Simultaneous Representation and Separation for Multiple Interference Allied with Approximation Message Passing

: — Broadband reliable communication is a competitive 5G technology for cognitive communication scenarios, but meanwhile introduces multiform interference to existing broadband transform domain communication system (TDCS) transmission. In order to facilitate the improvement of the anti-jamming performance for the coexistence of diverse interference and TDCS signals in wireless heterogeneous networks, it is important to separated and eliminate various interference to TDCS systems. In this paper, a novel sparse learning method-based cognitive transformation framework of interference separation is formulated for accurate interference recovery, which can be efficiently solved by iteratively learning the prior sparse probability distribution of the interference support. To further improve the separation accuracy and iterative convergence, the principal component analysis and Bayesian perspective in orthogonal base learning are exploited to singly recover the multiple interference and TDCS signals. Moreover, utilizing different sparsity states of spectrum analysis, the proposed novel interference separation algorithm is extended to simultaneous separation based on state evolving of approximation message passing, which iteratively learns the belief propagation posteriors and keeps shrunk by iterative shrinkage threshold. Simulation results demonstrate that the proposed methods are effective in separating and recovering the sparse diversities of interference to TDCS systems, and significantly outperform the state-of-the-art methods.

improved sparse method is represented for diversities based on principal component analysis and iterative weighted least squares.
3. An novel interference separation algorithm called ISAR-AMP based on state evolving is proposed for the sensed signal with known sparsity, which iteratively learns the belief propagation posteriors in the min-sum manner. To further reduce the number of messages caused by the extra burden of loopy graph connections per iteration, the remaining iterations are shrunk by the iterative shrinkage threshold, which achieves better separation accuracy and convergence rate.
The rest of this paper is organized as follows. Section II describes the statistical models of TDCS signals, the primary interference forms as well as the responding transformations. Section III presents the corresponding separation problems of single recovery and simultaneous elimination for diverse interference, respectively, according to the probable sparsity conditions on sensing spectrum. After that, an efficient learning method is proposed for the sensed signal with unknown sparsity in Section IV, and an improved simultaneous separation algorithm for the sensed signal with known sparsity is described in Section V. The simulation results are reported in Section VI with discussions in detail, followed by the conclusions in Section VII.

TDCS Signal
Different from direct sequence spread spectrum that mitigates interference at the receiver, transform domain communication system (TDCS) smartly synthesizes an adaptive waveform to avoid interference at the transmitter. The transmission sequence in TDCS has good auto-correlation and cross-correlation performance to achieve low LPI and orthogonality. Fig. 1 shows the processing steps of a general TDCS transmitter.

1) Spectrum Identification
In the first step, the available spectrum is estimated to ensure an interference-free transmission utilizing the period-gram and auto-regressive methods, and the interference is detected and eliminated by adaptive filtering or maximum likelihood according to the sensing spectrum. Then, a spectrum mask where N spectrum bands are separated from the entire spectrum band, T is the optimal transform domain for interference elimination, and miϵ{0, 1}.

2) Random Phase
The complex poly-phase vector is produced by a unique pseudo-random code selected randomly from an m-sequence with respect to the spectrum mask MT(w) element-by-element, 1

3) Base Function
The desired energy is distributed equally or diversely in the spectrum nulling to scale the magnitude [h1, h2, …, hN] according to limitations of the peak-to-average power ratio (PAPR), hi ≥0, i=1,2, …, N.
And then the appropriate inverse transform is performed for the energy-injected waveforms BH(w) to generate the buffered base function bn(t) in the time domain: (2)

4) Modulation
Generally, cyclic shift keying (CSK) is utilized for modulation in TDCS to adapt noise-like properties. For binary CSK (BCSK), the symbols are generated by circularly shifting the waveform over Ts/2 (half period). Then, the information bit stream [r1, r2, …, rN] is synthesized by mapping the buffered base function, riϵ{0, 1}, and the transmitted waveform with the normalized energy factor ξ can be expressed as  

