# Smooth Approximationl0-Norm Constrained Affine Projection Algorithm and Its Applications in Sparse Channel Estimation

Authors: Yingsong Li; Masanori Hamamura
Journal: The Scientific World Journal
 (2014)

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## Abstract

We propose a smooth approximationl0-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
l0-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.


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## Body

## 1. 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 [1–5]. 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 [7–9]. 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 [10–13], 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 [14–16]. 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 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, 22–27]. 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 [28–31], 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, l1-norm penalized APA has been proposed to render the standard APA suitable for sparse signal estimation applications [32]. However, these l1-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 approximationl0-norm 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 l0-norm obtained from a continuous function that is an accurate approximation of l0-norm. By exploiting the information of the sparsity channel and using the concepts of the smooth approximation of l0-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 l0-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 steady-state 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. Section2 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 l0-norm penalty on the cost function of the standard APA. Next, we provide a theoretical expression of the convergence analysis and the mean-square-error (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.


## 2. Conventional Channel Estimation Algorithms

In this section, we consider a sparse multipath communication system shown in Figure1 to discuss traditional channel estimation algorithms. The input signal x(n)=[x(n),x(n-1),…,x(n-N+1)]T containing the N most recent samples is transmitted over a finite impulse response (FIR) channel with channel impulse response (CIR) h=[h0,h1,…,hN-1]T, where (·)T denotes the transposition. The input signal x(n) is also used as an input for an adaptive filter h^(n) with N coefficients to produce an estimation output y^(n), and the received signal r(n)=y(n)+v(n) is obtained at the receiver.Figure 1
A typical sparse multipath communication system.


### 2.1. Affine Projection Algorithm (APA)

The channel estimation technique called the standard APA estimates the unknown sparse channelh using the input signal x(n) and the output signal y(n). In the standard APA, let us assume that we keep the last Q input signal x(n) to form the matrix U(n) as follows [28]:
(1)U(n)=[xT(n)xT(n-1)⋮xT(n-Q+1)]U(n)=[x(n)x(n-1)⋯x(n-N+1)x(n-1)x(n-2)⋯x(n-N)⋮⋮⋱⋮x(n-Q+1)x(n-Q)⋯x(n-N-Q+2)],
where Q denotes the projection order of the APA. Furthermore, we also define some vectors representing reusing results at a given instant n, such as the output y(n) of the channel, the output y^(n) of the filter, the received signal r(n), and the additive white Gaussian noise vector v(n) and these vectors are expressed as
(2)y(n)=U(n)h=[y(n)y(n-1)⋮y(n-Q+1)],(3)y^(n)=U(n)h^(n)=[y^(n)y^(n-1)⋮y^(n-Q+1)],(4)v(n)=[v(n)v(n-1)⋮v(n-Q+1)],(5)r(n)=[r(n)r(n-1)⋮r(n-Q+1)].From (1)–(5), the instantaneous error e(n) can be written as
(6)e(n)=[e(n)e(n-1)⋮e(n-Q+1)]e(n)=[r(n)-y^(n)r(n-1)-y^(n-1)⋮r(n-Q+1)-y^(n-Q+1)]e(n)=r(n)-y^(n).As for the channel estimation, the purpose of the APA is to minimize(7)∥h^(n+1)-h^(n)∥2subject  to:r(n)-U(n)h^(n+1)=0.The APA maintains the next coefficienth^(n+1) as close as possible to the current coefficient h^(n) 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 JAPA(n) of the APA:
(8)JAPA(n)=∥h^(n+1)-h^(n)∥2JAPA(n)=+[r(n)-U(n)h^(n+1)]TλAPA,
where λAPA is a Q×1 vector of Lagrange multiplier and λAPA=[λ0λ1⋯λQ-1]T. Equation (8) can be rewritten as
(9)JAPA(n)=[h^(n+1)-h^(n)]T[h^(n+1)-h^(n)]JAPA(n)=+[rT(n)-h^T(n+1)UT(n)]λAPA.Then, the gradient ofJAPA(n) with respect to h^(n+1) is given by
(10)∂JAPA(n)∂h^(n+1)=2h^(n+1)-2h^(n)-UT(n)λAPA.After setting the gradient ofJAPA(n) with respect to h^(n+1) equal to zero, we get
(11)h^(n+1)=h^(n)+12UT(n)λAPA.MultiplyingU(n) on both sides of (11), we have
(12)U(n)h^(n+1)=U(n)h^(n)+12U(n)UT(n)λAPA.By taking the constraint condition of (7) into consideration, we have
(13)r(n)=U(n)h^(n)+12U(n)UT(n)λAPA.Taking (3), (6), and (12) into account, we can get
(14)e(n)=12U(n)UT(n)λAPA.Then(15)λAPA=2[U(n)UT(n)]-1e(n).The update equation is now given by (11) with λAPA being the solution of (14) and is expressed as
(16)h^(n+1)=h^(n)+UT(n)[U(n)UT(n)]-1e(n)h^(n+1)=h^(n)+U+(n)e(n),
where U+(n)=UT(n)[U(n)UT(n)]-1. The above update equation corresponds to the conventional APA with unity convergence factor [28]. In the practical engineering applications, a convergence factor μAPA, 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 as
(17)h^(n+1)=h^(n)+μAPAUT(n)[U(n)UT(n)]-1e(n)h^(n+1)=h^(n)+μAPAU+(n)e(n).In general, the step-sizeμAPA should be chosen in the range 0<μAPA<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 Q=1.


