Anti-jamming technology of electronic communication based on improved intelligent algorithm

3.1 Description of anti-jamming technology in physical layer

The intelligent anti-jamming communication technology is embodied in both the communication device and the entire communication network. According to the idea of intelligent anti-jamming, its communication equipment should have the following functions: First, the rapid detection and identification of interference is performed in real time. Secondly, aiming at interference, the optimal anti-interference transmission waveform is reconstructed by intelligent real-time decision. Third, both the receiver and receiver have reliable signaling mechanism, and based on the reconstructed waveform, the information can be transmitted quickly and robustly [18–20].

In intelligent anti-jamming communication, interference cognition is the premise, real-time decision-making is the core, and waveform reconfiguration and fast and reliable transmission are the means. Figure 1 provides a schematic diagram of wireless information transmission, revealing the basic principles of intelligent anti-jamming communication technology. Among them, the main channel is between Alice and Bob, and Eve exists in the network as the eavesdropper, whose purpose is to eavesdrop on the information from Alice to Bob. So the channel between Alice and Eve is eavesdropping. OPEN IN VIEWER

Secrecy Capacity is the most important performance indicator for measuring information security communications in physical layers, as defined in (1): C s = ( C 1 d - C 1 e ) +(1)

Where ( x ) + = max ( 0 , x ) , C 1 d is the capacity of the main channel and C 1 e is the capacity of the eavesdropping channel. In the traditional network layer encryption security communication, the security and reliability of transmission are realized by encryption and decryption process and channel codec respectively. In the physical layer security communication, the security is actually integrated in the channel codec process, so the reliability and security of the total signal transmission are realized at the same time. Specifically, there is a security coding scheme, for code length n and transmission rate R, R < C _s and n→ ∞, bit error rate P _e  → 0, that is, to achieve the effectiveness of information transmission. H (W|Z ⁿ )/n → H (W) , I (X, Z) =0, that is, the security of information transmission is realized. Wyner’s security theory provides a theoretical basis for the optimal design of physical layer security transmission, which is to measure the maximum capacity of system security transmission through confidential capacity. It can also be seen as the maximum secure communication gain that the difference in wireless channels can bring to the system. In the traditional scenario, the feasibility of the physical layer security transmission performance is limited by the strong and weak relationship between the primary communication channel and the eavesdropping channel. As can be seen from the definition of the security capacity, when the quality of the eavesdropping channel is higher than the quality of the main communication channel, the confidentiality of the data transmission will degenerate to zero.

The imperfect CSI model is described, which consists of two factors: channel estimation error and channel expiration, both of which have a negative impact on the security performance of the system. In the actual communication system, if the base station is to obtain the CSI of the downlink, the channel interoperability needs to be used for the pilot estimation. In the case of Hugh, K users first send orthogonal pilot sequences to the base station, and then the base station uses the received pilot sequences to estimate the response CSI. Θ = [θ₁, θ₂, … θ _K ] ∈ C^τ×K is defined as a set of pilot sequences transmitted by the user, where θ _k  ∈ C^τ×1 represents the pilot sequence transmitted by user k and τ represents the length of each pilot sequence. The pilot sequence satisfies two constraints: the power of the pilot signal is normalized, and the different pilot sequences are orthogonal to each other. Therefore, there is θ k H θ k = 1 . Based on the above definition, the base station receives the pilot sequence sent by the user as: Y ( t ) = Θ H ( t ) + W ( t )(2)

W (t) ∈ C^{τ×N _t} represents additive white Gaussian noise at the CSI estimation stage. Each of these scenes obeys a Gaussian distribution with a mean of 0 and a variance of σ w 2 . The base station can obtain by multiplying the two ends of the received signal by the pilot matrix: Z ( t ) = Θ H Y ( t ) = H ( t ) + Θ H W ( t )(3)

For the kth user, there is: Z k ( t ) = h k ( t ) + θ k H W ( t )(4)

Using the MMSE channel estimation method, the channel information between the base station and the kth user can be estimated as: h ˆ k ( t ) = ( 1 + σ w 2 ) - 1 z k ( t )(5)