5) Transmission
The signal transmitted through the channel will be corrupted by additive Gaussian noise and external jamming; the channels are mainly dominated by the LOS component, and despite the possible limited multipath fade due to ground reflections, the propagation delay is mainly caused by path loss.
Meanwhile, given the potentially relative mobility, Doppler frequency shifts may affect the channel and, consequently, the spectrum allocation. Therefore, the Clarke flat fade channel model proposed in [26] was employed to estimate the influence of Doppler frequency shifts, and the received TDCS signal can be given by where κ is the fading factor, and φ is the Doppler shift.
In general, most signals processed in communication systems can be compressed by transforming them into coefficients in some respect. However, many of the transform coefficients of natural signals are close to zero [27]. Thus, the signals occupy only a small fraction of the entire transformed signal space and can be represented sparsely by a few elementary components out of a given collection.
Consequently, sparsity can also be observed in other classes of natural signals [28].
When the length of the original signal is N, there exist W=2 N possible forms of the transmitted signal in the entire signal space S. Only a small fraction of the space is occupied, which can be represented sparsely as where ΨT ϵR N×L is a sparse dictionary for signal space SϵR N×L , and most coefficients θT ϵR L×1 are apparently equivalent to zero.

Sparse Interference elimination
Practically, communication systems are subject to a wide range of interference in complex environments, and the types of interference signals can be divided mainly into three categories, which include impulse interference, carrier interference, and direct noise interference [29]. These latter two categories can be further divided as well. For example, carrier interference occurs in many forms, such as sine, square and saw-tooth waves, and direct noise interference can be divided into different categories according to its modulation mode, such as amplitude modulation interference, frequency modulation (or phase modulation) interference, and hybrid modulation interference. In addition, the above categories of interference can be divided into generally based on their spectrum characteristics into targeted, multi-frequency and blocking interference categories [30]. For instance, the narrow-band interference is representative of targeted interference. Here, targeted interference typically represents interference with a high degree of similarity to the interfered signals. Multi-frequency interference interferes with multiple carriers, and is characterized as frequency-division, time-division, and comprehensive multi-frequency interference. Among these, multi-tone interference is applied broadly to tactical communications. Blocking interference has the characteristics of wide broadband coverage, such as in the case of chirp interference and comb-spectrum interference. Considering the composition and application of jammer in practical applications, the Gaussian noise substituted for the interference source generator is investigated in this paper.
With the rapid development of signal processing theory, it provides many practical responses to these dilemmas in transforming analysis where it is urgent to improve capabilities of secure and reliable communication for anti-jamming and anti-interception performance, such as principal component analysis, linear discriminant analysis and sparse transformation analysis as well as the corresponding derived methods [31][32][33]. They convert the detected signal into a new form for pursing essential correlations, which will commendably adapt to the diversity of generalized signal forms and the selection of optimal transformation. Many methods have been proposed to suppress the interference, including filtering in the time domain and mitigating in the transform domain, which have achieved a good suppression effect on common interference, such as narrowband interference and impulse interference [34]. Sangho et al. utilized the spectrum sensing method for interference signals to opportunistically avoid the jamming spectrum; specifically, they designed a tactical data link with cognitive anti-jamming capability for improved bit-error-rate performance [35]. Yeh et al. verified the system performance of resisting impulse interference and continuous wave interference under the effect of multipath fading and shadow [36]. However, most of these conventional methods can only suppress a specific interference, and reconstruction of the interference cannot be achieved by varying the parameters; consequently, these methods fail to achieve effective and robust performance in communication systems.

