Application of high-order cumulants in source number estimation in underdetermined blind source separation

0 Introduction

Blind source signal separation (BSS) refers to the process of separating and recovering relatively independent source signals from observed multi-source mixed signals. Because little is known about the source signals and the mixing process, it is impossible to directly observe the useful information in the mixed signal. Only by separating them from the mixed signal through blind signal processing can the required signal be extracted. Because this technology has the ability to recover useful signals under relatively loose conditions, it has received more and more attention in the field of signal processing and has been widely used in communication, speech processing, seismic exploration, biomedicine, image processing, radar, and economic data analysis. The

usual blind source separation algorithms do not have the ability to estimate the number of unknown signal sources, and can only be calculated under the premise that the number of signal sources has been determined in advance. Therefore, the estimation of the number of sources in the processing process is of great significance to the development of blind separation technology and is also a problem that must be solved at present. At present, there are few special studies on the estimation of the number of blind separation sources in communication reconnaissance. This paper studies and optimizes a source number estimation algorithm based on the cumulant algorithm, which can estimate the number of signal sources under underdetermined conditions without prior knowledge.

1 Signal model and problem description

In the theory of blind source signal separation, the mixing process is divided into two categories: linear instantaneous mixing model and convolution mixing model. The requirements for the statistical properties of the source signal are also related to the blind separation algorithm used. This paper focuses on the estimation of the number of signal sources in the case of linear mixed blind signal separation.

There are n statistically independent n-dimensional signal vectors s1(t), s2(t), ..., sn(t) from the signal source, and m observed signals x1(t), x2(t), ..., xm(t) are obtained after linear instantaneous mixing through the m×n mixing matrix A.

The signal model is:

Where: aij is the mixing matrix coefficient; ni is the random observation noise; the vector and matrix expressions are:

Where n is an m×1 noise vector. This model is similar to the observed signal model of standard array signal processing, but in blind signal separation, the signal mixing coefficient does not have prior information such as the wave direction angle in the array signal model.

Therefore, the blind separation problem of the signal source can be described as calculating an n×m separation matrix W so that its output y(t)=Wx(t) is an estimate of s(t). Since the mixing matrix A and s(t) in the above formula are unknown, it is impossible to accurately identify the arrangement order and energy of each component of the source signal. This is the uncertainty problem in the blind signal separation problem. The first is the uncertainty of the arrangement order, that is, it is impossible to know which component of s(t) the extracted signal should be; the second is the uncertainty of the signal amplitude, that is, it is impossible to restore the true amplitude of the signal waveform. Since the information is mainly contained in the waveform of the signal, these two uncertainties do not affect the application of blind separation technology. However, in most practical problems of blind source separation, not only the waveform of the signal source is unknown, but also its number is unknown. This makes it impossible to determine the dimension of the separation matrix W, making the calculation impossible. Therefore, before blind separation, the number of signal sources must be estimated.

Currently, the commonly used methods for estimating the number of source signals are mostly based on the characteristic decomposition of the covariance matrix of the observed signal y(t). It is easy to obtain the covariance matrix of the observed signal y(t):

Where: Rs represents the covariance matrix of the source signal, and the eigenvalue of the covariance matrix Rx is λ1≥λ2≥…≥λn. Since the column A is full rank, the ^{rank of} _ARSAT is equal to k. After the eigendecomposition of Rx, k main eigenvalues ​​Λs=diag(λ1, λ2,…, λk) and (mk) noise eigenvalues ​​Λn=diag(λk+1, λk+2,…, λn)=σ2 are obtained in descending order. The number of information sources is equal to k, that is, m minus the number of the same minimum eigenvalues. The number of source signals can be determined only by observing the number of repetitions of the minimum eigenvalue. However, the covariance matrix of the observed signal y(t) is usually unknown. When Rx is estimated by a set of observation vectors, the probability that the eigenvalues ​​of Rx are different is almost 1. When the signal-to-noise ratio is relatively low, it is difficult to estimate the number of source signals by observing only the eigenvalues. Wax M and Kailath T proposed to use the AIC and MDL criteria in information theory to estimate the number of source signals. The above criteria are derived in standard array signal processing based on the basic assumption that the observed signals obey Gaussian distribution. In the case of standard array signal processing model and non-Gaussian source signals, HT Wu et al. gave an inspiring GDE estimate of the number of source signals. Many domestic scholars have also proposed many new algorithms. Traditional blind source separation algorithms assume that the number of observed signals is greater than or equal to the number of source signals. However, in some practical applications, the number of observed signals is less than the number of source signals, which is called underdetermined blind source separation, that is, overcomplete blind source separation. Since the mixed system is underdetermined, the mixed system is no longer reversible at this time, so the source signal cannot be obtained simply by inverting the mixing matrix. Because there is information loss in the mixing process, even if the mixing matrix A is known, the independent components of the signal cannot be completely restored. 2 Blind signal source number estimation algorithm based on high-order cumulants 2.1 Definition of fourth-order cumulants In practical problems, first-order and second-order statistics cannot fully describe the statistical characteristics of the signal. The use of high-order statistics can not only obtain better performance than second-order statistics, but also solve many problems that second-order statistics cannot solve. The important feature of the fourth-order statistics is its insensitivity to any form of Gaussian process, and it has many good properties in mathematical form, which are not possessed by the second-order moment. Therefore, it can effectively extract non-Gaussian signals from Gaussian processes or suppress Gaussian noise. This is especially important for Gaussian noise with unknown spectral characteristics. Since the correlation algorithm based on high-order cumulants is blind to Gaussian noise, it can not only correctly estimate the number of signal sources under white Gaussian noise, but also effectively suppress both colored Gaussian noise and correlated Gaussian noise, and still give a consistent estimate. For given random variables x1, x2, x3, x4, zero mean real random variables, define their fourth-order cumulants: 2.2 Construction of cumulant expansion matrix Under underdetermined conditions, the traditional source number estimation method is completely ineffective. By utilizing the array aperture expansion characteristics of the fourth-order cumulant, an appropriate fourth-order cumulant matrix is ​​constructed, and the covariance matrix is ​​expanded so that the information of the number of source signals is included in the matrix to estimate non-Gaussian signal sources that are greater than the number of observed signals, thereby improving the performance of the estimation algorithm and breaking through the limitation of the subspace algorithm on the number of incident signals.

