Abstract
Target signal extraction has a great potential for applications. To solve the problem of error extraction of target signals in the current constrained independent component analysis (cICA) method, an enhanced independent component analysis with reference (EICAR) method is proposed. The new algorithm establishes a unified cost function, which combines the negative entropy contrast function and the distance metric function. The EICAR method transforms the constrained optimization problem into unconstrained optimization problem to overcome the problem of threshold setting of distance metric function in constrained optimization problem. The theoretical analysis and simulation experiment show that the proposed EICAR algorithm overcomes the problem of the error extraction of the existing algorithm and improves the reliability of the target signal extraction.
1. Introduction
Target signal extraction is used to extract unknown source signals from multiple linear mixed signals, which has found a wide range of applications. Especially in the case of the complex electromagnetic environment, a substantial number of electromagnetic signals are interwoven together to interfere with each other [1]. When target signals are mixed with interference signals, multiple mixed signals are generated. These mixed signals that overlap and interconnect in the time domain and frequency domain lead to the communication failure. How to extract the target signal effectively from the mixed signals has become one of the hotspots and key points in the field of signal processing [2].
A current trend in target signal separation is the independent component analysis (ICA) approach, the core idea of which is to minimize the statistical relationship between all the signal sources [3, 4]. ICA can effectively separate all the signals, including target signals, interference signals, and background noise in the nonunderdetermined case, which is widely used in audio signal processing [5], mechanical engineering [6], and biomedical diagnosis [7–9]. A typical ICA optimization algorithm is the FastICA algorithm [10]. It should be pointed out that ICA is not suitable for underdetermined cases. For underdetermined cases, we can adopt sparse analysis [11–14], deconvolution [15, 16], and other methods. This paper only considers nonunderdetermined cases.
Although the ICA method can separate the mixed signals to some extent, the signal sorting order separated by ICA is only related to the nonGaussian of the source signal [2], so we cannot directly decide which one is the target signal, the background signal or the interference signal [4], while we are only interested in the target signal among the multiple separated signals.
In many practical applications, some characteristics of the target signal, such as the carrier frequency, modulation mode, and other prior information are known, which can be used for target signal extraction. If there is a frequency aliasing between the signals, the signal cannot be separated by the traditional filtering method. In this case, the constrained ICA (cICA) algorithm [17–19], incorporating prior information, can be used to extract the target signal [20, 21]. The cICA is also called ICA with reference (ICAR) [22].
However, in the process of optimization for the cICA algorithm, we need to set threshold parameters to distinguish target signals from other signals, which increases computation complexity and storage space and converges slowly. In some cases, the cICA algorithm cannot be converged [22–25].
Recently, Shi et al. proposed a new model of ICA with reference signal (ICAR), where an adaptive weighted summation method is introduced to solve the multiobjective optimization problem with a new fixedpoint learning algorithm [26, 27]. This method solves the threshold setting problem of cICA and effectively overcomes the problem of false extraction but faces the problem of determining the weight parameter.
Compared with the cICA algorithm or ICAR, the proposed enhanced ICA with reference (EICAR) directly contains the prior information into the ICA framework. By combining with the negative entropy contrast function and target signal distance metric function, the EICAR establishes a unified cost function so that the constrained optimization problem is transformed into an unconstrained optimization to overcome the problem of the threshold setting problem of distance metric function for cICA.
In the enhanced ICA with reference (EICAR) proposed in this paper, a priori information is directly contained in the ICA framework combined with the negative entropy contrast function and target signal distance metric function.
The EICAR puts forward four kinds of cost function to convert the constrained optimization problem into unconstrained optimization problem. By deductive analysis of the similarity of the four cost functions, EICAR establishes a unified optimization model, in which the model weight parameter is determined to meet four kinds of cost function at the same time. It not only overcomes the difficulty of setting the threshold for distance measurement function but also solves the difficulty of setting weight parameter.
In practice, the reference signal can be obtained in advance. Under the counter condition, the interference signal is not completely consistent with the target signal in frequency and may only overlap partially. Even if it is completely in the same frequency, the modulation mode of target signal and interference signal will be different. Moreover, interference signals are usually strong noises or direct background music and other unrelated signals, which are significantly different from target signals. In addition, the transmission time and mode of target signals have certain rules, while interference signals generally lack such rules. In general, continuous interference is adopted, or the same frequency interference is sent after the detection of target signals, which has obvious lag in time. We can predict in advance the precise frequency, modulation mode, and even the law of signal transmission of the target transmitter, which can be used as the basis for designing reference signals. Accordingly, interference signal does not have these characteristics.
