Satellite communication anti-jamming based on ABC blind source separation

The existing satellite communication anti-jamming technology mostly realizes anti-jamming communication from the perspective of interference suppression, and has defects such as low spectrum efﬁciency and limited anti-jamming capability. This paper proposes to use the independence of communication signal and interference signal to separate communication signal and interference signal in the waveform domain, and then realize communication anti-interference. This method can realize anti-jamming communication under strong jamming without reducing the spectrum efﬁciency. The blind source separation problem is essentially a multi-parameter joint optimization problem, and there is no analytical solution. This paper uses an artiﬁcial bee colony optimization algorithm to solve the blind separation problem and obtains a sub-optimal solution.

source signals is more than the number of receiving sensors), well-defined mixing (the number of source signals is equal to the number of receiving sensors) and over-determined mixing (the number of source signals is lower than the number of receiving sensors). The mixing model can also be divided into linear instantaneous mixing and convolutional mixing according to the transmission response characteristics of the mixed channel. Regardless of the mixed model, the current blind source separation theory is mostly based on independent component analysis (ICA) [1]. [1] gives three assumptions for using ICA method to realize BSS. One is that the source signals are independent of each other. Second, there is at most one Gaussian signal source in the source signal. Third, the channel transmission matrix is full-rank. The author of this article also summarizes the method of studying BSS based on ICA, that is, constructing a cost function based on certain criteria, and obtaining the separation parameters For example, signal sparse component analysis [2][3], dictionary learning [4][5], non-negative matrix factorization [6] [7], bounded component analysis [8] [9] and tensor decomposition [10][ method to search and obtain the global sub-optimal solution of the problem to be solved, so that the separation result has high robustness. This paper is organized as follows. First, the BSS problem is formulated in Section II. We then introduce the new anti-jamming algorithm based on ABC BSS in Section III. The performance of the algorithm is demonstrated with simulations in Section IV. Conclusions are drawn in Section V.

III. PROBLEM FORMULATION
The research of BSS theory originated from the well-known "cocktail party effect". Its original intention is to simulate the ability of both parties to accurately capture what they are interested in in a noisy venue, that is, to separate the desired signal from the interference signal and extract the desired signal. BSS mainly solves the problem of separation of desired signal and interference signal in this phenomenon. In solving this problem, the signal mixing model is unknown, the mixed source signal is unknown or there is only a small amount of prior information.
The satellite communication channel is generally modeled as AWGN channel. When there is jammer, the mathematical model of the communication signal s 1 (t) and the jamming signal s n (t) (n = 2, 3, · · · N ) is wherex i (t) is the signal received by the ith receiving antenna, a in is the channel coefficient from the mth signal source to the ith receiving antenna, ω i (t) is additive white Gaussian noise. Write the above formula in the form of a matrix where x (t) is the received signal vector, A is a channel mixing matrix of size M × N with full rank, s (t) is the source signal vector, ω (t) is the noise vector.
The goal of BSS is to find the separation matrix B, so that after the received signal vector x (t) is processed by the separation matrix, an effective estimate of the source signal vector s (t) is obtained,ŝ Ideally, the global matrix BA is the identity matrix. But it is difficult to realize this situation.
It is generally believed that as long as the global matrix BA is an approximate generalized permutation matrix, blind source separation can be achieved. The system model of BSS is shown in Fig.1. Regarding the research of BSS, the current mainstream research direction is based on the method of ICA. Its core idea is to assume that the source signals are independent of each other, and it is required that at most one signal in the source signal obey the Gaussian distribution, and then this independence assumption is combined with different criteria for measuring signal statistical characteristics to design the objective function, and the objective function is solved to obtain the desired separation matrix and source signal. The BSS algorithm design method based on ICA can be expressed by the equation shown in Fig.2.

Objective function
Optimization algorithm ICA-based BSS algorithm where the objective function affects the robustness and separation performance of the algorith-February 26, 2021 DRAFT m; the optimization algorithm determines the convergence speed and numerical stability of the ICA algorithm. Therefore, it can be concluded that the overall performance of the ICA algorithm depends on these two links. The criteria used by the ICA algorithm to design the objective function mainly include non-Gaussian maximization, mutual information minimization, maximum likelihood criteria, and so on. Commonly used optimization algorithms include stochastic gradient method, natural gradient method, and fixed point algorithm. The above-mentioned independence criteria have proved to be equivalent. The focus of current research is to properly model the signal and mixing system, and use efficient optimization algorithms to improve the accuracy and the efficiency of BSS algorithm.
In this paper, the objective function is set by maximizing non-Gaussianness as the independence criterion. The objective function is designed based on maximizing the absolute value of the kurtosis of the separated signal, and the objective function is obtained as represents the kurtosis of the separated signalŝ i (t). It is generally assumed that the variance of the source signal is 1 and the mean value is 0,