Problem Statement
Sparse representation builds the basis of emerging compressive sensing methods, which represents a sparse signal as a series of other forms and most of sparse coefficients close to zero. Thus, the sparse model can be illustrated by the various interference signals as  J J j Ψ θ (6) where ΨJ ϵR N×N is the complete dictionary for the particular interference jϵR N×1 , and θJ ϵR N×1 is the corresponding sparse coefficients. According to the characteristic distribution of sparse dictionary and coefficients, we will achieve it for the separation of different formed interference.
The detection of synthesized signal forms at the receiver is not affected in the spectrum sensing stage of TDCS because its signals are highly sparse and noise-like. And then, the synthesized forms may be determined whether the interference exists or not according to the obtained transformation domain. The problem based on hypothesis testing can be presented as where x is the perceived electromagnetic information, s is the TDCS communication signal, j (if exists) is electromagnetic interference, and n is the environmental noise.
If the assumption H0 is true, different interference forms, such as the narrowband interference and the multi-tone interference, should be considered. According to the dictionary for signal or interference obtained independently by orthogonal basis leaning, the spectrum estimation with sparse representation in transformed domains may be further given by y Ax A s j n A Ψ θ +Ψ θ n (8) where y' denotes the measured spectrum, A ϵR M×N denotes the measurement matrix, and n' is a measured noise in the compressive sensing procedure. When a TDCS system is put into the complex electromagnetic environment, the receiver will be situated in different spectrum states, and the received signal can be divided into the known sparsity and unknown sparsity states, which results in the diversity of separating for interference.

Signal Sensed with Unknown Sparsity
When the receiver loses connections with the transmitter resulting from the strong interference or some potential threats, the signal sparsity states will be changeable or unknown, and we will get into the dilemma without any information about sparse distribution of the signal. Therefore, we can only rely on the pre-collected sparsity library for the unknown signals by spectrum sensing at the receiver, and then achieve the recovery for interference separately.
According to the obtained dictionary for interference using multiple dictionary leaning, the spectrum estimation in multi-task compressed sensing with the noise-like TDCS signals is: where the compressed signal yk ϵR M×1 , Φk ϵR M×N is the common measurement matrix, ΨJk ϵR N×L is the interference dictionary, and the multiple coefficients θJk ϵR L×1 , and nek is a noise modeled as a Gaussian form with unknown variance σk 2 .
In the analytical transforming process, the diverse characteristics of interference signals present on the different transform domains. For the general transformed framework, the optimal sparse transformation    Τ will form the restricted conditions for the recovery of the interference signal j based on the assumptions of the separate characteristics for interference and sparse signals, which is illustrated as where ε denotes the recovery error, K denotes the sparsity of interference signal, and    Τ denotes the optimal sparse transformation. When the transformed framework    Τ is a strict framework and the measurement matrix A meets the restricted isomorphic property (RIP) about the framework, the unknown interference j can be reconstructed by the l1 analysis.

Signal Sensed with Known Sparsity
In this case, both the transmitter and the receiver keep communicating with each other when the encountered interference varies in its types or parameters, which will result in degrading the system performance. To address this issue, we can achieve the simultaneous separation of interference and signal based on the obtained orthogonal bases, respectively, which will further improve the anti-jamming performance.
According to the dictionary obtained by multiple dictionary leaning, the estimation in multi-task compressed sensing with sparse representation may be given by (11) where yk ϵR M×1 is the compressed signal; Φk ϵR M×N into the desired forms when the measured signal y and measurement matrix A are given as well as the sparsity is in the scope of the cardinality of support sets, which will achieve it for the allied sparse separation of interference j and signal s [37]. The problem can be described as Furthermore, based on the property of non-convexity for the l0 norm, the aforementioned problem can be resolved by the l1-split-analysis [38] when the transformed frameworks un-correlation with each other, which can be derived from Specially, the accurate recovery of unknown signals s and j can be achieved with great probability when A is a Gaussian random matrix, and the theoretical error between the optimal solution and real solution will be forced into where C0 and C1 are constants according to the signal recovery.
In the conventional perspective of l1-split-analysis, morphological component analysis (MCA) is deemed to be an effective approach for decomposing combined components [39]. For the indeterminate principle in compressive sensing, a sparse signal cannot be represented by more than two sparse base frameworks simultaneously, which means the number of nonzero elements in non-sparse coefficients is far larger than that in sparse coefficients. Therefore, the mixed components ji may be sparse representation with difference by designing the appropriate base framework, which is widely adopted in l1-split-analysis. And the separation method of base framework achieves separation singly utilizing the optimal sparse representation in the obtained over-complete dictionary, which has its advantage on flexibility and generality of sparse signals. For more complex models, however, the aforementioned method has the extra operating burden on transformation and analysis, which leads to the separation errors much more than optimum and is not suitable for the recovery of complex forms in sparse transforming separation. More practical solvers recently, the TFOCS [40] is provided with much computing complexity even in examples of simple linear combinations, and applied more complicatedly to our interference separation problem allied with multi-transforms. In these cases, the extra computing complexity O(L 2 ) will get exponential increase with the number of signals L. Hence, it requires to be efficiently solved by a novel algorithm for the separation problem of interference multi-transforms in the more complex electromagnetic situations.