Of course, the reference signal we designed is only an approximate version of the “expected” reference signal, but this does not affect the validity of the results. Because the reference signal is not required to be infinitely close to the actual target signal, only the distance measurement function with the target signal is the minimum. Since the reference signal is designed based on some features of the target signal, it is obvious that its distance measurement function with the target signal is smaller than that with other interference signals.
The research in this paper focuses on target signal extraction mainly targeted at nonunderdetermined system. In practice, some hybrid systems are underdetermined. In terms of the underdetermined system, scholars such as Woo et al. have conducted some fruitful researches [28–30]. It is one of our next research priorities that standing on the shoulders of giants and carrying out the rapid extraction of targets under indeterminate circumstances.
The rest of this paper is organized as follows. In Section 2, we summarize the mixed signal separation model and analyze the cICA algorithm. In Section 3, we propose the EICAR algorithm, establishing and solving and the cost function of unconstrained optimization. In Section 4, we carry out the experiment and show a series of numerical results of the EICAR algorithm on the monosyllabic frequency modulation signal to verify the extraction effect. Section 5 is the conclusion.
2. Mixed Signal Separation and cICA
2.1. Mixed Signal Separation Model
The Mdim ensional observation signals are produced by mixing the Ndimensional sources. It may be assumed that the mixture of signals is a linear mixture. Suppose the unknown source sources are ; the observed signals are ; and the mixed matrix is the mixing matrix. Thus, the signal mixing model can be obtained as shown in the following equation:
The signal separation model can simply be described as follows: for the observed signal , we can solve a dimension separation matrix by means of the optimization method to get the separated signal after separation:
In order to reduce the computation in the iteration process, we need to remove the correlation between the observed signals by whitening the data with before the iteration. The observation signal after albinism . So the separation model is changed intowhere are the estimated signals for the Ndimensional sources . It is impossible to determine which signal is the target signal from .
2.2. Constrained ICA and ICA with Reference
It is assumed that the target signal is and the corresponding separation vector is , so that the target signal is
Based on the prior information of the target signal, the reference signal is designed. In order to characterize the proximity between each separation signal and reference signal , the distance metric function between the separation signal and the reference signal is defined as follows:
As a result, the distance between the target signal and the reference signal is always minimal for any separation signal:where is any source signal outside the target signal.
For equation (6), in order to fully separate the different homologous signals, it is necessary to fully excavate their statistical independence. The FastICA algorithm based on the maximum negative entropy can separate the independent sources with a search direction of the maximum negative entropy. In the process of separation, when the nonGauss measure reaches the maximum, the separation of the independent components has been completed.
It is very difficult to calculate the negative entropy, so the nonquadratic function is often used to approximate the negative entropy [10]:where is a random variable with the standard Gauss distribution and can be shown as follows:
Accordingly, . As a result, the ICA algorithm can be expressed as
To extract the target signal from the separation signals, we only need to get the smallest through iteration of . From equation (6), the distance metric function between the target signal and the reference signal is always less than any other road vector. Therefore, there must be a suitable threshold parameter to satisfy
The cICA algorithm makes use of maximization negative entropy method to solve the target signal and the separation vector with the negative entropy function as the cost function and the distance metric function as the constrained condition:
It is difficult to set the threshold parameter in equation (11) for the existing cICA algorithm. When it is set too small, no separation vector conforms to the conditions; when it is too large, multiple separation vectors conform to the conditions. We abandon the method of setting threshold parameters and solve the problem directly from .
To solve the problem of setting threshold parameters , some improved ICAR algorithms are formulated [22, 26, 27]. This method solves the threshold setting problem of cICA and effectively overcomes the problem of false extraction but faces the problem of the weight parameter. For instance, Li used two cost functions at the same time to make two optimization operations, in which a rough pretreatment was carried out first, and then a fine postprocessing was carried out [22]. What this paper considers is to combine two kinds of functions into a cost function, which can be optimized once. In this way, four kinds of cost functions are produced, among which the problem of parameter setting is involved. By combining the four kinds, the problem of parameter setting is effectively solved.
