Smooth Approximation l 0-Norm Constrained Affine Projection Algorithm and Its Applications in Sparse Channel Estimation

We propose a smooth approximation l 0-norm constrained affine projection algorithm (SL0-APA) to improve the convergence speed and the steady-state error of affine projection algorithm (APA) for sparse channel estimation. The proposed algorithm ensures improved performance in terms of the convergence speed and the steady-state error via the combination of a smooth approximation l 0-norm (SL0) penalty on the coefficients into the standard APA cost function, which gives rise to a zero attractor that promotes the sparsity of the channel taps in the channel estimation and hence accelerates the convergence speed and reduces the steady-state error when the channel is sparse. The simulation results demonstrate that our proposed SL0-APA is superior to the standard APA and its sparsity-aware algorithms in terms of both the convergence speed and the steady-state behavior in a designated sparse channel. Furthermore, SL0-APA is shown to have smaller steady-state error than the previously proposed sparsity-aware algorithms when the number of nonzero taps in the sparse channel increases.


Introduction
With the development of wireless communication, there have been increasing demands for higher transmission rates in modern communication systems. This has led to the development of new standards for various wireless devices, such as smartphones, laptops, and iPads [1][2][3][4][5]. Given these requirements, broadband signal transmission is an essential technique for next-generation wireless communication systems [6]. In broadband wireless communications, a "hilly terrain" (HT) delay profile consists of a sparsely distributed multipath channel in which most of taps are zero or close to zero, while only a few taps are dominant [4]. In this paper, we consider the communication problems which involve the estimation and equalization of channels with a large delay spread but with a small nonzero support, which is also known as sparse channel estimation.
Recently, a rising method for sparse channel estimation has been proposed and extensively investigated by the use of compressed sensing (CS) to improve the performance of such sparse wireless communication channels [7][8][9]. We found that these CS channel estimation algorithms were sensitive to the channel interferences. Another effective class of methods that have been widely studied in channel estimation is adaptive filtering algorithms [10][11][12][13], such as least mean square (LMS), recursive least squares (RLS), and Kalman filter algorithms. However, these standard adaptive filtering algorithms cannot utilize the sparse property of the wireless communication channel and hence they perform poorly in dealing with the sparse signals. To utilize the sparse characteristic of such channels, some improved adaptive filtering algorithms by the use of partial updating techniques have been proposed and investigated in wireless communications [14][15][16]. However, this partial updating degraded the estimation performance in contrast to the standard LMS and RLS algorithms.
Motivated by the widely developed CS techniques [17,18], some efforts have been put into combining the CS technique into the adaptive filtering algorithms in order to improve the performance of standard adaptive filtering performance for sparse signal recovery. For example, a Kalman filter compressed sensing (KF-CS) algorithm has 2 The Scientific World Journal been proposed and applied in magnetic resonance imaging (MRI) by the combination of CS and standard Kalman filter [19]. In this algorithm, Kalman filter estimates the support set which has significant effect on the estimator errors. Furthermore, another algorithm denoted as least square compressed sensing (LS-CS) has been developed and well investigated by using the CS and RLS techniques [20,21]. Unfortunately, these algorithms are highly complex because of the computational complexity of Kalman filter and RLS algorithms. LMS algorithm has attracted much more attention in recent years due to its low computational complexity and reliable recovery capability. Inspired by the CS theory [17,18] and the KF-CS and LS-CS algorithms, several sparsity-aware LMS algorithms have been proposed with additional norm constrained terms in the cost function of standard LMS algorithms [6,[22][23][24][25][26][27]. It was found in these studies that these linear constrained sparsity-aware LMS algorithms can achieve faster convergence speed and better steady-state performance compared to the standard LMS algorithm. However, these sparsity-aware LMS algorithms are sensitive to the noise and the sparsity characteristics of the channel, which results in high steady-state misadjustment due to the estimation error that occurs in the adaptation. The affine projection algorithm (APA) is another popular method in adaptive filtering applications [28][29][30][31], with its complexity and estimation performance intermediary between the LMS and RLS algorithms. The APA reuses old data resulting in fast convergence, and is also an improved normalized LMS (NLMS) algorithm that converges faster than the standard LMS algorithm. Subsequently, 1 -norm penalized APA has been proposed to render the standard APA suitable for sparse signal estimation applications [32]. However, these 1 -norm penalized APAs impose the condition that the proportion of nonzero taps must be very small as compared to the proportion of dominant taps in the associated parameter vector in channel estimation.
In this paper, we propose a smooth approximation 0norm constrained affine projection (SL0-APA) algorithm for sparse channel estimation. The proposed SL0-APA is similar to the algorithms proposed in [32], which are known as zero-attracting affine projection algorithm (ZA-APA) and reweighted zero-attracting affine projection algorithm (RZA-APA). It differs by the regularization term which is a smooth approximation 0 -norm obtained from a continuous function that is an accurate approximation of 0 -norm. By exploiting the information of the sparsity channel and using the concepts of the smooth approximation of 0 -norm, we can improve the performance of the previous sparsity-aware APAs with respect to both the convergence speed and the steady-state performance. We also provide a convergence analysis and the mean-square-error analysis of our proposed SL0-APA. Furthermore, we experimentally investigate the effect of adding a smooth approximation 0 -norm penalty term to the cost function on learning the convergence behavior and the steady-state error performance of the SL0-APA. Accordingly, we experimentally illustrate that the SL0-APA is superior to ZA-APA and RZA-APA in terms of steadystate error and the convergence speed. Besides, the theoretical analysis is also presented and compared to the computer  simulation results. Finally, the computational complexity of the proposed SL0-APA is mathematically given and is experimentally evaluated.
The remainder of the paper is structured as follows. Section 2 briefly reviews the standard APA, ZA-APA, and RZA-APA based on a sparse multipath communication system. In Section 3, we first propose a SL0-APA by the use of a smooth approximation 0 -norm penalty on the cost function of the standard APA. Next, we provide a theoretical expression of the convergence analysis and the mean-squareerror (MSE) analysis of our proposed SL0-APA based on the energy-conservation approach. In Section 4, the proposed SL0-APA is experimentally investigated over a sparse channel to demonstrate the estimation performance of the SL0-APA, including the convergence speed, steady-state error, and the computational complexity. Finally, Section 5 is the conclusion.