Efficient Learning Allied with Base Orthogonality
The optimization problem in Eq. (10) has been solved based on a number of structured libraries investigated in previous studies, such as time-frequency dictionaries, wavelet packets and cosine packets.
Most of these libraries can be considered as corresponding to orthogonal bases in same cases. Herein, we denote these libraries as S, and an adaptive method [41] is utilized for selecting the optimal basis from S that delivers near-optimal sparsity representations on the order of Nlog(N) in terms of time. This method is formulated as follows.
Here, the term represents the level of entropy associated with information theory, and is an entropy function with scalar representation, which is defined as where the parameters represents the total number of distinct vectors occurring among all bases in the library. For example,

Mi=nlog(n) for the structured library based on wavelet packets.
According to the definition of information entropy [42], we can obtain the optimal transformation Here there exists the probability of occurrence exceeding (1 − ε /Mi ) when Eq. (17) satisfies the

Orthogonal base learning for representation
The above analyses indicate that one of the subproblems for various forms of the perceived TDCS signal s or interference j (unified as x ϵC N×1 ) between (10) and (13) would be adaptively solved by obtaining the optimal orthogonal dictionary Ψs , Ψj (unified as Ψ* ϵC N×N ) respectively through a transformation learning approach, while ensuring that the Ψ* obtained at each iteration is an orthogonal matrix. An iterative learning approach for the multiple sparse dictionary is presented in this section to ensure orthogonalization of the obtained Ψ* at each iteration.
Assuming that a random orthogonal dictionary Ψt ϵC N×N and a signal dataset x ϵC N×1 (whether s(t) and j(t) is separate or together) are given initially at the t th iteration, then the nested optimization problem is found as an optimum Θ* ϵC N×1 of For a temporary fixed Ψt , the optimal coefficient vector Θ* in (18) may be obtained by Moore-Penrose inverse as Supposing that the obtained coefficient * Θ is a short-term solution, we adopt SVD decomposition for the iterative representation of *  Ψ ϵC N×N , which is given by It is observed that the iterative forms are caught divergence away from the optimum in (19) and (20), whereas the stochastic descent approach motivates the process convergence. Based on the principal component analysis (PCA) [33] for the sparse recovery process, the optimization problem with weighted l2 analysis can be derived from (18) 1 However, the listed model in (21) is out of the unique condition but simple for solutions, which is depicted as: The above-mentioned depiction promotes the allied forms of sparse representation and estimation error whereas deserving to be settled separately. Then, the generalized cost function J(ꞏ) (also denoted as the overall error) may be obtained by the extra weight from (21) and (22) as follows: where w ϵC N×N denotes a diagonal matrix for the weighted error.
For minimizing the overall estimation error and simplifying the forms during the iterative procedure, the cost function is utilized by the iterative weighted least squares to avoid getting stuck into the local optima, which may be obtained by where W=ww H ϵC N×N , and αϵC N×1 denotes the iterative forms of estimation error, yielding the measurement at the ( t+1) th iteration Furthermore, the iteration stopping criteria are determined by (27) It is noted that the respective gradients of generalized cost function are given by Therefore, the minimum of (29) can be indicated as explicitly Then, the stochastic gradient descent method is employed to minimize the cost function in (23), and the global optimum Ψt+1 at the( t+1) th iteration may be formulated as (31) where xrs ϵC N×N is the randomly selected form from x with respect to descending distribution of the gradient function, βt+1 >0 is the learning step, and the Gram-Schmidt orthogonalization operator OR [•] is performed on the updated base.
To keep descent trend of the cost function, the one-dimensional search method is utilized for the criteria and the learning step in (31) should satisfy For further simplifying the generalized computation of the learning step, it can be exploited commonly as the exponential form with respect to βt , we can iteratively update βt+1 by (33) where β0 is the initial learning step; r is an adjustable parameter, r<1; and tmax is the iterations.
Furthermore, after normalizing the basis vector, the selected form xrs is updated by iterations After completing the iterative learning process, the obtained orthogonal basis is not required normalization until the sparsity achieves the predetermined settings.
Noticeably, after applying the Gram-Schmidt process for orthogonalization and normalization of the bases, relationships between are restricted with an arbitrary x at successive iterations. This provides the corresponding restriction: This means the convergent properties of the orthogonal base learning, and it will converge to the global minimum at the end.

