Classification Methods With Signal Approximation for Unknown Interference

There has been growing interest in the classification of interference types in communication systems, especially under large samples and unknown interference, which severely restrict anti-jamming performance of the system. In this paper, we present two signal approximation algorithms for classification restricted different conditions, and the transform learning label consistency (TLLC) is embedded into the evaluation owing to the imperfect performance for classification and feature library. First, the interference signals are converted into the signal feature space, and then the interference processing and feature extraction are conducted based on the Hilbert signal space theory. Second, the projection approximation (PA) for signal approximation is used to approximate the unknown interference, and the restricted projection property is demonstrated as well. Furthermore, in order to ease the restrictions, the sparse approximation (SA) for interference signals is demonstrated. Moreover, an unsupervised learning method and the unknown interference classifier are proposed based on the self-organizing map (SOM) neural network. Based on l1 minimization functions, we improve the accuracy of TLLC with sparse approximation, which is more suitable for general interference signals. Finally, the simulation results demonstrate that, compared with the traditional classification method, the proposed method improves the classification accuracy of known interference by 3.44%. In this case, the overall accuracy is close to that of the supervised learning method, and the speed of processing interference is increased by more than 10 times. When the SNR reaches 5 dB, the accuracy of unknown interference classification exceeds 94%. Finally, yet importantly, owing to the imperfect performance for classification and feature library at present, we acquire the final accuracy for them at 92.23% by intervening measures, and the time availability also has been obtained advantages on signal processing.

method for unknown interference in large samples in practice.
At present, many methods of interference classification have been proposed previously in various kinds of communication systems. Firstly, according to the characteristics of the interference signals, many researchers utilize intelligent methods [4]- [6] to autonomously perform feature detection and classification, whereas they mainly rely on the support of the original dataset and result in the limited processing speed in real-time applications. In order to improve the efficiency of processing and the rationality of characteristics, transformation analyses, such as the Fourier transforms [7], [8], Fractional order [9]- [11], and wavelet [12], [13], are used to identify inherent characteristics, which makes them more rational. Meanwhile, the corresponding spatialtemporal information, entropy feature, and high-order cumulates [14], [15] are also presented in the interference classification, and the system models are designed for a particular type. Furthermore, improved technologies are used to analyze the sparsity features in some aspects, including sparse decomposition [16], [17] and matching pursuit [18], and achieve the accurate recovery of signals. In these situations, the difficulty of hardware implementation is reduced to some extent by combining some simple classification methods.
However, owing to the variability of samples and the choice of interference modes, the efficiency of large-scale training samples remains low, and the applications are restricted in processing the actual large-scale. In the meantime, the existing work relies on the known or given interference type, whereas the actual interference is affected by many factors in practice. Therefore, it is sometimes difficult to satisfy the requirements of actual interference classification. Note that we focus on the classification of various signals from electromagnetic interference in our paper.
By introducing the theory of space and its metrics in linear algebra and functional analysis into the signal analysis and processing, the signal space can be obtained [19]- [21]. Since the signal satisfies the linear operation and its closure in a linear system, it is obvious that a linear space can be formed, which will reduce the complexity of interference processing. More importantly, machine learning methods pave the way for filling these gaps and developing many innovations such as supervised learning, semi-supervised learning, and unsupervised learning. Certainly, the changeless features cannot guarantee the completeness for various unknown interference, and as a result, the machine learning and intelligent methods are more adaptive solutions, where the proposed and updated characteristics preferably satisfied the request of identification by continuous learning and swarm intelligence. Among them, through the introduction of winning neurons and neighbor functions, the self-organizing map (SOM) neural network [22], [23] based on network topology simulates the side suppression phenomenon in biological neural networks with self-organizing features, and it is widely used in data clustering and pattern classification.
Nevertheless, it lacks adaptability to classification for varieties of interference.
Based on the interference signals and Hilbert signal space, this paper further improves the signal space and signal approximation properties. By utilizing the characteristics of the projection theorem (PT) and sparse approximation (SA), the improved SOM classification method based on signal feature space is proposed to process the unknown interference, and the specific processing flow of the classifier is designed. Finally, the simulation verifies the reliability of the proposed method for unknown interference classification, which will provide an important reference for anti-jamming performance in communication systems.
The rest of the paper is organized as follows. Section II presents the related work on interference model's details, enumerating the primary interference signals and classification procedure. Section III is subdivided into various sub-sections and introduces the Hilbert signal space theory with the projection theorem and the sparse approximation for supplement. Section IV designs the classifier as SOM classification based on signal feature space (SC-SFC). By utilizing compressed sensing with sparse approximation, Section V extends the approximation method in a supervised fashion and achieves the improvement needed for variable unknown interference. Section VI provides an analysis based on the classification accuracy of various classifiers and respective performance is improved. Finally, a brief conclusion is described in Section VII.

