Research on Single Antenna Co-frequency Mixed Signal Separation Based on Improved EFICA Algorithm

Signal separation under the condition of single antenna reception is a process of estimating the source signal component by using one-dimensional observation signal vector. Under the condition of single antenna receiving the same-frequency mixed signal, in view of the problem that the stability of the EFICA algorithm will be affected by the selection of a random initial iterative matrix, it is proposed to use the steepest descent method to select a suitable iterative initial matrix to improve the EFICA algorithm. An experimental comparison between the EFICA algorithm and the EFICA improved algorithm is presented. The simulation results show that the improved EFICA algorithm can achieve a better separation effect for the separation of mixed signals at the same frequency.


Mixed signal model
Consider a single antenna receiving two mixed signals of the same frequency. The general form of signal reception is: 1 2 ( ) ( ) ( ) ( ) y t x t x t v t = + + (1) In the above formula, v(t) is zero-mean additive white Gaussian noise with a power spectral density of 0 / 2 N , ( ) 1 x t and ( ) 2 x t are the two received signals with the same frequency, which can be further expressed as: The impulse response (including the transmitting filter, the channel filter and the receiving filter). If the transmitting filter matches the receiving filter, the equivalent filter will have a raised cosine impulse response, which will be a response with a completely determined rolling coefficient. Next, we assume that the rolling coefficient is known; the two signals have the same symbol period, T is the symbol period; 0≤ 1 ( ) t τ , 2 ( ) t τ ＜T is the time delay between the local reference clock and the received signal.
Assuming that the duration of the raised cosine waveform is ( ) (1) In the above formula ( ) 3. Improved EFICA algorithm to separate the same frequency mixed signal

Improved EFICA algorithm
The EFICA algorithm cannot eliminate the new influence of the selected random initial matrix on the convergence and stability of Newton's method. In this paper, the improved EFICA algorithm can select a suitable initial matrix for iteration, is insensitive to the initial value and the iteration speed is faster. There are six steps to improve the EFICA algorithm: Step 1: to average and whiten the mixed signal of the same frequency. It is to avoid the influence of signal strength, amplitude and other factors during the separation process. This treatment will not affect the separation effect, which is: (6) In the above formula, is the sample covariance matrix, is the average value of the mixing matrix, and Z is the observation signal including the preprocessing.
Step 2: Use the steepest descent method to select the appropriate initial iteration matrix. Randomly generate an initial iterative matrix, and orthogonalize it to get. Calculate the negative gradient value at W. If, is a very small positive number, it is used as the convergence threshold. If it converges, then W at this time is the appropriate initial iterative matrix. If it does not converge, then return to the step to calculate the negative gradient of W until convergence.
Step 3: Use the initial iterative matrix selected above to process the observation signal based on the maximum judgment criterion until it converges. The initial iterative matrix is obtained in the second step, and the iteration process is as follows: In the above formula, g and g' represent the first and second derivatives of G, 1N represents a column of vectors, and k is the number of iterations. And is a non-linear function often used.
Step 4: select an adaptive nonlinear function, first estimate the fourth moment of the signal sample, and determine whether the selected nonlinear function is close enough to the evaluation function. If it is not close to the evaluation function, a more suitable adaptive nonlinear function should be selected. If the signal distribution is a generalized Gaussian distribution, the evaluation function of the EFICA algorithm is: In the above formula, is the estimated value of the fourth-order matrix of the estimated source signal, (where 4 k m is the fourth-order moment of a single source signal) Step 5: Use to iterate, the iteration process is: Step 6: Carry out signal separation, the separated signal is y WZ The improved EFICA algorithm is compared with the EFICA algorithm. The steps of the improved algorithm after selecting the initial iterative matrix remain unchanged from the original algorithm. Although using the steepest descent method is equivalent to adding one more step to the original algorithm, it does not waste much time due to its fast initial iteration speed. And due to the suitability of the initial iterative matrix, the subsequent algorithm will proceed smoothly, and there will not be too many convergence times or even failure to converge. From the overall point of view of the improved algorithm, the speed will not necessarily be reduced, or even faster, but the stability will be improved.