Multiple Bayesian learning for Recovery
Because one measurement matrix implicitly measures all the coefficients when multiple compressive sensing is conducted for the interference, each compressive measurement is performed with random linear combination of the sparse basis in ΨJk, and each random projection corresponds to one measurement matrix [43]. Since the measured interference jk is compressive in the basis ΨJk, one measurement may approximate θJk (that is jk) accurately by solving an l1-regularized formulation from (10), thus achieving the single recovery for sensed interference with the obtained orthogonal base ΨJk: Despite many techniques available to perform the inversion for each compressive measurement separately, one measurement may conduct multiple sets of measurements, and many of the measurements are statistically related when repeated measurements are taken in similar scenes or for the interrelated tasks. Therefore, the inversion problem in (36) may be expressed from a Bayesian perspective with a prior belief.
In the context of a regression analysis from (36) and a temporarily-fixed interference sk with independence, we assume The likelihood function for θJk based on the observed yʹk ϵR M×1 is expressed as Due to the assumption of independence of the yʹk, the likelihood of yʹk for the coefficients θJk and parameters σk 2 of the observed yʹk can be written as The coefficients θJ are assumed to be drawn from the zero-mean Gaussian prior distributions with the hyperparameter vector α=[α1, α2,…, αL] independently shared by all related tasks, and we may To promote sparsity of the coefficients θJ, Gamma priors are defined by the hyper-priors over α, which are formulated as where parameters a and b are fixed to small positive values.
Given the measurements from the statistically-related samples by Bayesian rule, the posterior density function over the hypermeters α, coefficients θJk, and parameters σ 2 =[σ1 2 , σ2 2 , …, σK 2 ] of the observed Jk y θ α σ θ α σ θ α σ y y (42) Since the posterior   2 , , k p Jk θ α σ cannot be computed directly, we utilize the probability distribution of the observed yʹk: Jk Jk y y θ α σ θ α σ θ α σ (43) Then the posterior of   2 , , | k k p  Jk θ α σ y is decomposed as: We can obtain the posterior distribution over the coefficients θJk based on the conditional probability density functions: Furthermore, the posterior density function for the coefficients can be derived from (45) . Therefore, we obtain the final expressions of the recovered jk Above all, according to the updating parameters acquired, we can obtain the closed form solutions jk by iteratively alternating μk and ∑k concurrently until the convergence reaches a certain degree.

Interference Separation Altering with Relaxed AMP
Above we showed that, one measurement matrix implicitly measures all the coefficients when multiple compressive sensing is conducted for the signals, each compressive measurement is performed with random linear combination of the sparse basis, and each random projection corresponds to one measurement matrix, resulting in the problem of allied sparse separation in (11) is substantially more expensive than ordinary separation rules for the redundant computation even the orthogonal base is obtained efficiently. Therefore, multitask compressive sensing approach is a better option than other methods for recovery and separation of various forms of interference, which could be collected from different locations or transformations and extend the application of the methods to diverse interference forms [44].
After the base learning, the first step of the separation is to approximate the ideal components following the message passing rule in a factor graphical model where there exist a variable node set V (N variable nodes) and a factor node set F (M factor nodes) in each separation task, which is illustrated in Fig. 2.
Variable node set tasks Factor node set where R denotes the number of measured tasks, p(T1(s)) and p(T2(j)) are two priors in optimal transformation T1 and T2, respectively. Given the representation in the factor graph [45], the transformation of a particular distribution p(•) into a distribution domain T is defined as where d(•) denotes the Kullback-Leibler divergence.
To bypass the original global separation problem in (13) with marginalizing functions, we utilize an alternative quadratic approximation and Taylor series expansions for solving the separation problem.
Correspondingly, the proposed algorithm is divided into two phases-approximation and shrinkage, which is illustrated in detail as follow.