A. INTERFERENCE SIGNALS
The malicious man-made interference signals are often encountered for tactical communication, and according to their characteristics, they can be divided into blocking interference, multi-frequency interference and selective interference.
Single-tone interference is continuous wave interference in the time domain, and impulsive in the frequency domain. According to the carrier frequency characteristics of the interfered signal, the interference can be implemented by transmitting high-power continuous waves: where J 0 is the interference amplitude, f 0 is the interference frequency, and θ 0 is the phase. Multi-tone interference is actually a superposition of multiple single-tone interference. According to the carrier frequency characteristics, it can be conducted by transmitting multiple continuous waves simultaneously: where J i indicates the magnitude of the i th interference component, f i represents the i th interfered frequency, and θ i is the phase.
37934 VOLUME 8, 2020 Partial-band interference usually refers to the interference with narrow bandwidth (generally no more than 10% of the bandwidth of the interfered signal), which can be achieved through a narrow-band filter: where f H and f L are the upper and lower cutoff frequencies of the narrow-band filter, respectively. Chirp interference usually refers to the interference whose frequency varies linearly with time and has broadband characteristics in a certain period; besides, its power spectrum has non-stationary characteristics: where k i is the ratio of frequency adjustment. Impulse interference presents time-varying peak and impulse characteristics: where h(t) is the response for impulse. Comb-spectrum interference is a blocking interference whose envelope is constant in the time domain, and it may be generated by the superposition of saw-tooth interference and partial-band interference: where f 0 is the center frequency, t is the delay, and k adjusts the frequency.

B. INTERFERENCE PROCESSING
There is plenty of work on interference processing previously, and the suppression effect has been achieved on the commonly known interference. The authors of [24] realized the recovery of sparse signals by using the method of sparse dictionary, which can effectively improve the ability to deal with impulse interference and narrow band interference in OFDM system. S. Yeh et al. verified the system performance of resisting impulse interference and continuous wave interference under the effect of multipath fading and shadow [25].
With the combination of FrFT and OFDM, the performance of anti-interference was enhanced by the adjusting parameters in the condition of multi-tone interference and chirp interference [26]. The classification and suppression method for interference was proposed based on the modulation of signals, and achieved the anti-interference performance of frequency-hopping systems [27]. However, most of these previous literatures suppressed the particular interference by the classification and transformation analysis, they rely gravely on interference detection and characteristic parameters [28], which restrict the effectiveness and robustness in applications. Therefore, we seek the processing method for interference with transformation and classification simultaneously in this paper.

C. TRANSFORMATION ANALYSIS
Considering the fact that the diversity of interference signals includes the time-varying and non-stationary characteristics, the transform domains are divided into multiple domains (such as the time domain, frequency domain, fractional domain, and wavelet domain) for signal analysis and processing. The basis of each transform domain is t, e jwt , chirp signal, a,b (t), etc. They can be converted into each other, and at the same time, the signals can be processed independently [14], [15]. At present, the methods for digital signal processing based on transformation analysis technology are mainly as follows: Discrete Fourier Transform (DFT). DFT obtains the overall spectrum of signals through time-frequency domain transformation, which may be used to process stationary signals. The relationship of the time-frequency domain transformation is Here, ω N = e (−2πi)/N . Fractional Fourier Transform (FrFT). FrFT is primarily proposed for processing non-stationary signals, and the order p makes the transformation more effective than the traditional Fourier transform. However, the operations get more complex and time-consuming. The signal transformation relationship is derived as follows: Discrete Wavelet Transform (DWT). DWT decomposes the signal spectrum through the changing time window so as to extract the approximation coefficients and detailed coefficients, which can effectively deal with non-stationary interference in detail. The discrete wavelet function ψ j,k (t) and transformation coefficients are expressed as follows: Discrete Cosine Transform (DCT). DCT is similar to DFT, and it can use part of the coefficients to reconstruct signals: where Short-Time Fourier Transform (STFT). STFT can make the signals slip in time by a given window function with a narrow time width, which is suitable for the spectrum analysis of slowly time varying signals and has distinct effects on the local characteristics of non-stationary signals. The discrete formation is derived as where x(n + m)w(m) represents the short-time sequence. Fig. 1 shows the transformation effects of different interference signals by STFT, and the forms and characteristics of different interference are found variously. There obviously exist sparse characteristics, including the fact that the impulse interference after transformed by DFT, multi-tone (single-tone) interference and noise interference with amplitude modulation after transformed by FrFT, noise interference with frequency modulation after transformed by DWT, partial-band interference and chirp interference after transformed by DCT, and comb-spectrum interference after transformed by STFT. Consequently, the results may be used to reduce the complexity and improve the speed of signal processing.