Therefore, the real channel information can be written as: h k ( t ) = h ˆ k ( t ) + n k , MMSE ( t )(6)

Where n_k,MMSE (t) is the channel estimation error, which is independent of h ˆ k ( t ) . h ˆ k ( t ) ∼ CN ( 0 , σ h ˆ 2 ) and n k , MMSE ( t ) ∼ CN ( 0 , σ n MMSE 2 ) are noted, and further σ h ˆ 2 = ( 1 + σ w 2 ) - 1 and σ n MMSE 2 = ( 1 + σ w 2 ) - 1 σ w 2 . As described above, the base station estimates the CSI of the user channel using the pilot sequence, and then performs channel pre-coding using the estimated CSI and transmits through multiple antennas. Considering the hysteresis of the CSI when the base station transmits a signal and the CSI used for channel estimation, and the processing delay of the channel estimation by the large-scale antenna system of the base station, the CSI used for pre-coding will be expired. Error! Reference source not found will also have a negative impact on the system’s secure capacity. The delay between the expired CSI and the real CSI is defined as T _d . Its physical meaning is that if the base station receives the pilot sequence sent from the user at time t, the base station will complete the pilot channel estimation phase at time t + T _d and begin to send the total to the downlink user. Therefore, if H(t) is used to represent an expired channel, then the real channel is represented by H (t + T _d ). The correlation between the two is characterized by Jake’s model:

Where E (t + T _d ) represents random vectors independently distributed with H (t), and each component of which obeys the complex Gauss distribution of (0, σ² = 1). ρ _{T d} represents the correlation coefficient between two channels at different times, as follows: ρ T d = J 0 ( 2 π f D T d )(7)

Where f D = v c f c represents Doppler frequency shift (v represents the movement speed of the terminal and f _c represents the carrier frequency). The smaller ρ _{T d} is, the lower the correlation between the real channel and the expired channel. Conversely, if ρ _{T d}  = 1, the effect representing channel expiration is completely eliminated. Therefore, based on (6) and (7), for the kth user, the relationship between the real channel and the expired channel can be written as: h k ( t + T d ) = ρ T d h k ( t ) + 1 - ρ Td 2 e ( t + T d ) = ρ T d h ˆ k ( t ) + e ˆ k ( t )(8)

Where: e ˆ k ( t ) = ρ T d n k , MMSE ( t ) + 1 - ρ Td 2 e ( t + T d )(9)

According to the above analysis, it can be known that e ˆ k ( t ) obeys e ˆ k ∼ CN ( 0 , σ e ¯ 2 I N τ ) distribution, among which: σ e ⌣ 2 = ρ T d 2 σ n MMSE 2 ( t ) + ( 1 - ρ T d 2 ) = 1 - ρ Td 2 ( 1 + σ w 2 ) - 1(10)

In this section, a detailed theoretical analysis of the reachable traversal capacity of the MassiveM1M0 system under imperfect CSI will be performed. First, Lemma 1 is given, and in the proof of the propositions and theorems that follow, the conclusion of Lemma 1 will be widely used. Lemma 1: considering two variables x, y ∈ C^1×n, which satisfy x ∈ CN ( 0 , σ x 2 I n ) and y ∈ CN ( 0 , σ y 2 I n ) , the following four equations can be obtained: E [ xx H ] = n σ x 2(11) E [ xy H ] = 0(12) var [ xx H ] = n σ x 4(13) var [ xy H ] = n σ x 2 σ y 2(14)

In existing papers, zero-forcing (ZF) and regularized channel inversion (RCI) pre-coding can achieve better system performance than matched filter pre-coded (MF). However, in a large-scale antenna scenario, the computational complexity of the matrix inversion operation increases dramatically, and it is too time consuming for the base station to apply ZF or RCI pre-coding. Therefore, MF precoding scheme is considered in the research scenario. In fact, because the study of precoding schemes is not the focus of discussion, the comparison of more options is left for future work. As mentioned in the previous section, the base station transmits a signal at time t + T _d , while the pre-coding phase is performed at time t. Therefore, the MF pre-coding scheme used by the base station to transmit signals at time t + T _d can be expressed as: F ( t + T d ) = H ˆ ( t )(15)