Improved EFICA algorithm separation process
The first step uses the FastICA algorithm based on the largest negative entropy. This paper takes the largest negative entropy as the search direction, extracts the independent sources in order, and uses a fixed-point iterative optimization algorithm to make the convergence faster and more stable. The expression of the differential entropy of the random variable Y is: Is a Gaussian random variable with the same variance as the random variable, the negative entropy of the random variable Y is: When Y has a Gaussian distribution, The stronger the non-Gaussianness of Y, H(y) the smaller and N g (y) the larger the value, Therefore, it can be used as an evaluation estimate of the non-Gaussianness of random variables. Equation (12) needs to calculate the probability density distribution function when calculating the differential entropy, which is difficult to achieve in practical applications, so we can use the following equation for approximate calculation: Observe the matrix X, and use the FastICA algorithm to find the unmixing matrix W to have the largest Gaussian non-Gaussian property. The approximate Newton iteration formula of the unmixing matrix W is: The second step is the selection of non-linear function g m :  In the above calculation, Z represents the mixed signal after whitening,

Simulation results and analysis
The two signals with the same frequency received by the antenna at the receiving end have the same symbol period T, and the symbol period T=16; 0≤ 1 ( ) t τ , 2 ( ) t τ ＜T is the time delay between the local reference clock and the received signal. Different amplitude fading ( ) h t , the value of the amplitude A1=A2=1. With the same local carrier frequency w0, the carrier frequency offsets of the two modulated signals are w 1 and w 2 . Different initial phases 1 1 ϕ = , 2 2 ϕ = ,are selected as, respectively.
We conducted simulation experiments on the improved EFICA algorithm proposed in this paper and the EFICA algorithm at the same time. In the experiment, two signals with the same frequency are selected, a noise signal is randomly generated, and the three signals are mixed to separate the mixed signals. In order to improve the performance of the source signal, it is necessary to perform whitening before data separation. The input observation vectors after the whitening process are no longer relevant. Although correlation is a weaker statistical attribute than independence, whitening can simplify the source signal separation algorithm and make the iterative process simpler.   Through the separation results, we can clearly see that the improved EFICA algorithm is closer to the source signal, whether in amplitude or phase, and the improved EFICA algorithm has a better separation effect. In order to better compare and improve the separation performance of the algorithms, the average correlation coefficient and crosstalk error of the source signal and the separated signals of the two algorithms are selected as the measure of the separation effect. Use 10 consecutive experiments for measurement. It can be seen from Table 1 that although the steps of the improved algorithm are increased, it has little effect on the speed of the algorithm. From the average correlation coefficient and crosstalk error, it can be seen that the average correlation coefficient of the improved EFICA algorithm is higher than that of the EFICA algorithm, and the crosstalk error is lower than that of the original EFICA algorithm.  Figure 8 Crosstalk error between EFICA algorithm and its improved EFICA algorithm It can be seen from Figure 7 and Figure 8 that the average correlation coefficient and crosstalk error of the improved EFICA algorithm are more stable than those of the EFICA algorithm. Actual calculations show that the average crosstalk error of the improved EFICA algorithm is 9.14808 and the variance is 0.0086. The average crosstalk error of the EFICA algorithm is 9.94516, and the variance is 0.7408. The improved EFICA algorithm is better than the EFIC algorithm in terms of separation effect and stability.

Conclusion
This paper focuses on the separation of signals at the same frequency, and proposes signal separation based on the improved EFICA algorithm. By comparing the separation results with the EFCA algorithm, a communication mixed signal experiment is carried out on the Matlab platform. Experiments show that the algorithm is compared with the EFICA algorithm. Compared with the case where the running time is not much different, the improved EFICA algorithm can increase the average correlation coefficient of the algorithm, reduce the crosstalk error of the algorithm, improve the stability of the EFICA algorithm. Use the improved EFICA algorithm to separate the same frequency mixed signal, which has a better separation effect. Therefore, the improved EFICA algorithm will have a broader application prospect.