III. INTERFERENCE APPROXIMATION A. HILBERT SIGNAL SPACE
The derivation and description of the signal space under the linear causal system is illustrated in details as follow.
Define the signal space X as a linear space on the complex field K, if there is a mapping P, which makes arbitrary signals x, y ∈ X, α ∈ K. They meet the following conditions: x 0 ≤ p(x) < +∞. If p(x) = 0, and only when x = 0; y P(αx) = |a|P(x) z P(x + y) ≤ P(x) + P(y) Then the mapping P(x) is the norm on the signal space X, which is denoted by ||x||, and the signal space X constitutes the norm linear signal space by the norm ||x||.
If a discrete signal with the limited energy x = {x(k)} is an element in linear signal space l p , let p ≥ 1.
Next, prove that the signal space l p is a complete normed linear space.
Assume that x = {x (n) (k)} ∈ l p , and any ε > 0 if there exists N , which makes m, l > N . In this case, there is Thus, x = {x n (k)} ∈ l p is the Cauchy Sequence. It can be found by Cauchy convergence criteria that there exists x = (k) to make lim n→∞ x (n) (k) = x(k).
Just because x = {x(k)} ∞ k=1 , for any r by (14), when m, l > N , there exists Make l → ∞ then, Then, extend r → ∞, and there is As a result, x m − x p → 0. Therefore, the signal space l p is a complete norm linear space.
Define the signal space X as a linear space on the complex field K. There are x, y ∈ K for any signal x, y, z ∈ X, α, β ∈ K, and the following conditions are met.
x x, x ≥ 0. If x, x = 0 and only when x = 0; y x, y = y, x z αx + βy, z = α x, z + β y, z Then •, • is the inner products on the signal space X, and the signal space (X, •, • ) is for the inner product space on K. Furthermore, the inner product space defining the norms is the norm linear space.
If the signal inner product space (X, •, • ) is a complete norm linear space induced by norm ||x||, it is the Hilbert signal space.
At the same time, since the finite-dimensional subspace of the Hilbert signal space is still the Hilbert signal space, this provides a theoretical basis for practical digital signal processing and engineering applications.
The projection theorem For any element x of the sets M in the closed subspace of the Hilbert signal space X, there is one and only one y 0 ∈ M to make Based on the properties of the projection theorem, the six known interference types J 1 , J 2 , . . . , J 6 can be used for the linear combination. And according to the square mean, we can model the best approximation method: Obviously, J 1 , J 2 , . . . , J 6 are not the orthogonal bases and the maximal linearly independent groups. Note that the dimension of signal space X is variable depending on the dimension of its space basis vectors.
The effectiveness of the best approximation approach is judged by the linear fitting effect of RMS with power, as well as the maximum residual mode and maximum deviation. The maximum deviation is defined as follows: Suppose there is Y ∼ N (a + bx, σ 2 ) for x in an interval, where a, b and σ 2 are unknown parameters independent of x. ε = Y − (a + bx) is a random error, and we make a normal assumption for Y , The smaller the σ 2 is, and the smaller the mean square error of the approximation.
The empirical regression equation of y with respect to x iŝ y =â +bx, and the residual y i −ŷ i = y i − (â +bx i ). Besides, the squared sums of residuals Q e are Because Q e obeys the distribution Q e σ 2 ∼ χ 2 (n − 2), E Q e /σ 2 = n − 2, which derives the unbiased estimation E Q e n−2 = σ 2 . Then the significance test of the linear hypothesis is carried out by t test, and when the significance level of α (where α is always 0.05), the rejection domain (the regression effect is considered significant) is In order to overcome the strictly restricted conditions in signal space approximation, the multiple interference vectors y •k ∈ C N have been collected under different conditions (e.g., spatial, temporal, or other analytic transformations T • ) in many cases. Considering the fact that interference signals are not strictly sparse, which do not share the same common support, we need to jointly observe the unknown sparse matrix X ∈ C M ×K from the interference matrix Y ∈ C N ×K in different transformation domains and variable sensing matrix ∈ C N ×M , which is normally formulated as We can derive the formulation from (23) where there exists T −1 • (X) B2 = B2 T −1 • B1 X B1 . We utilize the decomposition for signals with structured approximation based on its sparse distribution, and the analytic transformationT −1 • (X) can be regarded as a row-sparse matrix embedded element-sparse matrix in its suitable domain, which does not force the target signals to share the same support and the optimization problem can be expressed by where α > 0 and β > 0 are respective weights for row-sparse partition and element-sparse partition. In [29], the convex l 1 norm and the convex l 1,2 norm are used instead for the problem (25), which results in solving the followings VOLUME 8, 2020 We can iteratively alternate to form the closed solution to (26) by few sub-problems: And the iterative version of (26) with no constraints is given by: Among them, the attenuation rate λ >0, and the gradient of the cost function ∇J(T −1 • (X),˜ * t ) may be derived from (24) Summarizing the above process, optimal solutions to interference approximation can be obtained by iterative optimization, and the SA algorithm for unknown interference is depicted by:

Algorithm 1 SA Algorithm for Unknown Interference
Inputs: datasets X, Y, dimensionality N , sparse weights α, β, attenuation rate α 0 , and iterations t max Outputs: analytic transformationT −1 • (X) Step 1 Initialize parameters: set the seed of random number generator, and make the initial orthogonal basis ψ (0) Step 2 Loop: Do optimal approximation for Y, as shown in (27) Step 3 Compute the gradient of cost function Step 4 Do update the orthogonal basis ψt according to (28); Step 5 Determine the optimal analytic transformation.

IV. UNSUPERVISED CLASSIFICATION
Considering the fact that the collected interference may not belong to the six common types of interference. How to deal with the classification and identification of unknown interference types has become an important factor restricting the performance of communication systems. Therefore, for the case of an unknown interference type, we utilize the advantages of the best approximation method of Hilbert signal space and the self-organizing map (SOM) neural network to achieve it.
In order to suppress the environmental noise in the samples and identify various types of interference accurately, we normalize and reduce noise for the interference signal. Moreover, we normalize their uniform dimension and orders of magnitude for different samples, which can prevent large-value interval attributes from affecting small-value interval attributes and reduce the numerical complexity in the calculation process. Since the amount of data after the pre-processing of the interfering signal is still large, identification cannot be directly performed, and the feature information of different interference signals needs to be extracted. The purpose of feature extraction is used to reduce the amount of computation and improve the classification efficiency by acquiring feature vectors with a small number of parameters.

