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An adaptive combination constrained proportionate normalized maximum correntropy criterion algorithm for sparse channel estimations
EURASIP Journal on Advances in Signal Processing volume 2018, Article number: 58 (2018)
Abstract
An adaptive combination constrained proportionate normalized maximum correntropy criterion (ACCPNMCC) algorithm is proposed for sparse multipath channel estimation under mixed Gaussian noise environment. The developed ACCPNMCC algorithm is implemented by incorporating an adaptive combination function into the cost function of the proportionate normalized maximum correntropy criterion (PNMCC) algorithm to create a new penalty on the filter coefficients according to the devised threshold, which is based on the proportionatetype adaptive filter techniques and compressive sensing (CS) concept. The derivation of the proposed ACCPNMCC algorithm is mathematically presented, and various simulation experiments have been carried out to investigate the performance of the proposed ACCPNMCC algorithm. The experimental results show that our ACCPNMCC algorithm outperforms the PNMCC and sparse PNMCC algorithms for sparse multipath channel estimation applications.
1 Introduction
In recent years, wireless communication is the fastest growing and most widely used technology in the field of information communication. As for wireless communication channels, radio wave propagation through the channels will produce a direct wave and a ground reflection wave, which are the most common waves in the propagation process. In addition, there will also be scattered waves caused by various obstacles in the propagation paths [1, 2]. These phenomena eventually result in the multipath effect [2–5], which often occurs and is often very serious. Such effect may lead to signal delays and to reduce communication quality. Furthermore, measurements show that the broadband channels are described as sparse multipath channels, which are very common in the field of wireless communication [6, 7]. The general characteristic of these sparse multipath channels is that most of the channel impulse response (CIR) coefficients are equal to zero. The very few nonzero coefficients are termed active coefficients [2, 7, 8]. Adaptive filters (AF) have been widely applied to the estimation of sparse channels to improve the quality of wireless communications [9–12]. The most traditional AF algorithm is the least mean square (LMS), which has been derived to minimize the mean square estimation error [9, 13]. It is widely employed in realworld applications due to its simplicity. However, LMS performance degrades in low signaltonoise (SNR) situations and in sparse channel estimation. Motivated by compressive sensing principles [14], a series of sparse LMS algorithms have been recently proposed by incorporating l_{0}norm, l_{1}norm, or l_{p}norm penalties into the LMS cost function [14–21]. These algorithms have been shown to lead to superior performances than LMS both in steadystate error (SSE) and in convergence speed for sparse multipath channel estimation. Nevertheless, the performances of such algorithms tend to degrade in nonGaussian white noise environments (NGWNE), a frequently encountered situation in the real wireless communication systems [22–24].
Some recent efforts have also been directed to improve the performance of adaptive filters in the presence of nonGaussian and impulsive disturbances. One of such efforts was to replace the MSE with information theoretic, entropyrelated, cost functions [25–28]. At present, most entropybased AF algorithms have been derived using the maximum correntropy criterion (MCC) and the minimum error entropy (MEE) criteria [25–27, 29–31]. Though MEEbased algorithms may lead to good estimation performance, they require a prohibitive computational effort for realtime application. MCCbased algorithms are less complex and have been successfully applied to channel estimation [26]. The MCC algorithm employs the correntropy as a local similarity measure, which tends to increase the robustness of the algorithm to outliers. It has been applied to channel estimation under NGWNE [26, 32, 33]. More recently, a generalized correntropy measure has been proposed which replaces the Gaussian kernel of the MCC with a generalized Gaussian kernel. The generalized MCC (GMCC) algorithm has been shown to outperform MCC in certain situations at the cost of an increase in computational complexity [34]. A normalized version of the MCC algorithm (NMCC) has also been proposed [35, 36] to make the convergence control less dependent on the input power.