Approximation
In the min-sum manner, the belief propagation posteriors of individual signal s are estimated by the local denote the passing message between the factor node and the variable node.
Then, in view of the opposite direction for passing message at the t th iteration, the update rules of the expectation propagation [46] at the (t+1) th iteration are expressed as In this case, each updated passing message may be approximated as a complex Gaussian distribution, and the closed marginal posteriors are denoted as Under the conditions of fixed point solutions for the cost function in (11)  where the iterative form Art is complied with an independent identically distribution of Ar , τr denotes a moderate threshold, which illustrates the overall degree of recovery error with the parameter ς>0.
Here, we define as Then, we neglect the high order term in the common AMP after the state evolving is altered the estimation, and the residual error er with Onsager term [49] is approximated through the quadratic Taylor extension, which is obtained by where bt denotes a reactive factor.
Furthermore, a universal framework of sparse separation for interference is needed to fit all types and present the block-sparse (clustered row-sparse), element-sparse, and finite-difference characteristics simultaneously whereas the relatively constant signal is deemed as infinite difference sparsity in a transitory period, such as partial-band interference around the center tone interfering will become a continuously clustered row-sparse (block sparse) interference, multi-tone interference with marked sparsity can be considered as an element-sparse sporadically interference, and comb-spectrum interference is decorated finite-difference with signal, which forms the diverse ISTs for sparse separation.

1) Separation of finite-difference and block sparsity
For the finite-difference signal s based on the exponential prior, a total-variation soft threshold [50] is applied to estimate the iterative optimum, which is evaluated by And for the block sparse interference j based on the block Gaussian prior, a block soft threshold [51] is applied to pursue the group sparsity, and the estimation is taken from where B denotes the number of clustered elements in row (also named block size).

2) Separation of finite-difference and element sparsity
Similarly, the iterative estimation for the finite-difference signal s is evaluated by a total-variation

Simulations and Results
In this section, we present exemplary comparisons to illustrate the effectiveness and complexity To generate a synthetic dataset, various signals and several types of interference are considered for diversity measurement. For signals, the information rate was 1 kbit/s, the sampling frequency was 1 kHz, the length of information code was 1,000, the modulation pattern was BCSK, the length of m-sequence was 1,024, and the base function was generated by hard threshold with mean amplitude.
Moreover, the initial signal-to-noise ratio was 3 dB, and the transmitter suffered slight narrowband interference with interference-to-noise ratio -3 dB. The typical six-tap multipath channel model was specified by cost207TUx6 for typical urban areas with the Rician fading. As for the sparse diversities of interference parameters, the three components of element-sparse multi-tone (Mt) interference with interval 5 kHz were represented as discrete interference in the simulations. The bandwidth of block-sparse partial-band (Pb) interference was 50 kHz, and its frequency overlap remained 5%.
Moreover, the initial carrier frequency of finite-difference chirp (Cp) interference was 75 kHz and its adjustable parameter was 130 whereas the initial carrier frequency of block-sparse comb-spectrum (Cs) interference was 25 kHz and its adjustable parameter was 160. More commonly, the response of element-sparse impulse (Ip) interference was (4σ(t), 2σ(t), 1σ(t)) and its power was -8 dBW under environment noise. The initial carrier frequency of finite-difference amplitude modulation for noise (An) interference was 80 kHz and its power was -5 dBW under environment noise. The Bernoulli matrix was chosen as the measurement in the numerical experiments. And the initial signal-to-noise-ratio is 3 dB whereas the initial interference-to-noise-ratio is 8 dB where the environment noise is assumed as a Gaussian white noise channel. The detailed settings of interference datasets considered in this subsection are listed in Table 1.  interference; (f) An interference.