A. SOM CLASSIFICATION BASED ON SIGNAL FEATURE SPACE (SC-SFC)
The SOM neural network is an unsupervised and competitive learning algorithm. It includes competition, cooperation, and synaptic adjustment. According to the given unlabeled training set in Hilbert interference signal space, there exist relationships among them where each element s i ∈ S, S is the input interference signal space, and each s i in the space is composed of n attributes.
There are M pre-classifications y i ∈ Y = {1, 2, · · · , M }. After finding a function g(s) that constitutes a classification function f (s) = sgn(g(s)), it can be used to infer the classification of y values corresponding to any one of the interference signals s. Then, a six-dimensional signal feature space H C is introduced and the interference signal space S can be transformed into another low dimensional signal feature space H C : s → ϕ(s). Furthermore, the training sets T S of the original input interference signal space S is converted into a new training set in the signal feature space H C : T C = s 1 , s 2 , · · · , s l = ϕ s 1 , ϕ s 2 , · · · , ϕ s l (31) The self-organizing map neural network based on signal feature space is a two-layer feed-forward neural machine, which is composed of two parts, the input layer and the competition layer. The typical two-dimension structure is shown in Fig. 2.  The bottom is the input layer, which is used for receiving the training sets T C in signal feature space. In this layer, the number of neuron nodes is L, which is the same length as the training set T C , and the datasets in nodes are projected to the upper layer, i.e., the competition layer. The competition layer is also called the Kohonen layer, and it contains seven sets of neuron nodes (the same number of expected categories) as the output layer. Moreover, there are no connection weights in the input layer and output layer.
In the competition process, the vector s i is randomly selected from the training sample T C and normalized as an input. Then, the inner products between all neurons in the output layer and s i are calculated.
During the cooperation, the neurons in the neighborhood of the winning neurons become excited neurons w i , and their positions are determined by y i . A commonly used neighbor function is the Gaussian function.
In the update process, assuming that the excited neuron in the λ th iteration is w j (λ), the mapped winning neurons and the excited neurons in the topological neighborhood are at the λ + 1 th order: where η(λ) is the learning rate of the λ th order, which decreases as the number of iterations increases. f (y i ; σ , λ) indicates the topological neighbors in the λ th order. According to the principle of SOM classification algorithm demonstrated above, the multi-class network is designed as shown in Fig. 3.
The SOM network adjusts the network weights through self-organization and finally makes each node of the output layer become a neuron sensitive to a specific mode. The corresponding weight vector becomes the central vector of each input mode class so that the sample layer may be reflected in the output layer. The ordered feature map is used to classify the spatial samples of unknown interference signals. Summarizing the above process, the SOM classification algorithm based on signal feature space is illustrated as follows.

Algorithm 2 SOM Classification Based on Signal Feature Space (SC-SFC)
Inputs: Initialize parameters T S , Y, the number of neuron nodes L, the number of iterations Outputs: Class y(index), classification accuracy Step 1 Initialize the weight vector of ownership, set the initial width parameter, the learning rate, and the maximum number of iterations; Step 2 Take the interference signal space training sets shown in (30), and convert to the interference signal feature space training set according to (31); Step 3 For the λ th iteration, find the winning neurons y i from the training sample set T C ; Step 4 Update the winning neuron and its topological neighbors according to (32); Step 5 Determine whether the number of iterations λ exceeds the maximum number of iterations: if so, execute Step 2.

V. TRANSFORM LEARNING CONSISTENT LABEL WITH SPARSE APPROXIMATION
Due to the strictly restricted conditions on signal space approximation, although it is simple, flexible, and timeconsuming for special interference samples, we extend the approximation method by the sparse characteristics of interference based on the sparse model in a supervised way. According to label-consistent dictionary learning, we improve the approximation and classification accuracy for changeable unknown interference as well as prove that the approximation with classification is suitable for various sparse interference signals.

A. SPARSE REPRESENTATION
The specific process of compressed sensing includes sparse representation (signals with sparse or transformed into the sparse), sparse measurement (dimensionality reduction by the matrix not related to the signals), and reconstruction VOLUME 8, 2020 approximation (achievement of the signals recovery). The framework is shown in Fig. 4.
The compressed sensing in Fig. 4 may be expressed as follows: The sparse basis = [ψ 1 , ψ 2 , · · · , ψ N ] (also expressed as the sparse dictionary) is selected for the signal x ∈ C N , which is sparse in the order K N: Among them, represents the projection coefficient matrix, and 0 = K . Given an M -dimension signal y by measurement matrix M ×N , which satisfies the relationship M N, there exists where A = T is the sensing matrix. In the end, the original signal x can be reconstructed accurately by solving the following optimization problems: Sparse representation [30], [31] is key to many signal processing tasks involving linear inverse problems, and in many cases, we need to estimate the sparse signal by the unrestricted form. This is normally present as follows: When the measurement vectors maintain some degree of dependence with the associated coefficients, the optimization problem can jointly estimate the effect of sparse representation.

