Blind Separation of Weak Object Signals Against the Unknown Strong Jamming in Communication Systems
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Abstract
To obtain the mixed weak object signal against the super power signal (jamming) is still an challenging task in modern communication systems. In this paper, a novel framework is designed for weak object signal blind separation against the strong interference signal. To extract the strong interference signal,firstly, we separate the mixed signals with the optimized FastICA algorithm, then, an improved Interference Cancellation algorithm is proposed as reference signal based on the separated strong signal. Next, we separate the weak mixed signals by the improved FastICA algorithm again. Finally, we discuss the performance of the proposed method and verify the novel method based on several simulations. The experimental results demonstrate the effectiveness and robustness of the proposed method.
Keywords
Blind signal separation Gauss channel Kmeans clustering FastICA algorithm1 Introduction
In weak signals against the strong signal interference system, the components are composed of two parts: interference signal and mixed object signals. Since the interference signal is strong (strong interference signal) compared with the object signals (weak object signal), it is very difficult to obtain the weak object signals against the strong interference signal by using the classical blind source separation methods. Meanwhile, the classical blind source separation methods are lack of robustness. There are several existing algorithms that are partially related to the object signal detection, such as the Relax algorithm by Li etc [6], CLEAN technology by Sao etc. [7], FFT signal separation method by Gough [8] and FastICA algorithm by Hyvarinen etc [9, 10, 11]. Although these algorithms are partially related to the weak signal separation, their performances on passive communication system are still not sufficient for the practical applications. Hence, it is still necessary to develop more efficient object signal separation algorithm for the weak signal blind source separation against the strong interference. In this article, we proposed a novel method to separate the weak mixed signals against the strong interference. This method has a better separation performance and robustness than the traditional methods.
In [12], we first cancel the strong interference by using the Interference Cancellation algorithm (ICalgorithm), then, separate the mixed weak signals in Passive Radar System. The method has a good performance but has some limitations. It is only in the Passive Radar System, that is, the strong interference is known to us.
In this paper, an improved FastICA algorithm is proposed with the Kmeans clustering algorithm, which has lower complexity and better stability than classical FastICA method. By exploiting the separated strong signal, the channel parameters are estimated. From the above knowledge, a novel framework is designed for the weak object signal separation against the strong signal interference in the modern communication system. We first separate the mixed signals with the improved FastICA algorithm, then, an improved Interference Cancellation algorithm (ICalgorithm) is proposed based on the separated strong signal as reference signal. Next, we we separate the weak mixed signals by using the improved FastICA algorithm again. Finally, We verify the proposed method based on several simulations. The experimental results demonstrate the effectiveness of the proposed method.
The rest of the paper is organized as follows. In Sect. 2, we introduce the improved FastICA algorithm with Kmeans cluster and the Interference Cancellation algorithm (ICalgorithm). Section 3, we introduce the experimental process. In Sect. 4, we discuss the properties of the above method, including performance comparison, robustness analysis, complexity analysis, convergence analysis etc.. Finally, the conclusion is drawn in Sect. 5.
2 Blind Separation of Weak Signals Against the Strong Signal Interference
In this section, we introduce the blind separation of weak signals against the strong interference signal.
2.1 BSS Model
2.2 Framework of Blind Separation of Weak Signals Against the Strong Signal Interference
2.3 FastICA Algorithm
In this section, we separate each interesting object signals with the improved FastICA algorithm combined with kmeans clustering.
 1.
Standardized data
 2.
Choose the original vector \(W_0\) and set \(\Vert W_0\Vert =1\)
 3.Select a nonquadratic function, e.g.,$$\begin{aligned} g_1(y)=\tanh (a_1y),\quad g_2(y)=y \exp \left( \frac{y^{2}}{2}\right) ,\quad g_3(y)=y^{3} \end{aligned}$$(5)
 4.Let$$\begin{aligned} W_p=E\{Zg(W_p^{T})\}E\{g^{\prime}(W_p^{T})\}W_0 \end{aligned}$$(6)
 5.$$\begin{aligned} W_p=W_p\sum (W_p^{T}W_j)W_j,\quad j=1,2,\ldots ,p1 \end{aligned}$$(7)
 6.$$\begin{aligned} W_p=\frac{W_p}{\Vert W_p\Vert } \end{aligned}$$(8)
 7.
If \(W_p\) is convergence, go to (8). Otherwise, return (4)
 8.
Let \(p=p+1\), if \(p\le m\), return (2).
2.4 Improved FastICA Algorithm with KMeans Algorithm
As the above statement, the FastICA algorithm results depend on algorithm original vector \(W_0\). However, in a blind context, it is hard to tell which original vector gives the best results, as the selection of the original vector is not a sufficient condition to have the optimal solution. We thus propose, for all sampling points, reduce the scope of the original vector with Kmeans clustering algorithm.
Kmeans has a rich and diverse history as it was independently discovered in different scientific fields by Steinhaus (1956) [14], Lloyd (proposed in 1957, published in 1982) [15], Ball and Hall (1965) [16], and MacQueen [17], it is the most popular and the simplest partitional algorithm.
 1.
Give a cluster number K for starting;
 2.
Compute the squared Euclidean distance d from each object to each cluster and assign each object to the closest cluster;
 3.
Minimize WithinCluster Sum of Squares (WCSS) in (9) and Update the cluster center for each cluster;
 4.
Recalculate the squared Euclidean distance d based on the new memberships;
 5.
Repeat steps 3 and 4 until there is no possibility to move the objects to clusters.
The output of the Kmeans is the means vector \(\mu _1,\mu _2,\ldots ,\mu _K\). The examples are shown in Figs. 4 and 5. It is seen that \(\mu _i(i=1,2,\ldots ,K)\) is the cluster centers and stands for the general feature of the corresponding class. So, we choose the original vector \(W_0\) in \(\mu _1,\mu _2,\ldots ,\mu _K\).