The constructed cumulant expansion matrix can be expressed by Kronecker product, and the real signal model is easily obtained:

Then the new signal model is easy to obtain:

The coefficient positions in the new mixing matrix are very similar to those in the original mixing matrix, but the new matrix can still maintain the independence of the original signal and meet the basic conditions for blind signal separation. N signal sources, M channels, M>N, then after decomposing the GX features, N2 large eigenvalues ​​and M2-N2 small eigenvalues ​​are obtained.

2.3 Algorithm proposed

The cum-singular value algorithm is studied by combining the high-order cumulant optimization process. According to the project conditions, the number of blind sources is estimated under dual-channel conditions. The calculation process is as follows:

Suppose there are n signals (independent) mixed through m channels:

(1) Collect data x=[x1 ₍ n), x2 ₍ n),…, xm ₍ n)]T;

(2) Construct the fourth-order cumulant matrix Cx through high-order cumulant optimization _; (

3) Apply the cum-singular value algorithm to Cx to obtain m2 _eigenvalues ^, and arrange these eigenvalues ​​from large to small σ1≥σ2≥ _{… ≥σm} ; ₍ 4 ) The number of main eigenvalues ​​εδ ₌ k0 ₊ 1, the value of _k0 is εδ=γ( _kmax ), γk ₌ σk + ₁ /σk _+2, which is the number of signal sources required. 3 Simulation experiment The simulation experiment is carried out under the underdetermined condition of 2-channel 3-source signal. The simulation selects two groups of three independent signal sources according to the project background, namely AM, FM, BPSK as the first group of signals; BPSK, QAM, LFM as the second group of signals. Among them, the carrier frequency of AM and FM is 20 MHz and the bandwidth is 10 MHz; the carrier frequency of BPSK and QAM signals is 20 MHz and the code rate is 5 MHz; the initial frequency of the linear frequency modulation signal is 20 MHz and the bandwidth is 20 MHz. 3.1 Experiment 1: Graph of the performance of the source estimation algorithm as the mixed signal SNR changes Randomly select 500 points of data, the linear variation range of SNR is -10 to 10 dB, the step size is 1 dB, and 100 Monte Carlo simulations are performed for each SNR point. The accuracy of target number estimation is shown in Figure 1. After optimization under underdetermined conditions, the cum-singular value algorithm can not only estimate the number of sources, but also increase the probability of correct estimation with the increase of SNR. However, compared with other classic algorithms, it is still the most stable algorithm with the shortest time. Figure 2 shows the correct estimation probability of the second group of signals. It can be seen that the performance of each algorithm has obviously declined, indicating that the performance of the algorithm is different for different signal models. The estimation of the number of sources in blind separation has certain requirements for the model signal. The probability of correct estimation of different mixed signals at different SNRs is very different, and there are not many general methods. 3.2 Experiment 2: Graph of the performance of the source estimation algorithm as the number of mixed signal sampling points changes The simulation is performed at an SNR of 8 dB, with the data length increasing from 50 to 450, the step length is 50, and 100 Monte Carlo simulations are performed for each point. As shown in Figure 3, the fourth-order cumulant SVD algorithm has outstanding performance under small sample conditions and has a higher accuracy than the AIC and MDL algorithms. However, as the number of snapshots gradually increases, the performance of the algorithm based on information theory begins to improve. Figure 4 shows that under the same experimental conditions, different combinations of sources bring different algorithm performance.

4 Conclusion

Starting from the basic assumption of blind signal separation, this paper studies the estimation method of the number of dual-channel blind signals in communication reconnaissance. By referring to the array signal processing theory, it is proved that the high-order cumulant optimization process under underdetermined conditions can be applied to the problem of source number estimation, and the influence of the changes in the two parameters of signal-to-noise ratio and number of sampling points on the performance of the optimized algorithm is discussed respectively. The results show that the performance of the source algorithm estimation will be different under different signal mixing conditions, because it will be affected by multiple factors such as the source model and the propagation environment, which will cause the performance to degrade. The diversity of signal models and the limitations of the source estimation algorithm and the situation of more numbers need to be further studied, and the solution will be discussed in the subsequent correction algorithm.