To address the issue of sparse channel estimation in nonGaussian noise, sparse MCC algorithms have been proposed by incorporating different norm penalties into the MCC cost function. Similar to the zero attracting LMS (ZALMS) algorithm [16], in [33], an l_{1}norm penalty was introduced into the MCC cost function as a zero attractor. The resulting algorithm was called zeroattracting MCC (ZAMCC) algorithm. Then, a reweighting controlling factor ξ was incorporated into the ZAMCC algorithm, yielding the reweighting ZAMCC (RZAMCC) algorithm [33]. The ZAMCC and RZAMCC were shown to improve the steadystate and convergence performances of MCC in estimating multipath channels. Different zero attractors were added to MCC in [33, 37, 38] using an l_{0}norm and a correntropyinduced metric (CIM).
One drawback of the ZAMCC algorithms is that it uniformly attracts all the channel coefficients to zero, which may result in a rather large estimation error for less sparse channels. Also, the selection of a good learning rate is not simple for these algorithms. Zeroattracting NMCC (ZANMCC) and reweighting ZAMCC (RZANMCC) algorithms have also been developed in [39]. Moreover, a soft parameter function has also been introduced into the NMCC algorithm to attenuate the uniform attraction of coefficients to zero. The resulting algorithm is called SPFNMCC [40]. The SPFNMCC algorithm improved the steadystate performance of NMCC, but tends to converge slowly due to the large discrepancy between coefficient values in sparse channels. To remedy this drawback, proportionatetype [41, 42] NMCC algorithms were proposed. The proportionate NMCC (PNMCC) algorithm was developed in [43, 44] and led to faster convergence speeds than the MCC and NMCC algorithms since the filter coefficients are proportionately updated according to the magnitude of the estimates of the channel coefficients. Unfortunately, however, the convergence of the PNMCC algorithm tends to slow down an initial convergence period [43, 44], a known effect of the proportionate type of adaptation.
In this paper, we propose a new adaptive algorithm by effectively combining some of the aforementioned strategies to derive a method that combines the properties of the l_{0}norm, the l_{1}norm, and the proportionate adaptation. This is done by integrating an adaptive combination function (ACF) to the PNMCC cost function to create a new adaptive zero attractor. The resulting algorithm is called Adaptive Combination Constrained Proportionate Normalized Maximum Correntropy Criterion (ACCPNMCC) algorithm. The performance of the proposed ACCPNMCC algorithm is investigated through several different simulation experiments, and it is compared to the MCC, NMCC, PNMCC, and sparse PNMCC algorithms in terms of the SSE and convergence speed. The simulation results illustrate the potential of the ACCPNMCC algorithm to provide a superior estimation performance for sparse multipath channel estimation under nonGaussian and impulsive noise environments.
The paper is organized as follows. The NMCC and PNMCC algorithms are briefly reviewed in Section 2. In Section 2, we derive the proposed ACCPNMCC algorithm in the context of sparse multipath channel estimation. Simulation experiments are presented in Section 3 to illustrate the algorithm performance, and we finally summarize our work in Section 4.
2 Channel estimation algorithms
2.1 The NMCC and PNMCC algorithms
Consider the estimation of an unknown channel with impulse response given by g_{0}=[g_{1},g_{2},g_{3},⋯,g_{M}]^{T}. The Mdimensional input vector is \(\textbf {x}\left ({n}\right) = {\left [{x\left ({n}\right),x\left ({n  1}\right),x\left ({n  2}\right), \cdots \!,x\left ({n  M + 1}\right)} \right ]^{^{T}}}\), and the observed channel output is d(n)=x^{T}(n)g_{0}+r(n), where r(n) is the additive noise. Denoting \(\hat {\mathbf {g}}\left (n \right)\) the channel response estimate, the estimation error is given by \(e\left ({n}\right) = d\left ({n}\right)  {\textbf {x}^{{T}}}\left ({n}\right)\hat {\mathbf {g}}\left ({n}\right)\).
It has been show in [36] that the NMCC weight update equation is the solution of the following problem:
where ∥·∥^{2} represents the Euclidean norm of a vector, \(\hat e\left (n \right) = d\left (n \right)  {\textbf {x}^{{T}}}\left (n \right)\hat {\mathbf {g}}\left ({n + 1} \right)\), the parameter σ>0 is the bandwidth of the Gaussian kernel used to evaluate the instantaneous correntropy between the observed signal and the estimator output [26], and α_{NMCC}>0 is a design parameter that affects the quality of estimation.
Solving (1) using the Lagrange multiplier technique [36] leads to the NMCC weight update equation
where α_{NMCC} is the NMCC step size. If α_{NMCC}=β∥x(n)∥^{2}, (1) becomes the MCC algorithm update [26].
One way to modify the NMCC algorithm to improve adaptation performance for sparse channels is to adjust each adaptive weight using a different step size, what can be done through a diagonal step size matrix
The proportionate NMCC (PNMCC) algorithm assigns the step sizes m_{i} such that
where γ_{i}(n), i=1,…,M are given by
with κ and η positive constants with typical values of κ=5/M and η=0.01 [42–45]. Parameter κ is used to prevent the coefficients from stalling when they are much smaller than the largest one, while η avoids the stalling of all coefficients when \(\hat {\mathbf {g}}\left (n \right) = {\textbf {0}_{M \times 1}}\) at initialization. The weight update equation for the PNMCC algorithm is
where θ is a regularization parameter, and α_{PNMCC} is the PNMCC step size. Although the PNMCC algorithm improves the performance of the NMCC, its convergence rate tends to slow down after an initial period of fast convergence, a well known issue affecting the conventional (MSE based) PNLMS algorithm [21]. This basically happens due to the dominant effect of the inactive taps, whose step sizes tend to be excessively reduced as their weight estimates get closer to zero, reducing the convergence speed.
Among the several different strategies proposed to increase the PNLMS convergence rate after the initial convergence period, the l_{1} norm penalty recently proposed in [21] has led to very interesting results. In [21], the l_{1} norm of the weight vector premultiplied by the inverse of the step size matrix has been used. This makes the l_{1} penalty to be governed primarily by the inactive taps. The constrained optimization problem solved in [21] to yield the ZAPNLMS algorithm is
One problem of enforcing sparsity through an l_{1} norm constraint is that the resulting algorithm may lose performance when identifying responses with different levels of sparsity. One approach recently proposed in [46] to mitigate this problem for the LMS algorithm is to employ a socalled nonuniform norm constraint, inspired in the pnorm frequently used in compressive sensing. The sparsity of the weight vector is enforced through an adaptive combination function (ACF) that is a mixture of l_{0}norm and l_{1}norm and whose general form is
where 0≤q≤1. We can see that the definition of the ACF is different from the classic Euclidean norm [47]. When q→0, it becomes to be
which approximates to be l_{0}norm that can count the number of nonzero coefficients in the sparse multipath channels. Besides, the ACF is the l_{1}norm, which is described as
2.2 The proposed ACCPNMCC algorithm
To address the problems of estimating a sparse response with possibly different levels of sparsity, obtaining a good convergence rate from initialization to steadystate, and providing robustness to impulsive noise, we propose to integrate an adaptive combination function (ACF) into the PNMCC’s cost function. This approach leads to the design of a new correntropybased zero attractor algorithm that combines the techniques of [21, 46–50]. The proposed ACCPNMCC algorithm is derived in the following.
Integrating the ACF into the cost function of the PNMCC algorithm with the weight scaling as in [21], we propose the following optimization problem:
where γ_{ACC} is a tradeoff parameter between the steadystate performance and the ACF penalty. Using Lagrange multipliers, the cost function associated with this optimization problem is
where λ_{ACC} is the Lagrange multiplier.
As a first step to minimize (11) with respect to \({\hat {\mathbf {g}}\left ({n + 1} \right)}\), we impose
which yields the recursion
Now, imposing the equality constraint
and solving for λ_{ACC} yields
Substituting (15) in (13) and rearranging the terms yields
In (16), we observe that the elements of the matrix \({  \frac {{\textbf {x}(n){\textbf {x}^{T}}(n)\textbf {G}(n)}}{{{\textbf {x}^{T}}(n)\textbf {G}(n)\textbf {x}(n)}}}\) tend to be very small for reasonably large values of M. Hence, we simplify the algorithm recursion using the approximation \(\textbf {I}{  \frac {{\textbf {x}(n){\textbf {x}^{T}}(n)\textbf {G}(n)}}{{{\textbf {x}^{T}}(n)\textbf {G}(n)\textbf {x}(n)}}}\approx \textbf {I}\). The same approximation has been successfully used in [21]. Then, (16) becomes