Sparsity Measurement and Transformation Learning
We investigated the self-adaption and superiority of our proposed ELOBA algorithm for different sparsity of interference, and comparisons of transformation learning were divided into two cases -the diversity of algorithms and base orthogonality. The representative types of interference with distinct sparsity were selected, which includes the element sparse Mt interference and Ip interference, the block sparse Cs interference and the finite-difference An interference. More obviously, the obtained bases of aforementioned four types of interference were converted into gray-scale images, and the middle parts (6╳6 diamonds) of the base images were present in Fig. 2, which display the sensitivity of interference for particular frequency and spatial locations. More specifically, Fig.3 presents the relative estimation errors for the representative interference in transformation analysis, which is obtained by the noiseless and noise measurement, respectively.
The relative estimation error is defined as ||j * -j||F / ||j||F, and data statistics were analyzed in Table 2.
These results demonstrate the recovery diversities of ELABO applied to multiple forms of interference and the relative errors were obtained ultimately in a convergent way. Obviously, their recovery performances were affected by the measured noise and degraded a lot.   and 11.1%, respectively, which present the superiority of performance for interference representation.

Sparsity Analysis and Interference Separation
We studied the performance of our proposed ISAR-AMP algorithm for different distribution sparsity of interference, and comparisons of interference separation were divided into three cases according to the intrinsic sparsity which includes the element sparse interference, the block sparse interference and the finite-difference interference. In addition, we keep TDCS signals constant without the impact of interference at receiver and they only vary with the surrounding spectrum at transmitter.
The projection matrix is constructed by a Gaussian distribution when the signal-to-noise ratio is 5 dB, and then the matrix is normalized the rows to a unit value. For recovery indicator, we select the relative overall estimation error Oval_err to evaluate the threshold τr of interference separation in (61), and the relative separation distance error Dist_err during the iterations is defined by Dist_err =(||s t -s||F +|| j t -j||F)/||s + j||F . Furthermore, the relative separation error for interference and signal j_err , s_err are normalized respectively by j_err =|| j t+1 -j t ||F/|| j t+1 ||F , s_err =||s t+1 -s t ||F/|| s t+1 ||F.
The estimation and separation results for block sparse interference and finite-difference signal in  We present another separation case for element-sparse interference and finite-difference signal in Fig. 8-9. The results show that, whenever the Mt interference or the Ip interference is selected, the separation examples achieved the bound similarly. Moreover, the distance error Dist_err and the interference error j_err decreased gradually and rapidly converged while the overall error Oval_err and the signal error s_err are not readily discernible for the ideal state. The tendency difference of iterative results is coming from the fact that the spectrum with random distribution for Mt interference requires more learning steps than the clustered distribution for Ip interference, which indicates that the detailed frequency allocations will promote the Mt interference separation. More generally applicable results of finite-difference interference and finite-difference signal are presented in Fig. 10-11, which presents the relative estimation and separation errors obtained with respect to iteration epochs when applying ISAR-AMP algorithm. We note that the proposed method achieved the finite-difference interference separation eventually and the distance error Dist_err converged to the ideal state. Furthermore, whenever the Cp interference or the An interference is confirmed, the interference error j_err and the signal error s_err are impacted by random searching steps during the learning period without obtaining the priors in advance, which results in the overall error Oval_err with great fluctuation.  Figure 11. Relative estimation and separation errors when applying ISAR-AMP algorithm to the finite-difference sparse An interference: (a) relative overall estimation error; (b) relative separation distance error; (c) relative separation error for interference;(d) relative separation error for signal.