3. Enhanced ICA with Reference
3.1. EICAR Cost Function
With the combination of (9) and (10), the target signal needs to meet two optimization problems at the same time:
In order to describe the two optimization problems in a unified way, the maximization and minimization can be transformed into each other. can be converted to or ; similarly, can be converted to or . So we can describe the problem of target signal extraction in two directions and get four EICAR solutions.
Direction 1: combining the distance metric function and the negative entropy contrast function, we reduce the constrained conditions and establish two forms of cost functions , according to the different transformation forms of :where and are the positive scaling factors. The corresponding EICAR schemes are shown as follows:
Since the selection of and has a great impact on As, in equation (14) is related to the selection of appropriate parameters and , and the optimization result of equation (16) depends on and . This problem does not exist in equation (15), whose optimization result is reliable. Therefore, and need to be set appropriately so that the final optimization result of equation (16) is consistent with equation (15).
Direction 2: combining the distance metric function and the negative entropy contrast function, we reduce the constrained conditions and establish two forms of cost functions , according to the different transformation forms of :where and are also arbitrary positive scaling factors. The cost function in equation (17) and the cost function in equation (13) are reciprocal relations, while the cost function in equation (18) is positive and negative with the cost function in equation (14). Then, the corresponding EICAR scheme is shown in the following expressions:
The scheme of (19) and (20) is corresponding to the scheme of (15) and (16). It also faces the problem of division operation or scaling factor setting iteration. Similarly, and need to be set appropriately so that the final optimization result of equation (20) is consistent with that of equation (19).
According to the comprehensive analysis of equations (12)–(20), the optimization results of equations (15) and (19) are definite and reliable. The four equations combine the two cost functions according to four different combinations, so it is reasonable to produce four optimization algorithms.
Now, we will try to find out the common features of equations (15), (16), (19), and (20).
Theorem 1. The gradient of the cost function for the four EICAR schemes , , , and can be expressed in a similar form.
Proof. the gradient of the cost function for and is shown as follows:They can be uniformly expressed asThe gradient of the cost function for and is shown as follows:They can be uniformly expressed asSo they can be expressed in a similar form:It can be obtained by using Theorem 1 that we can take a particular as the cost function that is more certain. Therefore, the new description of EICAR is shown in the following equation:where , , and are the distance metric functions and negative entropy contrast functions, respectively.
3.2. Optimal Solution for Cost Functions
The gradient of in equation (26) is expressed as follows:where and . Define and as the derivative of . Since , we get
For , we get
From equations (27)–(29), we can obtain
Thus, the gradientbased learning algorithm is shown in the following equation:where . The corresponding iterative algorithm process is shown in Algorithm 1.

4. Simulation Experiment and Performance Analysis
4.1. Experimental Signal
In the experiment, 10 groups of analog signals with different systems were selected. A total of 1000 experiments were conducted. One group of signals, in which the frequency of each signal was close to each other, is shown as follows: the source signals and are the single tone FMmodulated signals, while the source signal is the carrier signal. For instance, a set of experimental signals is as follows:
For the convenience of displaying signals, we take the frequency as , , , , and . These signals overlap each other in the frequency domain and cannot be separated and extracted by filtering. In addition, a random white Gaussian noise signal is produced as . As a result, the 4 source signals and corresponding spectra are shown in Figures 1 and 2.
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An () mixed matrix is generated randomly to mix the source signals, and then the mixed signals and corresponding spectra are shown in Figures 3 and 4.
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For the target signal we need, this prior information is desirable. For example, in the case of 4 × 4 mixture, we need to analyze the number of the target signals. Take communication signals as an example, one of which is our normal communication signals and the other is antijamming signals and unintentional jamming signals. Then, only our normal communication signals are our target signals. The prior information of transmitter signal of our communication object can be known, which is sufficient to extract the target signal we need.
The reference signals should carry the prior information of the expected source signals with nonGauss characteristic. There are many kinds of reference signal design, and the most typical method is the pulse method. We select the pulse signals with the same frequency as the source signals as the reference signals.
4.2. Experimental Result
In practice, for example, only one of the 4 signals need to be extracted, which means that only one signal is needed to extract the source signal.
In the simulation experiment, in order to compare the performance with cICA, we designed the reference signal for each signal and extracted each signal.