Evaluation of the gradient vector in (17) yields
where ⊙ denotes the Hadamard product, and
As (18) is a function of \(\hat {\mathbf {g}}(n+1)\), it substitution in (17) would not yield a recursive update for \(\hat {\mathbf {g}}(n)\). To obtain an implementable recursive update, we replace \(\hat {\mathbf {g}}(n+1)\) in (18) with \(\hat {\mathbf {g}}(n)\). This approximation tends to be reasonable for practical step sizes. The resulting weight update equation is then
Though recursion (19) should fulfill the objective of minimizing (11) at convergence, it is still too complex for realtime implementations due to the last term on the r.h.s., which is responsible for the zeroattracting property of the algorithm. Since we propose to use either an l_{0}norm or an l_{1}norm, q must assume a value equal to zero or one, respectively. For q=0, the last term in (19) is equal to zero. For q=1, it is equal to \(\gamma _{\textrm {ACC}}\, {\boldsymbol {\psi }}(n) \odot \text {sgn}\left (\hat {\mathbf {g}}(n)\right)\), which is a vector whose ith element is given by \(\gamma _{\textrm {ACC}}\,\text {sgn}\left (\hat {g}_{i}(n)\right)\).
A final decision to be made is how to choose the value of q. According to the definitions (8) and (9), it should affect the norm of the whole vector \(\hat {\mathbf {g}}(n)\). However, there is no clear measure to set the value of q to obtain good performance in practical applications. We propose instead to determine a different value of q for each element of \(\hat {\mathbf {g}}(n)\) and based on a threshold that depends on the recent behavior of each weight estimate:
Similar solutions have been employed in [47, 49, 50] to regularize different model misfit cost functions.
Given (20), and denoting becomes q_{i} the value of q applied to q_{i}(n), we set q_{i}=0 if \({{\hat {g}}_{i}}(n) > {h_{i}}(n)\), and q_{i}=1 if \({{\hat {g}}_{i}}(n) < {h_{i}}(n)\). Hence, those coefficients for which \({{\hat {g}}_{i}}(n) > {h_{i}}(n)\) will contribute to this term as if the norm l_{0} had been used. Conversely, those coefficients for which \({{\hat {g}}_{i}}(n) < {h_{i}}(n)\) will contribute to this term as if the norm l_{1} had been used.
We implement this strategy by defining a diagonal matrix
with elements
so that the last r.h.s. term of (19) becomes \(  {\gamma _{{\textrm {ACC}}}}\,\textbf {F}\,{\text {sgn}} \left ({\hat {\mathbf {g}}(n)} \right)\).
Finally, it has been verified [50] that the zeroattracting ability of \(  {\gamma _{\textrm {ACC}}}\textbf {F}{\text {sgn}} \left ({\hat {\mathbf {g}}(n)} \right)\) can be improved by a reweighting factor used in the RZAMCC algorithm. By doing that, the updating equation of the proposed ACCPNMCC algorithm becomes
where θ>0 is a very small positive constant, ξ is the reweighting factor, and α_{ACC} denotes the step size.
The last r.h.s. term in (23) is the newly designed zero attractor. Each coefficient is treated as if a different norm penalty had been applied to the cost function. This is done by comparing each channel coefficient estimate to the designed threshold. In the following, we will show that the proposed zero attractor speedsup the convergence of the small coefficients after the initial convergence so that the proposed ACCPNMCC algorithm outperforms the PNMCC algorithm. In addition, a better sparse multipath channel estimation performance can be obtained by properly selecting the value of the reweighting factor ξ.
3 Results and discussions
In this section, several different simulation experiments are carried out to investigate the performance of the proposed ACCPNMCC algorithm. The input signal x(n) is a random signal, while the noise signal r(n) is a mixed Gaussian signal with distribution [33, 37, 40]
where \( N\left ({{\mu _{i}},\nu _{i}^{2}} \right)\left ({i = 1,2} \right)\) denotes a Gaussian distribution with mean μ_{i} and variance \({\nu _{i}^{2}}\). The mixing parameter χ is used to balance the two Gaussian noises. In our simulation experiments, these parameters are set to be \(\left ({{\mu _{1}},{\mu _{2}},\nu _{1}^{2},\nu _{2}^{2},\chi } \right) = \left ({0,0,0.05,20,0.05} \right)\).
The performance of the proposed ACCPNMCC algorithm is evaluated through the steadystate mean square deviation (MSD), which is defined as