Separation Performance and Running Time
Since our proposed ISAR-AMP algorithm is designed for the interference separation with different distribution sparsity, the experiments were also including three sparse separation cases above, and the performance of estimation was compared with MIX-AMP, IMIX-AMP and SE-AMP algorithms. Here, the selected datasets are same as section 6.2 and the projection matrix is normalized from the i.i.d.-Gaussian distribution when the signal-to-noise ratio is 5 dB. For estimation indicator, we select the normalized mean squared error for signal and interference NMSE_s, NMSE_j are respectively given by NMSE_s =(∑||ŝr-sr||F /||sr ||F )/R, NMSE_j=(∑||ĵr-jr||F/|| j r||F)/R.  More generally, the estimation and separation results for element sparse interference and finite-difference signal are illustrated in Fig. 13, which are applied with ISAR-AMP, MIX-AMP, IMIX-AMP and SE-AMP algorithms. We note that the final overall error of the proposed ISAR-AMP algorithm is obtained by 1.735e -08 by estimating the discrete difference of the element sparse Mt interference, which improved the separation accuracy greatly than the SE-AMP. In addition, due to similar reduction for the number of messages during the iterations, it is also shown that the proposed algorithm achieves the approximated convergence speed with the ISAR-AMP algorithm whereas they still achieve much higher gains than MIX-AMP or IMIX-AMP algorithm. Moreover, it can be observed that the complexity of proposed approach demonstrates its superiority over the existing      Fig. 16-18.
The MSEs of block sparse Pb interference separation in Fig. 15 show that the performance of ISAR-AMP is impacted by frequency, amplitude and INRs diversely in some degree. With the increase of the interference frequency in Fig. 15 (a), the separation performance present periodical regularity due to the bandwidth of Pb interference keeps constant. In contrast, the separation performance is get improvement with the increase of interference amplitude whereas the increase of the MSEs is much slower with a positive correlation between INRs. Hence, it can be concluded that the proposed IASR-AMP algorithm will greatly reduce the separation errors of measurement data required for accurate recovery when encountered high power or low INRs of Pb interference, achieving higher anti-jamming efficiency than conventional counterparts. The separation performance of the proposed method for element sparse Mt interference versus the frequency, amplitude and INRs is depicted in Fig. 16. We noted that the separation performance cannot present obvious regularity with the increase of the interference frequency whereas the proposed algorithm reaches less separation errors when the interference amplitude and INRs grow more, respectively. It is thus validated that the proposed method can accurately separate the Mt interference with much higher power and high INRs under the conventional spectrum estimation. Since the measured Mt interference only occupies 3 locations with 25-30 kHz intervals in the frequency spectrum, it can be inferred that the proposed method is capable of effectively separating and recovering at least 30 kHz out-band Mt interference. To further exploit more complex distribution of sparsity interference, we compared the MSE performance of the proposed method for block sparse Cs interference under different impact factors, which is illustrated in Fig. 17. Apart from the influence of interference amplitude, the separation performance of the ISAR-AMP method achieves slight improvement or degradation with the increase of frequency or INRs. Furthermore, it can be observed that the proposed algorithm for dealing with the high power components of Cs interference significantly outperforms the state-of-the-art counterparts and even presents positive reciprocal relation with interference amplitude. The reason is that the component accumulations of Cs interference have enough discriminating difference between various signals even though lacking unknown parameters. This implies that the Cs can be more effectively separated and recovered in the proposed message passing framework of sparse approximation using the iterative learning.

Conclusions
In this paper, a novel sparse learning method-based cognitive transformation framework of interference separation is formulated for the coexistence of diverse interference and TDCS systems.
The combinatorial optimization problem of multiform interference separation is efficiently and accurately solved by the proposed sparse learning algorithms of ELABO and ISAR-AMP, which iteratively learns the prior sparse probability distribution, i.e., the sparse pattern, of the interference support by minimizing the loss function of state evolution. By imposing spectrum analysis on the sparsity states, the intricate interference separation problem is generalized from single separation and simultaneous separation. Furthermore, the principal component analysis and Bayesian perspective in orthogonal base learning ELABO is exploited to simultaneously recover the multiple interference and TDCS signals, which achieves better separation accuracy and iterative convergence. Moreover, the proposed novel interference separation algorithm based on state evolving iteratively learns the belief propagation posteriors in the min-sum manner, and the number of passing messages is shrunk by iterative shrinkage threshold. It is verified by theoretical analysis and numerical simulation results that the proposed algorithms outperform state-of-the-art counterparts in separation accuracy and computational complexity, which is especially suitable for interference separation and elimination of broadband communication.

Conflicts of Interest:
The authors declare no conflict of interest.