In order to compare the separation effect conveniently, we carry out the separation experiment by means of the EICAR method proposed in this paper and the cICA method.
Firstly, we use the method of cICA to carry out simulation experiments [27]. In general, in cICA we do not know the correct threshold parameter . In order to ensure that the solution of the required independent components is in the feasible region of the inequality, the initial value of is given to a larger value so that the feasible region is large enough. For the normalized signal and , . If the initial value is set as 1, all independent components in the possible region can be obtained; that is, all target signals can be extracted simultaneously. In this case, the range of needs to be narrowed repeatedly until no signal is extracted when . Taking as the threshold , then 1000 experiments were repeated.
According to the experiment using the cICA method, different reference signals appear many times and the same target signal is extracted. For 1000 experiments with the cICA method, about 1/10 was erroneously extracted. Some results of these experiments are shown in Figures 5–8.
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The key to the cICA algorithm is the setting of the threshold that cannot guarantee the accuracy of the extraction signal. If the threshold is too large, there are lots of source signal vectors satisfying the inequality, then the output of the system cannot be just interested; on the contrary, if the threshold is too small, there is no separate vector algorithm satisfying the inequality.
Then, we make use of the EICAR method proposed to carry out the experiment. The result of 1000 times shows that no extraction error occurred, as shown in Figures 9 and 10.
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The experimental results of Figures 5–10 indicate that the EICAR method in this paper overcomes the above problems with a good extraction effect for different target signals. The separated target signal corresponds to the reference signal one by one without error extraction phenomenon.
4.3. Performance Analysis
On this basis, we continue to study and compare the separation and extraction performance. First, we study the SNR of the extracted target signal, and the SNR of each target signal extracted is as follows:where is the th source signal and is a separate signal corresponding to the target signal. For the EICAR method, are in onetoone correspondence with source signals ; for the cICA method, when eliminating the extraction results error, are all in onetoone correspondence with the source signals . For the FastICA method, since the corresponding relationship is random, we take the form of highest SNR for each . The average SNR of the 1000 independent experiments is shown as shown in Table 1.
Table 1 shows that the antinoise performance of the EICAR method proposed in this paper is less than that of the FastICA method, but it is superior to the cICA method. This is because the EICAR method and the cICA method all join the constraint conditions and the cumulative errors in the iterative process are also increased accordingly. The EICAR method overcomes the error extraction of the cICA method, so the mean SNR of the EICAR method is greater than that of the cICA method.
The corresponding run time is shown in Table 2.
Table 2 shows that the separation time of the EICAR signal is less than that of cICA, and the time of separating all three signals is longer than that of FastICA, but the time of separating the single signal by EICAR is less than half of that of FastICA. For the single target signal, the EICAR separation efficiency is the highest. That is determined by the computational complexity of the respective algorithms. Tables 1 and 2 also show that this algorithm improves the separation signal quality and separation efficiency while overcoming the error extraction of the target signal. This simple example would be suffice to show that FastICA can only separate multiple sources but cannot tell which is the target signal; the cICA algorithm can extract the target signal, but there is a problem of false extraction, which is not reliable in practical application. The algorithm in this paper can not only separate the source signal but also effectively extract the target signal and overcome the problem of false extraction.
5. Conclusion
Extracting the target signal accurately from the mixed signal is one of the difficulties in the field of signal processing. Based on the existing cICA algorithms, we propose an enhanced independent component analysis with reference (ICAR) to overcome the shortcomings of random and false extraction of separate signal sequence in the existing hybrid signal separation algorithm. By combining the negative entropy contrast function and the distance metric function of the target signal, we establish a unified cost function, which transforms the constrained optimization problem into an unconstrained optimization problem. The EICAR algorithm proposed in this paper not only overcomes the threshold setting problem of distance measurement function but also solves the problem of weight parameter setting. Theoretical analysis and simulation results show that the proposed ICAR algorithm outperforms the existing algorithms in extracting the target signal.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Disclosure
The initial research of this paper was published in the Conference summary of the 5th International Conference on Information Science and Control Engineering (ICISCE) in 2018.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All the authors made theoretical and experimental verification and analysis of the content.
Acknowledgments
This paper has been supported by the National Natural Science Foundation of China (grant nos. 61602511 and 61401513).