Conventional Channel Estimation Algorithms
In this section, we consider a sparse multipath communication system shown in Figure 1 to discuss traditional channel estimation algorithms. The input signal x( ) = [ ( ), ( − 1), . . . , ( − + 1)] containing the most recent samples is transmitted over a finite impulse response (FIR) channel with channel impulse response (CIR) h = [ℎ 0 , ℎ 1 , . . . , ℎ −1 ] , where (⋅) denotes the transposition. The input signal x( ) is also used as an input for an adaptive filterĥ( ) with coefficients to produce an estimation outputŷ( ), and the received signal r( ) = y( ) + k( ) is obtained at the receiver.

Affine Projection Algorithm (APA).
The channel estimation technique called the standard APA estimates the unknown sparse channel h using the input signal x( ) and the output signal y( ). In the standard APA, let us assume that we The Scientific World Journal 3 keep the last input signal x( ) to form the matrix U( ) as follows [28]: . . .
where denotes the projection order of the APA. Furthermore, we also define some vectors representing reusing results at a given instant , such as the output y( ) of the channel, the outputŷ( ) of the filter, the received signal r( ), and the additive white Gaussian noise vector k( ) and these vectors are expressed as . . .
As for the channel estimation, the purpose of the APA is to minimizeĥ subject to: The APA maintains the next coefficientĥ( + 1) as close as possible to the current coefficientĥ( ) and minimizes the a posteriori error to zero at the same time. Here, the Lagrange multiplier is used to find out the solution that minimizes the cost function APA ( ) of the APA: where APA is a ×1 vector of Lagrange multiplier and APA = [ 0 1 ⋅ ⋅ ⋅ −1 ] . Equation (8) can be rewritten as Then, the gradient of APA ( ) with respect toĥ( + 1) is given by After setting the gradient of APA ( ) with respect toĥ( + 1) equal to zero, we get Multiplying U( ) on both sides of (11), we have By taking the constraint condition of (7) into consideration, we have Taking (3), (6), and (12) into account, we can get Then The update equation is now given by (11) with APA being the solution of (14) and is expressed aŝ The Scientific World Journal The above update equation corresponds to the conventional APA with unity convergence factor [28]. In the practical engineering applications, a convergence factor APA , also known as step-size, is adopted to tradeoff the mean square misadjustment and convergence speed, and thus, the update equation (16) can be rewritten aŝ In general, the step-size APA should be chosen in the range 0 < APA < 2 to control the convergence speed and the steady-state behavior of the APA. It is worth noting that the APA becomes familiar normalized least mean square (NLMS) when the = 1.