Next, using lemma 1, the power limitation of the pre-coding matrix can be expressed as: ξ ˆ = E [ tr ( H ˆ H ˆ H ) ] = σ h ˆ 2 N t K(16)

Therefore, the signal that user k receives at time t + T _d is: y ˆ k ( t + T d ) = P B ξ ˆ H ( t + T d ) H ˆ H ( t ) s ( t + T d ) + n k ( t )(17)

The capacity of user k under imperfect CSI can be expressed as: C k imp = E [ log 2 ( 1 + γ | h k ( t + T d ) h ˆ k H ( t ) | 2 ξ ˆ + γ ∑ j ≠ k | h k ( t + T d ) h ˆ j H ( t ) | 2 ) ](18)

It is difficult to directly know the closed expression of (15), therefore, the lower bound of C k imp is derived in Theorem 1. Theorem 1: under imperfect CSI, when the number of transmitting antennas of base station approaches infinity, the tight lower bound of channel capacity of user k can be expressed as: C k imp ⩾ log 2 { 1 + γ ρ T d 2 σ h ˆ 2 N t K + γ ( K - 1 ) }(19)

The degree of compaction of the lower bound will be verified in the simulation. In this section, a tight upper bound on the traversal capacity of the eavesdropper channel is also derived. At moment t + T _d , the signal received by the eavesdropper is: y ˆ e ( t + T d ) = P B ξ ˆ G ( t + T d ) H ˆ H ( t ) s ( t + T d ) + n e ( t + T d )(20)

The traversal capacity of the eavesdropping channel can be expressed as (22): C e imp = E [ log 2 ( det ( I N e + γ ξ ˆ G H ˆ H HG ˆ H ) ) ](21)

In the following part, theorem 2 is given to represent the tight upper bound of eavesdropper traversal capacity under imperfect CSI. Theorem 2: under definitions m = min(K, N _e ) and n = max(K, N _e ), the tight upper bound of the eavesdropper traversal capacity under imperfect CSI can be expressed as: C e imp ⩽ m log 2 ( γ K ) + log 2 ( m ! ) + log 2 { L m n - m ( - K γ ) }(22)

Where L n α ( x ) denotes the n-order generalized Legendre polynomial, the specific form can be expressed as: L n α ( x ) = 1 n ! e x x - α d n dx n ( e - x x n + α ) = ∑ m = 0 n ( - 1 ) m C n + α n - m x m m !(23)

C n m = n ! ( n - m ) ! m ! represents the number of combinations. The degree of compaction of the upper bound will be verified in the simulation. First, in theorem 3, the lower bound of system traversal safety capacity is given. Theorem 3: Under imperfect CSI, the system below the lower bound of the lower bound of the safe tolerance can be expressed as: C s imp ⩾ { K log 2 { 1 + γ ρ T d 2 σ h ˆ 2 N t K + γ ( K - 1 ) } - ( m log 2 ( γ K ) + log 2 ( m ! ) + log 2 { L m n - m ( - K γ ) } ) }(24)

Specifically, as T _d increases, the correlation coefficient ρ _{T d} becomes smaller and smaller. Similarly, the reduction of σ h ˆ 2 means that the MMSE channel estimation error becomes larger. Therefore, both of these factors will lead to the reduction of system traversal security capacity. Conversely, when ρ _{T d}  = 1 and σ h ˆ 2 = σ h 2 = 1 , or CSI, are perfect, the system traverses the security capacity to reach the maximum value. Imperfect CSI only affects the capacity of legitimate links, but does not affect the capacity of eavesdropping channels. This is because, for eavesdroppers, its demodulation phase does not depend on the CSI between the base station and the user. In the research scenario, it is assumed that the eavesdropper has a strong eavesdropping capability and is able to learn the pre-coding matrix of the legitimate user channel and its own CSI. Therefore, it can demodulate the received user signal without being affected by imperfect CSI. In order to further understand how different factors affect the system traversal security capacity, the inference 1 is given below. Corollary 1: In addition to imperfect CSI, there are four other factors that affect the traversal security capacity of the system, including the number of transmitting antennas N _t , the number of receiving antennas N _e , the number of transmitting antennas SNR _γ and the number of users K. As N _t grows, the system traversal security capacity will increase, and as N _e grows, the system traversal security capacity will decrease. As SNR increases, two scenarios are considered: When K = 1, the system traverses the safe capacity at the beginning will grow rapidly, and then slow growth, and eventually converge to a constant value. When K>, 1, the security capacity of the system traversal will increase at the beginning, and decrease when it reaches a certain value.