B. TRANSFORM LEARNING WITH LABEL CONSISTENCY (TLLC)
Transform learning [32]- [34], as the transformed dictionary learning, is a new representation learning technique, and has been widely used for solving inverse problems based on the label consistency criteria. In recent times, sparse representation has been forced into the sparsity constraint on the coefficients, and the most popular method of K-SVD for solutions may be represented as According to the types of unknown interference predicted initially prior, we could utilize the predicted varieties to supervise the transform learning. The mathematical expression is given by where W represents the map between the coefficients θ and the binary class labels Q. Furthermore, as expressed in (35), transform learning obtains the coefficients θ by a transformation basis ψ. The formulation is mathematically derived without the constraints: Among them, log (det ψ H ) is the transformed factor in preventing the solution from degenerating, and || ψ H || 2 F controls the scale of penalty function. Combining (38) with (39), the following formulation is obtained We can separate (40) from few easier sub-problems through the alternating method as follows: The coefficients θ and transformation basis ψ in the update process are utilized with the Cholesky decomposition and SVD by the pseudo-inverse: Ultimately, the corresponding coefficients may be transformed in the training: Obviously, feature generation is superior to dictionary learning owing to the closed-form update rather than the l 1 -minimization problem.

Algorithm 3 Transform Learning With Label Consistency (TLLC)
Inputs: Dataset x, class labels y (0) , dimensionality N , λ, µ, ε > 0, number of iterations Outputs: ψ, θ , W, y test Compute the Cholesky decomposition for L: Do update the transformation Compute the full SVD: Compute the Cholesky decomposition for T:

Produce the approximate class label y test ≈ W H Z End
According to the characteristic of TLLC for unknown interference classification, when the accuracy cannot satisfy the requirement for classification strictly by adding the feature library, we can utilize the initial classification labels to update the unknown but segregated types of interference datasets by embedding it. This will further improve the accuracy for approximation and classification. The integrated process is depicted briefly in Fig. 5.

VI. NUMERICAL RESULTS AND ANALYSIS
In order to verify the effectiveness of the proposed algorithms, experiments for multi-classification were evaluated in interference datasets. We compared the existing work [7], [12], [18] for interference classification where the accuracy of support vector machine (SVM) multi-classifiers and probabilistic neural network (PNN) are more suitable for classification. Therefore, we chose to compare with SVM and PNN. The simulation hardware platform is an Intel(R) Core(TM) i7 CPU (3.40 GHz), 4G memory PC. All experiments are conducted in the MATLAB R2013b environment. For different datasets, the cross-validation method was utilized for the selection of parameters.
For the six kinds of interference signals, 200 Monte Carlo simulations were performed with the change in interference frequency when the signal-to-noise ratio was −2dB to 10dB, and 1200 interference signal samples were obtained. Of these, 600 samples were selected for classifier training, and 600 were used for interference signal classification tests. In the interference signal simulation, the channel was assumed to be a Gaussian white noise channel, the sampling frequency f s = 512 MHz, and the initial signal-to-noise ratio was 3 dB.

A. CLASSIFICATION RELIABILITY OF KNOWN INTERFERENCE
For the multi-classification problem of known interference, the traditional SVM classifier based on feature parameters is compared in this paper. The training datasets are used to drive a series of classifications, and the LIBSVM Toolbox [35] and RBF kernel function are conducted in the experiments. Considering the fact that the selection of penalty factor C and radial basis kernel function parameter γ have great impacts on the performance of SVM classifier, C and γ are determined by a ten-fold cross-validation testing. From this analysis, the values of C and γ are set at 1 and 0.1, respectively. Meanwhile, the parameters of SOM are determined where the topology function is hexagon and distance function is linked. The mean value and standard deviation of classification accuracy for each interference type is presented in Fig. 6-7, and the statistical details of accuracy for dealing with the known different interference signals are shown in Table 1 and Fig. 8. It is found that the SOM classification algorithm based on signal feature space has better performance than the traditional SVM classifier, and the   classification accuracy has been significantly improved. The overall classification accuracy has increased by 3.44%, and the learning process does not prescribe supervision.