Zero-Attracting Affine Projection Algorithm (ZA-APA).
To improve the performance of the standard APA and to utilize the sparsity property of the sparse multipath communication channel, an 1 -penalty term is cooperated into the cost function of (8), which is known as zero-attracting affine projection algorithm (ZA-APA) [32]. In the ZA-APA, the cost function is defined by combining the cost function APA ( ) of standard APA with 1 -penalty of the channel estimator and is given by where ZA is the vector of Lagrange multiplier with × 1. ZA > 0 is a regularization parameter to balance the estimation error and the sparse 1 -penalty ofĥ( +1). In order to minimize the cost function ZA ( ), we use the Lagrange multiplier to calculate its gradient, which is expressed as where sgn[⋅] is a component-wise sign function defined as As is known to us, the minimum is obtained by letting ZA ( )/ĥ( + 1) = 0. Thus, we can get Multiplying both sides by U( ), we can obtain Considering the constraint condition of (7), we can get the following expression: From the above discussion, we know that e( ) = r( ) − U( )ĥ( ). Thus, the Lagrange multipliers vector ZA is obtained: Substituting (24) into (21) and assuming that sgn[ĥ( + 1)] ≈ sgn[ĥ( )], we can obtain the update function of the ZA-APA:ĥ To balance the convergence speed and steady-state error, a step-size ZA is introduced and integrated into (25). Then, (25) can be rewritten aŝ Comparing the update equation (26) of the ZA-APA with the update (17) of the standard APA, we find that there are two additional terms in (26) which attract the tap coefficients to zero when the tap magnitudes of the sparse channel are close to zero. These two additional terms are zero attractors whose attracting strengths are controlled by ZA . Intuitively, the zero attractor can speed the convergence of ZA-APA when the majority taps of the channel of h are zero or close to zero, such as sparse channel.

Reweighted Zero-Attracting Affine Projection Algorithm
(RZA-APA). Unfortunately, the ZA-APA cannot distinguish the zero taps and the nonzero taps of the sparse channel, and it exerts the same penalty on all the channel taps, which forces all the taps to zero uniformly [22,32]. Therefore, the performance of the ZA-APA is degraded when the channel is a less sparse one. In order to improve the performance of the ZA-APA and to solve this problem, a heuristic approach first reported in [33] and employed in [22,32] to reinforce that the zero attractor was proposed and was denoted as reweighted zero-attracting affine projection algorithm (RZA-APA). In the RZA-APA, The Scientific World Journal 5 of ‖ĥ( )‖ 1 . Thus, the cost function of the RZA-APA can be written as where RZA > 0 is a regularization parameter, RZA > 0 is a positive threshold, and RZA is the vector of the Lagrange multiplier with size of × 1. The Lagrange multiplier is used for calculating the minimization of RZA ( ) and the gradient of RZA ( ) can be expressed as Let RZA ( )/ĥ( + 1) = 0 and assume sgn[ĥ( , and then we can get By multiplying U( ) on both sides of (29), the following equation can be obtained: Taking (7) and (30) into consideration, we can get Thus, the Lagrange multiplier vector RZA is obtained: where e( ) = r( ) − U( )ĥ( ). Substituting (32) into (29), we can get the update equation of the RZA-APA: Similarly, a step-size RZA is introduced and cooperated into (33) to balance the convergence speed and the steady-state error of the RZA-APA. Then, (33) can be rewritten aŝ From the analysis and the a priori knowledge of the sparse channel, we know that the RZA-APA is more sensitive to taps with small magnitudes. Note that the reweighted zero attractor mainly affects taps whose magnitudes are comparable to 1/ RZA while it has less shrinkage exerted on |ĥ( )| ≫ 1/ RZA . Thus, the RZA-APA can improve steadystate performance compared to the ZA-APA.