From the updating Eq. (23), we notice that there are several key parameters which may affect the ACCPNMCC performance and thus must be properly selected. To better select these parameters, we experimentally analyzed their effect on the MSD performance of the ACCPNMCC algorithm. Herein, the regularization parameters include α_{ACC}, γ_{ACC}, and ξ. In this paper, each parameter is optimized to obtain small MSD. At each optimization, only one parameter is changed while other parameters are set to the optimal values. In the experiments designed for this study, the number of nonzero coefficients is 1 and the total length of the unknown channel is set to be M=16. Firstly, γ_{ACC} is studied under different SNRs [51]. In this simulation, we have set α_{ACC}=0.27 and ξ=5. The simulation results are shown in Fig. 1. It is observed that the MSD with different values of γ_{ACC} decreases as the SNR increases from 1 to 40 dB. The SNR increases with different slopes for different values of γ_{ACC}. This means that the effect of γ_{ACC} is dependent on the SNR. As can be verified from Fig. 1, γ_{ACC}=5×10^{−4} yields the smallest MSD for a SNR of 30 dB. Hence, we study the effect of the step size value α_{ACC} on the MSD performance for SNR = 30 dB and γ_{ACC}=5×10^{−4}. The corresponding MSD is shown in Fig. 2. We can verify that the MSD gradually decreases as the step size increases from 1×10^{−3} to 5×10^{−2}, while the MSD increases for step sizes greater than 5×10^{−2}. This indicates the importance of a proper choice of step size value. The effect of reweighting factor ξ is studied for a simple case on the zero attractor term. The results are shown in Fig. 3. From Fig. 3, we note that the reweighting factor mainly attracts to zero the coefficients that are smaller than the defined threshold, while the zero attractor term becomes zero for the coefficients with values greater than the threshold. In addition, the shrinkage ability of the reweighting factor decreases with ξ ranging from 0 to 25. Thus, the reweighting factor in the proposed ACCPNMCC algorithm exerts strong zero attraction on the relatively small coefficients to improve their convergence after the initial transient.
Based on the above parameter selections, the MSD performance of the proposed ACCPNMCC algorithm was verified for different sparsity levels at SNR = 30 dB. For comparison purposes, the conventional MCC, NMCC, PNMCC, and sparse PNMCC algorithms were also considered in the simulations. The sparse PNMCC (ZAPNMCC and RZAPNMCC) algorithms were obtained by using the l_{1}norm and are briefly described in the Appendix. In this experiment, the sparsity level of sparse multipath channel was set as K. In other words, K represents the number of nonzero coefficients in the sparse multipath channel. Firstly, there is only one nonzero channel coefficient (K=1), which is randomly distributed within the unknown sparse multipath channel. To obtain the same initial convergence speed for all the compared algorithms, the related parameters for the MCC, NMCC, PNMCC, ZAPNMCC, and RZAPNMCC algorithms were set as follows: α_{MCC}=0.03, α_{NMCC}=0.4, α_{PNMCC}=α_{ZA}=α_{RZA}=0.3, γ_{ZA}=5×10^{−5}, γ_{RZA}=1×10^{−4}, α_{ACC}=0.27, γ_{ACC}=5×10^{−4}, and ξ=5. α_{MCC}, α_{ZA}, and α_{RZA} are the step sizes of the MCC, ZAPNMCC, and RZAPNMCC algorithms, respectively. Moreover, γ_{ZA} and γ_{RZA} are the tradeoff parameters of the ZAPNMCC and RZAPNMCC algorithms, respectively. The MSD performance of the proposed ACCPNMCC algorithm for K=1 is given in Fig. 4. It is observed that ACCPNMCC achieves the lowest steadystate MSD when all the compared algorithms have the same initial convergence speed. Then, the corresponding steadystate MSDs for K=2, 4, and 8 are shown in Figs. 5, 6, and 7. We note that the steadystate MSDs of the ZAPNMCC, RZAPNMCC and ACCPNMCC algorithms increased with the sparsity level increasing from K=1 to K=8. However, it is worth noting that the MSD of the ACCPNMCC algorithm is still lower than those of the other algorithms. In addition, the effects on the ACCPNMCC algorithm with a sparsity level K/M are shown in Fig. 8 to more intuitively understand the effect of sparsity level on the MSDs of compared algorithms. It is found that the MSDs of ZAPNMCC, RZAPNMCC, and ACCPNMCC gradually increase as the sparsity level K/M increases from 0.0625 to 0.5, which is similar to the results in Figs. 4, 5, 6, and 7. The MSDs of the MCC, NMCC, and PNMCC algorithms remain almost invariant, as they do not have any zero attractor term that is sensitive to sparsity K/M. However, one should note that the MSDs of ZAPNMCC and ACCPNMCC are almost equal to 1×10^{−4} when K/M=0. This is because the ACCPNMCC algorithm uses l_{1}norm when K/M=0, like the ZAPNMCC algorithm. When K/M is greater than 0.0625, the ACCPNMCC algorithm is always better than the other algorithms. Thus, we can say that the ACCPNMCC algorithm is suitable for sparse multipath channel estimation in practical applications.
