model, it can be seen that the model is too complex, assuming too many combinations, and it is very difficult to solve. Therefore, it is necessary to simplify the above model. Through the above derivation, the difference of each hypothesis is still reflected in the variance of observed values.
We assume that the variance of the signal received from the j-th satellite is σ 2 j , and the variance of the noise is σ 2 0 . The variance of the illegal user is unknown as σ 2 x .
Then the above hypothesis can be changed into: where H 0 means the channel is idle, H 1∼j means the channel is occupied by the first to j-th primary user, H j+1∼k−1 means the channel is occupied by multiple primary users, H k∼M −1 means the channel is occupied by illegal users, and H M means the channel is occupied by both primary users and illegal users. The above model is too complicated to solve. Since the purpose of this paper is to detect spectrum opportunity and further detect whether there are illegal users, the above multiple hypothesis test models can be merged without affecting the detection results. Then, the above hypothesis can be further simplified into a ternary hypothesis test model: , · · · , σ 2 j , σ 2 1 + σ 2 2 , σ 2 1 + σ 2 3 · · · , σ 2 1 + σ 2 2 + · · · + σ 2 j , H 2 : σ 2 > σ 2 0 , σ 2 = σ 2 1 , σ 2 2 , · · · , σ 2 j , σ 2 1 + σ 2 2 , σ 2 1 + σ 2 3 · · · , σ 2 1 + σ 2 2 + · · · + σ 2 j , where H 0 means that the spectrum is not used, H 1 means that the spectrum is occupied by legitimate users (namely the primary users), and H 2 means that illegal users exist.

B. DETECTION METHOD WHEN THE NUMBER OF AUTHORIZED USERS EXCEEDS ONE
Since the model with multiple satellites as the primary user is also built as a ternary hypothesis test model, the analysis method in the previous section is used for processing first.
As can be seen from the above analysis, this ternary hypothesis test problem can also be optimized and decomposed into two sub-problems: and max R 1 ,R 2 Pr (H 2 |H 2 ) , 50408 VOLUME 8, 2020 Similarly, the final solution is the intersection of two subproblems. Therefore, the original problem can be simplified by decomposition. Moreover, the two problems have significant physical significance, which can be called detection subproblems and identification subproblems.

1) DETECT SUBPROBLEMS (IDLE OR BUSY)
The subproblems in the equation can be regarded as derived from the following binary hypothesis test problem: For detection subproblems, the number of primary users has no effect on idle or busy detection of the channel, so the detection method for detection subproblems is the same as before, which is not repeated here.
Therefore, the judgment threshold is: where, −1 (·, ·) is the inverse function of the incomplete Gamma function. Therefore, the detection subproblem is determined as:

2) RECOGNITION SUBPROBLEM (LEGAL OR ILLEGAL)
The second subproblem in the equation is believed to come from the following binary hypothesis test problem: First, the maximum likelihood estimation of unknown variance σ 2 x is calculated Generalized likelihood ratio can be obtained in Eq.64, as shown at the bottom of this page.
Due to the complexity of the detector, it is difficult to carry out theoretical analysis. So you need to think about simpler detectors.
In order to simplify the above detection methods and design a detection method that is easy to theoretical analysis, which can well realize the detection purpose with multiple satellites as the primary user, this section proposes a two-step detector. The core idea is to select the most likely variance and then determine whether there are illegal users.
Specifically, the first step is to adopt the maximum likelihood criterion The second step is to derive the generalized likelihood ratio Through the maximum likelihood criterion of variance, the above formula can be well simplified. Similarly, this is also a bilateral detection problem. The detection threshold can be deduced by continuing to use the above method. ) + · · · + 1 (2π(σ 2 1 +σ 2 2 +···+σ 2 j )) (64) VOLUME 8, 2020 The decision thresholds η 1 and η 2 are the solutions of the following two equations The verdict is So the total decision domain is Similarly, there is also the possibility of overlap in the decision domain, and the derivation and analysis of overlap are the same as above, which will not be repeated here. Finally, if there is overlap, the threshold for η 2 The decision domain becomes where η 0 is calculated from Eq. 59, η 0 = η 1 , and η 2 is obtained from Eq. 71.

V. SIMULATION RESULTS AND ANALYSIS
All satellite and earth station parameters are referred to O3b and OneWeb, respectively. The downlink transmission frequency is 18.48GHz, satellite 1 is a GEO satellite with an altitude of 35678Km, the satellite transmission antenna gain is 52dBi, the earth station antenna gain is 55.4dBi, the earth station antenna diameter is 2.4m, and the earth station noise temperature is 200k. Satellite 2 is an 8062Km NGEO satellite with a transmission gain of 35dBi and a transmission antenna gain of 37.2dBi for the earth station. Satellite 3 is an NGEO satellite with an altitude of 6000Km, the satellite transmission gain is 31.5dBi, and the earth station transmission antenna gain is 36.7dBi. Satellite 4 is an NGEO satellite with an altitude of 10000Km, the satellite transmission gain is 37.3dBi, and the earth station transmission antenna gain is 38.4dBi. The spectrum bandwidth is 6MHz, the sampling number is generally N = 300, and the sampling time is 0.05ms. The noise variance is σ 2 0 = 10 −5 Watt.   x when the number of satellites is from 1 to 4, where α = β= 0.8. As can be seen, when σ 2 x is close to the variance of the primary user, the detection performance decreases significantly. This is because the similarity between various states is too high, which makes it difficult to distinguish. Therefore, when the number of satellites increases, the number of detection performance degradation will increase obviously due to the combination of variance. On the other hand, when the distance between the variances of the primary users increases, the detection performance can be significantly improved. Fig. 7 shows the relationship between detection performance and constraint parameter α and β, where α = β and N = 300. As can be seen, the detection performance will decline with the improvement. Fig. 8 shows the relationship between detection performance and sample number N, where α = β= 0.8. It can be seen that when N increases, detection performance is improved. Fig. 9 and Fig. 10 show the detection performance under different performance constraints and samples when the number of primary user satellites is 2, 3 and 4, respectively. As can   be seen from the figures, the number of satellites has no impact on the detection performance. This is because the maximum likelihood criterion is adopted for variance, and the variance closest to the estimated variance is chosen for judgment. Therefore, when the estimated variance remains unchanged, changing the number of satellites will not affect the detection performance.

VI. CONCLUSION
In this paper, we investigate the detection of illegal spectrum access behavior through spectrum perception in the satellite-terrestrial spectrum sharing system. First we establish a model of multiple hypothesis test, and use the GMNP criteria to solve the problem. Then, we deduce the judgment threshold formula. Finally, through simulation, the performance of the detection method is evaluated and analyzed. The results show that the method has a high detection rate for illegal spectrum access behavior.

APPENDIX PROOF OF THEOREM 1
Firstly, we consider the critical point (α c , β c ) which makes the overlapping just happens, i.e., η 0 = η 1 = η * 1 . First, based on (36), η * 1 is formulated Substitute η * 1 for η 1 in (43), and we have So, we can obtain η * 2 that is the solution to the equation above, η 2 = η 1 . Then, Pr (H 1 |H 1 ) is formulated as follows Further, it is found that when β increases, λ 1 increases, η 1 decreases, and the area of the overlapping region increases. On the other side, when α increases, η 0 decreases and the area of the overlapping region increases. This means that once β is over the value in Eq.73 the overlapping happen. In conclusion, the overlapping condition is obtained just as formulated in Eq.49.
This completes the proof.