Here’s a simple proof: As can be seen from (19) and (22), with the growth of N _t , the accessible user capacity always increases, while the eavesdropper’s capacity remains unchanged, because it has nothing to do with N _t . With the growth of N _e , the decoding ability of eavesdroppers is increasing, which will increase the capacity of eavesdropping channel and thus reduce the security capacity of the system. Two cases are discussed respectively: when K = 1, the system traversal security capacity can be simplified as follows: C s imp = { E [ log 2 ( 1 + γ | h k ( t + T d ) h ˆ k H ( t ) | 2 ] - E [ log 2 ( 1 + γ Ψ H Ψ ) ] } +(25)

According to Appendix B, there is | h k ( t + T d ) h ˆ k H ( t ) | 2 ≈ ρ Td 2 σ h ˆ 4 N t , therefore, the above formula can be further expressed as: C s imp ≈ { log 2 ( 1 + γ ρ T d 2 σ h ˆ 4 N t ) - E [ log 2 ( 1 + γ Ψ H Ψ ) ] } +(26)

At low SNR, the formula is expressed as follows: E [ log 2 ( 1 + γ Ψ H Ψ ) ] ⩽ log 2 ( 1 + γ E [ Ψ H Ψ ] ) = log 2 ( 1 + γ N e ) ≈ 0(27)

There is: C s imp > ≈ log 2 ( 1 + γ ρ T d 2 σ h ˆ 4 N t )(28) OPEN IN VIEWER

It indicates that the system traverses the secure capacity: it grows as the SNR increases. At high SNR, (27) can be expressed as: C s imp ≈ { E [ log 2 ( 1 + γ ρ T d 2 σ h ˆ 4 N t 1 + γ Ψ H Ψ ) ] } + ≈ { E [ log 2 ( ρ T d 2 σ h ˆ 4 N t Ψ H Ψ ) ] } +(29)

It indicates that the system traversal safety capacity tends to be stable when the SNR is large. When K >  1, it is easy to observe that the user capacity tends to grow at a low SNR. Since the user’s received power and the down-interference power increase in proportion to the increase in SNR, when the SNR is high, the user capacity approaches a constant value. However, for the eavesdropper capacity, it always increases as the SNR increases. More specifically, when the SNR is higher than the SNR, the eavesdropper capacity increases by m = min(K, N _e ) bits when the SNR increases by 3 dB.

4.1 Experimental thoughts

In this part, the physical layer security performance of the multi-user Massive MIMO system under perfect CSI is evaluated, and the proposed channel prediction algorithm is verified. First, an independent cellular communication system is considered, which includes a base station, K users and a malicious eavesdropper. For convenience, only small-scale fading channel models are considered, and large-scale fading factors can be considered as 1. The channel correlation within the small 1H frame is depicted by Jake’s model, where each frame is a length of T = 0.01s, the carrier center frequency is f _c  = 2GHZ, and the average moving speed of the user is v. {H (t) , H (t - T) , … , H (t - (L - 1) T) } is used to represent the real CSI of user channel in different frames. In order to ensure the correlation between CSI in different frames, when CSI is generated, H (t - (L - 1) T) is generated first, and then H (t - (L - 2) T) is generated according to the autoregressive model described in literature [95, 96]. By analogy, all channel information is generated according to the following formulas: H ( t - ( L - i ) T ) = ρ T H ( t - [ L - ( i - 1 ) ] T ) + 1 - ρ T 2 E ( t - ( L - i ) T )(30)

Since the goal is to predict H (t + T _d ), in order not to lose generality, it is assumed that T _d and T are equal. Therefore, the channel information at time t + T _d can also be generated using the same model. In the following sections, the effect of several factors on the system traversal security capacity is evaluated by simulation.