Reduce iteration times
If the separation results are out of the acceptable range or the FastICA algorithm is nonconvergent, we must replace the initial value. The improved FastICA algorithm reduces iteration times and improve the stability of the convergence.

Improve the stability of the algorithm
The original vector \(W_0\) is in \(\mu _1,\mu _2,\ldots ,\mu _K\), which have universality. Then, the process can improve stability of the algorithm.
2.5 Interference Cancelation Algorithm (ICAlgorithm)
After the introduction of improved FastICA algorithm, we next introduce our Interference Cancellation algorithm (ICalgorithm).
3 Simulation and Blind Source Signal Separation Results
In this section, we verify the proposed method. We first introduce the parameter setting used in our experiments. We set the sample rate as \(fb=2*10^{4}\) Hz, the transmission bit rate as \(fb=10^{3}\)bps, the modulation frequency as \(f_0=2*10^{3}\)Hz, the bit numbers as \(m=80\), and the original signal numbers as \(MK=4\).
3.1 Effectiveness of Interference Cancellation algorithm (ICAlgorithm)
 1.Vector Standardization, Suppose the vector is \(a=(a_1,a_2,a_3)\), the standardization vector is$$\begin{aligned} \frac{a}{\parallel a\parallel }=\left( \frac{a_1}{\parallel a\parallel },\frac{a_2}{\parallel a\parallel },\frac{a_3}{\parallel a\parallel }\right) \end{aligned}$$(19)
 2.Error function,where \(\hat{a}=(\hat{a_1},\hat{a_2},\hat{a_3})\) is the estimation of \(a=(a_1,a_2,a_3)\).$$\begin{aligned} Error=\left\ \frac{\hat{a}}{\parallel \hat{a}\parallel }\frac{a}{\parallel a\parallel }\right\ _2 \end{aligned}$$(20)
3.2 Extracting the Strong Interference Signal with the Improved FastICA Algorithm
3.3 Strong Interference Signal Cancelation Using the Interference Cancelation Algorithm (ICAlgorithm)
3.4 Blind Signals Separation with the Improved FastICA Algorithm
After transitions from the Gauss channel, the received mixed signal waveforms are shown in Fig. 10 (Received Composite Signal). Here, we consider four channels to fully simulate the realistic signal transmission, which are shown from top row to the bottom row in Fig. 10, respectively.
4 Discussions on the Properties
4.1 The First Comparative Experiment of Effect
We can see that blind sources signals can be efficiently separated with the proposed method and it has a better performance than the classical FastICA algorithm.
4.2 The Second Comparative Experiment of Effect
From Fig. 13, we can see that it has a better performance than the classical JADE Separation Algorithm too.
4.3 Termination Criteria
4.4 Robustness Analysis
4.5 Complexity Analysis

Estimate the channel parameters with the reference strong jamming.
In (4), the coefficient matrix is \(L\times K\), the source signal matrix is \(K\times N\), then, the multiplicative complexity is \(O(K \times N \times L)\), addition complexity is \(O(K \times N \times L)\).

Calculating the strong interference signal can be separated from the mixed signal based on the channel parameters.
So, the overall complexity can be determined as \(O(K\times N\times L)\).
4.6 Convergence Analysis
In this subsection, we discuss the convergence of the proposed Interference Cancelation Algorithm (ICalgorithm). Due to the influence of the noise, the vector space \(\widehat{\mathscr {L}}\) has a ambiguity. Obviously, the ambiguity will hamper the algorithm convergence due to the arbitrary vectors influencing the iterative process [25]. An convergence point is assumed to be unstable under the Interference Cancelation Algorithm (ICalgorithm) update rules if a small perturbation on the convergence procedure may cause the Interference Cancelation Algorithm (ICalgorithm) to diverge away from the convergence point [26]. However, these can be easily avoided if each iteration make \(\parallel L_{i}\widehat{L_{i}}\parallel\) minimize. The following statement discuss the convergence point of the Interference Cancelation Algorithm (ICalgorithm).
Theorem 1
Let \(\widehat{\mathscr {L}} = \{\widehat{L_{1}},\widehat{L_{2}},\widehat{L_{3}},\widehat{L_{4}}\}\) denote the vector estimation space, then, for any initialization of the Interference Cancelation Algorithm (ICalgorithm), the limit \(lim_{i \longrightarrow \infty }\) exists, that is, the Interference Cancelation Algorithm (ICalgorithm) converges [27].
Proof
5 Conclusions
In this paper, we propose a novel blind source signal separation method. We first separate the mixed signals with the improved FastICA algorithm. Then, an improved Interference Cancelation Algorithm (ICalgorithm) is proposed based on the separated strong signal as reference signal. Next, we separate the weak mixed signals using the improved FastICA algorithm again. In the following, We verify the proposed method based on several simulations. The experimental results demonstrate the effectiveness of the proposed method. At last, we discuss the properties of the above method, including performance comparison, robustness analysis, complexity analysis, convergence analysis etc.
Notes
Acknowledgements
This work is fully supported by a grant from the national High Technology Research and development Program of China (863 Program) (No. 2012AA01A502), and National Natural Science Foundation of China (No. 61179006), and Science and Technology Support Program of Sichuan Province (No. 2014GZX0004), and the Opening Project of Artificial Intelligence Key Laboratory of Sichuan Province (2017RZJ01), the Opening Project of Key Laboratory of Higher Education of Sichuan Province for Enterprise Informationalization and Internet of Things (2017WZJ01), and Sichuan University of Science and Engineering talent introduction project (2017RCL11).
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