The third experiment considered the application of ACCPNMCC to echo cancelation. The typical echo path of a 256tap channel with 16 nonzero coefficients is shown in Fig. 9, which is also known as blocksparse path [52]. However, the data reusing, sign, and memory schemes have been considered in [52], which has high complexity. Additionally, the sparsity is not given by a formula. Herein, the sparsity characteristic of the echo channel is measured by \({\vartheta _{12}} = \frac {M}{{M  \sqrt {M}}}\left ({1  {{{{\left \ \textbf {g} \right \}_{1}}} / {\sqrt {M}{{\left \ \textbf {g} \right \}_{2}}}}}\right)\) [40, 45, 49]. In this experiment, we use 𝜗_{12}=0.8222 for the first 8000 iterations, while 𝜗_{12}=0.7362 are set for the after 8000 iterations. Other simulation parameters are α_{MCC}=0.0055, α_{NMCC}=1.3, α_{PNMCC}=α_{ZA}=α_{RZA}=0.9, γ_{ZA}=1×10^{−6}, γ_{RZA}=1.5×10^{−6}, α_{ACC}=0.8, γ_{ACC}=5×10^{−6}, and ξ=5. The corresponding steadystate MSDs are shown in Fig. 9. It can be seen that ACCPNMCC outperforms the other algorithms in terms of both steadystate MSD and convergence speed. Although the sparsity is changed from 0.8222 to 0.7362, the performance of the proposed ACCPNMCC algorithm is still superior to that of the other algorithms, indicating the effectiveness of the ACCPNMCC algorithm for echo cancelation.
At last, the computational complexity of the proposed ACCPNMCC algorithm is presented in Table 1 in comparison with the corresponding algorithms. Herein, the numerical complexities of the algorithms include multiplications, additions, exponentiations, divisions, and comparisons. Form Table 1, it can be seen that the computational complexity of the developed ACCPNMCC algorithm is slightly higher than that of the ZAPNMCC and RZAPNMCC algorithms, which is owing to the calculation of the ACF. However, the proposed ACCPNMCC algorithm obviously increases the convergence speed and reduces the MSD.
Based on the above experiment analysis, we infer that the proposed ACCPNMCC algorithm has a superior steadystate MSD performance and convergence speed for sparse multipath channel estimation applications. This is because ACCPNMCC utilizes the l_{0}norm penalty for the channel coefficients which are larger than our designed threshold, while it exerts the l_{1}norm penalty on the channel coefficients that are smaller than the designed threshold to attract relatively small coefficients to zero to improve the convergence. Thus, the proposed ACCPNMCC algorithm performed better than the other algorithms for sparse channel estimation.
4 Conclusions
In this paper, an ACCPNMCC algorithm has been proposed for sparse multipath channel estimation. The proposed ACCPNMCC algorithm exploits inherent sparsity features of sparse multipath channels by utilizing the designed zero attractor. The performance of the algorithm was investigated and compared with the performances of the MCC, NMCC, PNMCC, and sparse PNMCC algorithms for sparse multipath channel estimation. Experimental results illustrated that the proposed ACCPNMCC algorithm is superior to the competing algorithms in terms of both the steadystate MSD and convergence speed.
5 Appendix
Similar to the ZAMCC and ZALMS algorithms [16, 33, 37, 38], the l_{1}norm is introduced into the cost function of the PNMCC algorithm to develop the zeroattracting PNMCC (ZAPNMCC) algorithm. The cost function of the ZAPNMCC algorithm is
where λ_{ZA} is Lagrange multiplier. By using LMM, the updating equation of the ZAPNMCC can be written as
where α_{ZA} and γ_{ZA} are the step size and tradeoff parameter of the ZAPNMCC algorithm, respectively. Similarly to the RZAMCC and RZALMS algorithms [16, 33, 37, 38], a reweighting factor is introduced into the ZAPNMCC algorithm to develop the reweighting ZAPNMCC (RZAPNMCC) algorithm, the corresponding updating equation is
where ξ_{1} is reweighting factor, α_{RZA} denotes step size and γ_{RZA} represents tradeoff parameter of the RZAPNMCC algorithm.
Abbreviations
 ACCPNMCC:

An adaptive combination constrained proportionate normalized maximum correntropy criterion
 ACF:

Adaptive combination function
 AF:

Adaptive filter
 CIM:

Correntropy induced metric
 CIR:

Channel impulse response
 CS:

Compressive sensing
 GMCC:

Generalized maximum correntropy criterion
 LMS:

Least mean square
 MCC:

Maximum correntropy criterion
 MEE:

Minimum error entropy
 MSD:

Mean square deviation MSE: Mean square error
 NGWNE:

NonGaussian white noise environments
 PNMCC:

Proportionate normalized maximum correntropy criterion
 RZAMCC:

Reweighting zero attracting MCC
 SNR:

Signaltonoise
 SSE:

Steadystate error
 ZALMS:

Zeroattracting LMS
 ZAMCC:

Zeroattracting MCC
 ZANMCC:

Zeroattracting NMCC
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Acknowledgements
I would like to express our great appreciation to Prof. Felix Albu for his valuable suggestions for computing numerical complexities of the proposed ACCPNMCC algorithm in this research work.
Funding
This work was supported by the PhD Student Research and Innovation Fund of the Fundamental Research Funds for the Central Universities (HEUGIP201707), National Key Research and Development Program of China (2016YFE0111100), National Science Foundation of China (61571149), the Key Research and Development Program of Heilongjiang (GX17A016), the Science and Technology innovative Talents Foundation of Harbin (2016RAXXJ044), the Opening Fund of Acoustics Science and Technology Laboratory (grant vo. SSKF2016001) and China Postdoctoral Science Foundation (2017M620918).
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YW devised the code and did the simulation experiments, then she wrote the draft of this paper. YL helped to check the codes and simulations, and he also put forward to the idea of the ACCPNMCC algorithm. JCMB and XH helped to check the derivation of the formulas and improved the English of this paper. All the authors wrote this paper together, and they have read and approved the final manuscript.
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Wang, Y., Li, Y., M. Bermudez, J.C. et al. An adaptive combination constrained proportionate normalized maximum correntropy criterion algorithm for sparse channel estimations. EURASIP J. Adv. Signal Process. 2018, 58 (2018). https://doi.org/10.1186/s1363401805815
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DOI: https://doi.org/10.1186/s1363401805815