Figures 3 and 4 show the relationship between system traversal capacity (including user traversal capacity, eavesdropper traversal capacity and system traversal security capacity). The real-time anti-interference communication intelligent decision-making module is based on the result of the local interference detection, the interference detection result of other nodes from the signaling transmission layer, and the link performance and other indicators. The best anti-interference transmission mode and corresponding parameter configuration (such as frequency, power and modulation and coding mode) should be adopted in real-time intelligent decision making. By signaling the transmission channel to the corresponding receiving node, each node reconstructs to produce anti-jamming waveform, and uses reliable waveform transmission mechanism to complete the transmission of information between the sending and receiving sides. OPEN IN VIEWER

OPEN IN VIEWER

Fig. 4

Relationship between user number and traversal security capacity.

The system parameters are set as follows: SNR = 10 dB , N t = 20 , N e = 10 , L = 10 , σ w 2 = 1 , v = 1 m / s(31)

As can be seen from Fig. 4, under different scenarios of CSI (perfect CSI, non-perfect CSI, and predictive CSI), the system traverses the security capacity will increase with N _t . This is because according to the conclusion in inference 1, with the increase of Nt, the user capacity increases, while the eavesdropper capacity remains unchanged. From the point of view of electromagnetics, with the increase of the number of base station antennas, base station can concentrate the transmitted energy more on the user’s direction, while eavesdropper’s direction will not receive excess energy. From Figs. 2 and 3, the following conclusion can be drawn: the system safety capacity under perfect CSI is much larger than the imperfect CSI. However, with our predictive solution, the negative impact of non-perfect CSI can be overcome to some extent to enhance the system’s safe capacity, but this negative impact cannot be completely eliminated. For eavesdroppers, whether the user channel CSI is perfect or not is irrelevant, and this result is also proved in Theorem 2, which is more accurate.

Figures 4 and 5 describe the relationship between channel security capacity and the number of users. The system parameters are set as follows: SNR = 10 dB , N t = 200 , N e = 10 , L = 10 , σ w 2 = 1 , v = 1 m / s(32) OPEN IN VIEWER

As the number of users K increases, the system security capacity increases. Intuitively, it can be explained as follows: For multi-user massive MIMO systems, because the channels between different users are orthogonal to each other, increasing the number of users 0 can better tap the advantages of freedom brought by large-scale antennas, serving more users to increase capacity. However, when the number of users increases to a certain extent, as the interference between users becomes more and more serious, the channel capacity of users increases more and more slowly. On the other hand, the number of given eavesdropping antennas is becoming increasingly difficult for eavesdroppers to investigate information about more users.

In Figs. 6 and 7, the relationship between eavesdropper capacity and the number of eavesdropper antennas N _e is studied. The system parameters are set as follows: SNR = 10 dB , N t = 200 , N e = 20 , L = 10 , σ w 2 = 1 , v = 1 m / s(33) OPEN IN VIEWER

OPEN IN VIEWER

Fig. 7

Relationship between number of antennas and user capacity and eavesdropping capacity.

In fact, for a given number of base station antennas N _t and the number of users K, the increase in the number of eavesdropper antennas does not increase the legitimate user capacity, but only allows the eavesdropper to enhance their eavesdropping ability. As can be seen from the figure, whether for perfect CSI, imperfect CSI or predicted CSI, the system security capacity will become 0 when the number of eavesdropper antennas grows to be large enough. In the face of multi-antenna eavesdropping by eavesdroppers, it is necessary to take some other means (such as artificial noise, etc.) in order to reduce the negative impact on system security.