B. CLASSIFICATION ACCURACY OF UNKNOWN INTERFERENCE 1) CLASSIFICATION BY ADOPTING THE SC-SFC
In order to verify the classification accuracy of unknown interference, the identification of interference in the unknown  interference intensity range with good approximation effect is selected based on the known accurate identification of interference. The Hilbert signal feature space is still used. With 200 interference samples obtained with the change in interference intensity, a five-fold testing is utilized for the cross-validation of radical basis function for classifier training, and the selected parameter σ = 1.100 samples were selected for classifier training, and 100 samples were used for testing. Our method is compared with PNN on the unknown interference training datasets.
The classification results and the required training time are given in Table 2. The proposed method can effectively identify all kinds of interference signals that can be optimally approximated in the Hilbert signal space. The overall classification accuracy is 91.14%, which is close to the supervised learning PNN classification accuracy. Meanwhile, the running time in data processing is increased by more than 10 times, and it is more suitable for practice and timeliness when there are many interference samples and a large number of datasets in the complex electromagnetic environment.
To better reflect the superiority of the proposed algorithm, the classification accuracy of the unknown interference signals under different SNRs is shown in Fig. 9.  It can be found that with the increase in signal-to-noise ratio, the average classification accuracy of interference signals gradually increases. The unknown interference classification algorithm proposed in this paper has improved the accuracy of various types of interference classification under different SNRs. When the signal-to-noise ratio is lower than 1 dB, the unknown interference is close to the Gaussian white noise of the known interference superposition, and it is easy to identify the error; therefore, the classification accuracy of the interference signal is low. When the signal-to-noise ratio is in the range of 1∼4 dB, the classification accuracy has been greatly improved. Furthermore, above 5 dB, the classification accuracy is stable at about 94%, which satisfies the basic interference classification requirements. Thus, the proposed algorithm can accurately identify each interference type, and its good interference classification performance is verified.

2) EVALUATION FOR MUITI-CLASS EMBEDDED WITH TLLC
When the accuracy of classification cannot achieve the expectation, there are series of signal samples for unknown interference and the initial labels have been obtained by the method above as listed in Table 3. Owing to the imperfect performance for classification and feature library, we need to embed the TLLC into the evaluation. The dataset is obtained by varying the frequency, amplitude, and width in a random way when SNR is 5 dB, and a ten-fold cross-validation is utilized for testing the parameters λ = 50, µ = 1, and ε = 0.05. It can be pointed out that Class 1 has no ability to develop a new class and it should be modified further. After the intervening measures by TLLC, we acquire a final accuracy for the unknown class of interference signals about 92.23% by combing Class 1 with Class 6 as a whole, which achieves the improvement on performance about 1.59%. Furthermore, the details of consuming time for ten-fold testing are illustrated in Fig. 10 and the time availability also has advantages in signal processing.

VII. CONCLUSION
The classification of interference is an important factor in restricting the anti-interference performance for tactical communication. In this paper, the classification problem of unknown interference is solved initially by the theoretical basis of Hilbert signal space and projection approximation methods. The SC-SFC for unknown interference is then proposed. In order to satisfy the requirement for accuracy or the feature library in practice, the TLLC can gain effective improvement for unknown interference multi-classification. It is hoped that the framework and models described in this article will help pave the way for researchers to analyze the interference characteristics with compressed sensing for separation and detection in the future.

APPENDIX PROOF OF THEOREM 1
Proof: Firstly, based on the given sets of the selected known interference, we can construct the non-orthogonal basis and the maximal linearly independent group correspondingly. Note that the dimensions of the signal space X are variable, and its spatial basis vector is provided in this paper. According to the obtained subset J 1 , J 2 , . . . , J M of complex domain K, we can regard the dimensions of the interference signal space as G, namely the time domain, frequency domain, spatial domain, and other suitable transformation domain XQ, and the basis vector is X T , X F , X S , · · · , X Q T , respectively.
By the definition of linear space, there exists For the given general interference x = x (2) (k) ∈ l 2 in the Hilbert signal space, we can obtain the linear combination of the available basis vector X T , X F , X S , · · · , X Q T , and design the optimal approximation model based on the meanings of mean square, which is expressed as: Under the restrictive conditions |P| =0, we can derive the formulation from (A.2) Considering the fact that the complex domain K is a complete space, that is, the linear combination of the M known interference types J 1 , J 2 , . . . , J M can be used to establish the optimal approximation model according to the square mean, which is depicted by Obviously, there exists the coefficients vector β 1 , β 2 , . . . , β M meeting the conditions Therefore, the existence and uniqueness of solutions are proved.