A. Data preprocessing
Before processing the separation algorithm, it is usually necessary to preprocess the received signal, including de-averaging and whitening the received signal. De-averaging is to remove the DC component in the received signal so that the received signal vector of each antenna is zeroaveraged. Whitening is to remove the correlation between the signals received by each antenna, so that the signals received by each antenna tend to be independent in a statistical sense, and the signal vector variance after whitening is 1. Through these two steps, the calculation and analysis process of subsequent processing can be simplified. The calculation process of de-averaging and whitening is as followsx the whitening signal, and the autocovariance matrix is the identity matrix.
The whitening matrix Q can be obtained by eigenvalue decomposition of the autocovariance matrix R.
where D is a diagonal matrix, and the diagonal elements are the eigenvalues of R, E is the eigenvector matrix of R. The blind source separation processing of the observation signal x (t) is equivalent to the processing of the whitening signal z (t). Let the new separation matrix be W, then the separated signalŝ Considering that the variance of the source signal is 1, that is E [ŝ 2 i (t)] = 1,we can get E is the identity matrix, so the ith row vector of the separation matrix ∥W i ∥ 2 = 1, where ∥·∥ is the Frobenius norm.
Therefore, the objective function can be described as

B. BSS based on artificial bee colony optimization
The objective function in Eq. (10) is a constrained multi-parameter optimization problem, which is usually solved by gradient descent method, but this method has the disadvantages of slow convergence speed and low accuracy. This paper uses artificial bee colony algorithm to optimize the objective function. The ABC algorithm is a bionic intelligent optimization algorithm with good global search capabilities and fast convergence speed, which is very suitable for solving multi-parameter joint optimization problems. The basic idea is to simulate the group behavior of bees in the process of collecting honey. The bee colony is mainly composed of worker bees, observation bees and reconnaissance bees. The number of worker bees is the same as the number of food sources, and the quality of the food sources corresponds to the objective function of the problem to be optimized. Assuming that the initial population of bee colony, i.e. the number of candidate solutions, is SN , each solution is a vector w i , i = 1, 2, · · · SN composed of parameters to be optimized. The worker bees first search for nearby food sources, and then update the location information of the food sources according to Eq. (11) where φ is a random number in the interval and d ̸ = i. After the worker bees have calculated all the new solutions, the reconnaissance bees use roulette to decide whether to keep the solution, so as to determine the new location information of the food source Where P i is the probability of choosing the ith solution, f it i is the fitness function of the ith solution, f i is the objective function. If a certain solution has not been optimized after continuous limit iterations, and the objective function value of the solution is not the current optimal value, then the solution is abandoned. The worker bee corresponding to this solution is transformed into a reconnaissance bee, and a new solution is generated through Eq. (13) The satellite communication anti-jamming algorithm flow based on ABC blind source separation can be expressed as follows. (1) Use Eq. (5) and Eq. (6) to preprocess the received signal.
(2) Initialization parameters: initial separation vector w i , i = 1, 2, · · · SN , limit, the maximum number of iterations iter max. In the optimization process of the ABC algorithm, the variation range of the parameters to be optimized needs to be restricted, and the variation range of the elements of the parameter w to be optimized in the above algorithm is infinite. Under constraint ∥w∥ 2 = 1, the Hough transform of w can further constrain the parameters to be optimized. Hough transform is as follows.  where α 1 , α 2 , · · · α M −1 ∈ [0 2π]. After this transformation, only M-1 parameters with limited value range need to be optimized.

V. RESULTS AND DISCUSSION
In order to verify the effectiveness of the algorithm, a BPSK modulation signal is used as the communication signal and a tone signal as the interference signal. BPSK modulation signal adopts root raised cosine roll off shaping filter with roll off factor of 0.35 and symbol rate of 128kbps. The center frequency of single tone jamming is 16kHz. Fig. 3, Fig. 4 and Fig. 5 show the waveforms of the source signal, the received signal and the separated signal respectively when the signal to jamming ratio (SJR) is -20dB and the signal to noise ratio (SNR) is 20dB. It can be seen from the waveforms in the figures that the algorithm can separate the communication signal well from the jammed signal when the SJR is -20dB, so as to achieve the goal of communication anti-jamming.