-Norm Constrained Affine Projection Algorithm (SL0-APA)
On the basis of the discussion of the ZA-APA and RZA-APA, we find that the RZA-APA can improve the performance of ZA-APA for sparse channel estimation because ∑ =1 log(1 + RZA |ĥ ( + 1)|) is more similar to 0 -norm [22,32,33]. On the other hand, solving 0 -norm ‖ĥ( + 1)‖ 0 is a NP-hard problem [18]. Fortunately, smooth approximation 0 -norm (SL0) with low complexity has been proposed as an accurate approximation of ‖ĥ( + 1)‖ 0 to reconstruct sparse signals in CS theory [34,35]. Inspired by the SL0 algorithm and in order to exploit the sparse characteristic of the multipath channel in a more accurate way, a smooth approximation 0norm constrained affine projection algorithm (SL0-APA) is proposed by exerting the SL0 on the cost function of standard APA to further improve the performance of the RZA-APA. 6 The Scientific World Journal 3.1. Proposed SL0-APA. Similar to the ZA-APA and RZA-APA discussed above, the cost function of the SL0-APA is written as where SL0 is the vector of the Lagrange multiplier with size of ×1 and SL0 > 0 is a regularization parameter to tradeoff the estimation error and the sparse 0 -penalty ofĥ( +1). Here, the smooth approximation of 0 -norm ‖ĥ( + 1)‖ 0 is a continuous function defined as follows: where is a small positive constant which is used for avoiding division by zero, and the gradient of this continuous functions for SL0 is obtained: To obtain the minimum of the SL0 ( ), we use Lagrange multiplier to calculate the gradient of SL0 ( ). Then the gradient of the cost function of the SL0-APA is written as Let the left-hand side of (38) be equal to zero. We can get the following equation: Multiplying U( ) on both sides of (39), we can get By taking (7) into consideration, (40) can be rewritten as From the discussion of the ZA-APA and RZA-APA, we can get the Lagrange multiplier vector SL0 from (41) by taking e( ) = r( ) − U( )ĥ( ) into account: Substituting (42) into (39) and assuming that sgn(ĥ( + 1))/(|ĥ( + 1)| + ) 2 ≈ sgn(ĥ( ))/(|ĥ( )| + ) 2 , the update function of the SL0-APA can be achieved: where T( ) = sgn(ĥ( ))/(|ĥ( )| + ) 2 . Similar to the ZA-APA and RZA-APA, a step-size SL0 is introduced into (43) to create a balance between the convergence speed and steadystate error of the SL0-APA: It is important to mention that our proposed SL0-APA is superior to APA, ZA-APA, and RZA-APA for sparse channel estimation because we utilize a smooth approximation of ‖ĥ( + 1)‖ 0 , which is proved to be an approximate and nearaccurate approximation of 0 -norm in comparison with the sum-log function ∑ =1 log(1 + RZA |ĥ ( + 1)|) in the RZA-APA. Moreover, it is easy to calculate the gradient, as we can easily find a continuous gradient for this smoothed 0 -norm function.

Analysis of the Proposed SL0-APA.
In this section, we analyze the mean-square-error (MSE) behavior of the SL0-APA. Here, energy-conservation approach [36][37][38] is employed to obtain the theoretical expressions for the MSE of the SL0-APA. Let us consider the received signal r( ) that is derived from the following linear model: where h is the sparse channel vector of the multipath communication system that we wish to estimate and k( ) is The Scientific World Journal 7 the additive Gaussian noise at instant . Our objective is to evaluate the steady-state MSE performance of the proposed SL0-APA. The steady-state MSE is defined as where E[⋅] denotes the expectation and is the estimated error at time . Taking (45) and (47) into account, we obtain Subtracting h from both sides of the SL0-APA update function (44), we get the misalignment vector: Substituting (48) into (49), we can get Taking expectations on both sides of (50), we get We assume that the additive noise k( ) is statistically independent of the input signal x( ), and hence we have E[U + ( )k( )] = 0. Therefore, (51) can be simplified as From previous studies on sparse LMS algorithms [22,39], in the steady state, we have Thus, the E[T( )] in (52) can be written as In addition, when the channel length is far larger than 1, ≫ 1, the E[U + ( )U( )] can be written as [37,40,41] Since E[x ( )x( − 1)] = 0 for sparse channel estimation, the inner expectation reduces to . . .
The Scientific World Journal Here, we define where 2 is the power of the input signal. Thus, where Tr(⋅) is the trace of matrix and I is the × identity matrix. Moreover, we can obtain Then we can approximate E{U Therefore, (52) can be rewritten as It is found that the matrix T( ) is approximately bounded between − I and I . Therefore, we see that such convergence is guaranteed only if (I − SL0 R/ 2 ) is less than 1 [28], which is given by where max is the maximum eigenvalue of the autocorrelation matrix R of x( ). We can observe that the stability condition of the SL0-APA is independent of the parameter SL0 . We assume that the estimated vectorĥ( ) converges when → ∞. Then, (61) can be rewritten as From (63), we can obtain which can be regarded as Note that (65) implies that the optimum solution of the SL0-APA is biased, as was also shown for zero-attracting least mean square (ZA-LMS) algorithms [22]. We then proceed to derive the steady-state MSE for our proposed SL0-APA. Firstly, multiplying both sides of (44) by U( ) from the left, we can get Furthermore, Additionally, we define the a posteriori error vector e ( ) and the a priori error vector e ( ) as Combining (67) On the basis of the discussion mentioned above, we notice that U( )U + ( ) = U( )U ( )[U( )U ( )] −1 = I. By considering the power of both sides of (73), using the steady- and assuming that e ( ), e ( ), andĥ( ) are independent of x( ) in the steady state, we get Substituting (71) into the left-hand side (LHS) of (74), we have Moreover, substituting (70) into the right-hand side (RHS) of (74), we have By combining (75) and (76), we get We also assume that the additive Gaussian noise k( ) is statistically independent of the input signal x( ). Thus (77) can be simplified as Here, we also assume that the U( ) is statistically independent of e( ) at the steady state. Moreover, we use the definition of E[e ( )e( )] = E| ( )| 2 S [36], where where 1 = [1 0 ⋅ ⋅ ⋅ 0] and ( ) is the top entry of e( ) [36]. Then, the LHS of (78) can be rewritten as Similar to the calculation of (80), the first term in the RHS of (78) can be written as 10 The Scientific World Journal In addition, the second term of RHS of (78) can be rewritten as Then the last term of the right-hand side of (78) can be expressed as When the SL0 is small, we can get Therefore, the MSE of the proposed SL0-APA with small step-size SL0 can be written as When the step-size SL0 is large, S ≈ 1 ⋅ 1 [36]. In this case, Thus, the MSE of the proposed SL0-APA with large step-size SL0 can be written as

Results and Discussions
In this section, we present the computer simulation results to illustrate the performance of the proposed SL0-APA over a sparse multipath communication channel. Moreover, the simulation results for predicting the mean-square error of the proposed SL0-APA are also provided to verify the effectiveness of the theoretical expressions obtained in Section 3.2.
In addition, the computational complexity of the SL0-APA is presented and compared with past sparsity-aware algorithms, namely, the ZA-APA, RZA-APA, and standard APA, NLMS algorithms. the proposed SL0-APA in comparison with the previously proposed sparse channel estimation algorithms including the APA, ZA-APA, RZA-APA, and NLMS algorithms. In the setup of this experiment, we consider a sparse multipath communication channel h whose length is equal to 16 and whose number of dominant taps is set to two different sparsity levels, namely, = 1, = 4, similarly to [6,22,25,26]. The dominant channel taps are obtained from a Gaussian distribution subjected to ‖h‖ 2 2 = 1, and the positions of the dominant channel taps are random within the length of the channel. The input signal x( ) of the channel is a Gaussian random signal, while the output of the channel is corrupted by an independent white Gaussian noise k( ). An example of a typical sparse multipath channel with a channel length of = 16 and a sparsity level of = 3 is shown in Figure 2. In the simulations, the power of the received signal is = 1, while the noise power is given by 2 V . In all the experiments, the difference between the actual and estimated channels based on the sparsity-aware algorithms and the sparse channel mentioned above is evaluated by the MSE defined as follows:

Performance of the
In this subsection, we aim to investigate the convergence speed and the steady-state performance of the SL0-APA. The simulation parameters used to compare the convergence speed while maintaining the same MSE are listed as follows: NLMS = 0.25, APA = 0.125, ZA = 0.165, RZA = 0.18, SL0 = 0.21, ZA = 5 × 10 −5 , RZA = 8 × 10 −5 , SL0 = 3 × 10 −6 , RZA = 10, SL0 = 0.001, = 2, and 2 V = 10 −3 , where NLMS is the step-size parameter for NLMS algorithm. It can be seen from Figure 3 that our proposed SL0-APA possesses the fastest convergence speed compared to the previously proposed channel estimation algorithms at the same steady-state error floor. In addition, all the affine projection algorithms, namely, APA, ZA-APA, RZA-APA, and SL0-APA, converge much more quickly in comparison with NLMS algorithm, because the affine projection algorithms reuse the old data signal that is implemented by the use of parameter . Thus, we discuss the effects of the affine projection order for SL0-APA and compare it with the APA and NLMS algorithms. The computer simulation results with different values of are shown in Figure 4. It reveals that the convergence speed is improved by the increment of the affine projection order . However, the steady-state performance has deteriorated from = 2 to = 8. Thus, in our proposed SL0-APA, the affine projection , the step-size SL0 , the regularization parameter SL0 , and SL0 should be take into account to balance the convergence speed and the steady-state behavior.
Next, we show the effects of the sparsity levels on the steady-state performance of the proposed SL0-APA at = 1 and = 4. To obtain the same convergence speed, the simulation parameters used in this experiment are listed as follows: NLMS = 0.095, APA = ZA = RZA = SL0 = 0.05, ZA = 5 × 10 −5 , RZA = 8 × 10 −5 , SL0 = 4 × 10 −6 , RZA = 10, = 0.01, and 2 V = 10 −3 . We can see from Figure 5 that our proposed SL0-APA has the best steady-state performance compared to the ZA-APA, RZA-APA, APA, and NLMS algorithms. The SL0-APA can achieve 10 dB smaller MSE than the RZA-APA for = 1 and = 2 shown in Figure 5(a). When the sparsity level increases to 4, it is seen in Figure 5(b) that our proposed SL0-APA still outperforms other algorithms, while its steady-state error increases in comparison with that of = 1. When the affine projection order increases to = 3, we can see from Figure 6 that the convergence speed is significantly improved compared to that of = 2 shown in Figure 5. However, the steady-state error is also slightly increased when the increases. Furthermore, our proposed SL0-APA still has the best convergence speed and lowest steady-state error. Finally, we use the theoretical expressions obtained in Section 3.2 to predict the mean-square-error (MSE) of the proposed SL0-APA with different SL0 and compare the theoretical results with the simulation ones. The MSE comparisons of the SL0-APA as a function of the step-size SL0 for the designated sparse multipath communication channel with the simulation parameters of SL0 = 4 × 10 −6 , = 0.01, 2 V = 10 −3 , = 3, and = 1 are shown in Figure 7. The theoretical results are obtained from (85) to (87) for small values of SL0 and large values of SL0 , respectively, while the simulation results are obtained by averaging 50 independent trials. We can see that the simulation results exhibit good agreement with the theoretical expressions with different step-size SL0 . In addition, we can see that the steady-state misadjustment between the computer simulation and the theory predicting is becoming larger with the decrement of the SL0 for small SL0 shown in Figure 7(a), but the steadystate error is becoming lower. For the large SL0 , both the steady-state error and the convergence speed are deteriorated by the increment of the step-size SL0 . Generally speaking, as SL0 increases, the MSE increases. Although a large zero attractor can help the SL0-APA to converge faster, it will lead to a higher misadjustment. Thus, in the most cases, we should choose the step-size SL0 carefully in order to balance convergence speed and steady-state performance.

Computational Complexity.
In this subsection, we present the computational complexity of the proposed SL0-APA and compare it with the conventional sparsityaware channel estimation algorithms, including the APA, ZA-APA, and RZA-APA. It is worth noting that when the affine projection order is equal to 1, these three affine projection algorithms converge to familiar NLMS, ZA-NLMS, and RZA-NLMS algorithms, respectively. Here, the computational complexity is the arithmetic complexity, which includes additions, multiplications, and divisions. We assume nonzero taps in a sparse channel model as an FIR filter with coefficients, and the order of these affine projection algorithms is . The computational complexity of the proposed SL0-APA and the relevant sparsity-aware algorithms are shown in Table 1.
From Table 1, we see that our proposed SL0-APA with the best steady-state performance and fastest convergence speed needs more calculations than the RZA-APA. The additional computational complexity comes from the continuous function for SL0 approximation, which can be reduced by proper selection of this continuous function. Furthermore, the complexity of all the APAs is higher than the NLMS algorithms. In addition, the sparsity property of the channel can also help to reduce the computational complexity of the proposed SL0-APA.

Conclusion
In this paper, we proposed an SL0-APA to exploit the sparsity of sparse channel and to improve the performance on both  14 The Scientific World Journal the convergence speed and steady-state error of the APA, ZA-APA, and RZA-APA. This algorithm is mainly developed by introducing a smooth approximation 0 -norm, which has a significant impact on the sparsity due to the incorporation of SL0 into the cost function of the standard APA as an additional constraint. The improvement can evidently accelerate the convergence speed by exerting such additional regularization term on the zero taps of the sparse channel. Then, we provided a mathematical analysis for predicting the mean square error of our proposed SL0-APA. We also showed the convergence behavior and the steady-state performance in comparison with the standard APA and relevant sparsityaware channel estimation algorithms. In summary, the simulation results demonstrated that the proposed SL0-APA with moderate computational complexity accelerates convergence speed and improves steady-state performance in a designated sparse channel.