### 2.2. 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, anl1-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 JAPA(n) of standard APA with l1-penalty of the channel estimator and is given by
(18)JZA(n)=∥h^(n+1)-h^(n)∥2+[r(n)-U(n)h^(n+1)]TλZAJZA(n)=+γZA∥h^(n+1)∥1,
where λZA is the vector of Lagrange multiplier with Q×1. γZA>0 is a regularization parameter to balance the estimation error and the sparse l1-penalty of h^(n+1). In order to minimize the cost function JZA(n), we use the Lagrange multiplier to calculate its gradient, which is expressed as
(19)∂JZA(n)∂h^(n+1)=2h^(n+1)-2h^(n)-UT(n)λZA∂JZA(n)∂h^(n+1)=+γZAsgn⁡[h^(n+1)],
where sgn⁡[·] is a component-wise sign function defined as
(20)sgn⁡[x]={x|x|,x≠00,x=0.As is known to us, the minimum is obtained by letting∂JZA(n)/∂h^(n+1)=0. Thus, we can get
(21)h^(n+1)=h^(n)+12UT(n)λZA-12γZAsgn⁡[h^(n+1)].Multiplying both sides byU(n), we can obtain
(22)U(n)h^(n+1)=U(n)h^(n)+12U(n)UT(n)λZAU(n)h^(n+1)=-12γZAU(n)sgn⁡[h^(n+1)].Considering the constraint condition of (7), we can get the following expression:
(23)r(n)=U(n)h^(n)+12U(n)UT(n)λZAr(n)=-12γZAU(n)sgn⁡[h^(n+1)].From the above discussion, we know thate(n)=r(n)-U(n)h^(n). Thus, the Lagrange multipliers vector λZA is obtained:
(24)λZA=[U(n)UT(n)]-1{2e(n)+γZAU(n)sgn⁡[h^(n+1)]}.Substituting (24) into (21) and assuming that sgn⁡[h^(n+1)]≈sgn⁡[h^(n)], we can obtain the update function of the ZA-APA:
(25)h^(n+1)=h^(n)+U+(n)e(n)h^(n+1)=+12γZAU+(n)U(n)sgn⁡[h^(n)]h^(n+1)=-12γZAsgn⁡[h^(n)].To balance the convergence speed and steady-state error, a step-sizeμZA is introduced and integrated into (25). Then, (25) can be rewritten as
(26)h^(n+1)=h^(n)+μZAU+(n)e(n)h^(n+1)=+12γZAU+(n)U(n)sgn⁡[h^(n)]h^(n+1)=-12γZAsgn⁡[h^(n)].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 γZA. 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.


### 2.3. 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, ∑i=1Nlog⁡(1+εRZA|h^i(n)|) is adopted instead of ∥h^(n)∥1. Thus, the cost function of the RZA-APA can be written as
(27)JRZA(n)=∥h^(n+1)-h^(n)∥2JRZA(n)=+[r(n)-U(n)h^(n+1)]TλRZAJRZA(n)=+γRZA∑i=1N‍log⁡(1+εRZA|h^i(n+1)|),
where γRZA>0 is a regularization parameter, εRZA>0 is a positive threshold, and λRZA is the vector of the Lagrange multiplier with size of Q×1. The Lagrange multiplier is used for calculating the minimization of JRZA(n) and the gradient of JRZA(n) can be expressed as
(28)∂JRZA(n)∂h^(n+1)=2h^(n+1)-2h^(n)-UT(n)λRZA∂JRZA(n)∂h^(n+1)=+γRZAsgn⁡[h^(n+1)]1+εRZA|h^(n+1)|.Let∂JRZA(n)/∂h^(n+1)=0 and assume sgn⁡[h^(n+1)]/(1+εRZA|h^(n+1)|)≈sgn⁡[h^(n)]/(1+εRZA|h^(n)|), and then we can get
(29)h^(n+1)=h^(n)+12UT(n)λRZA-12γRZAsgn⁡[h^(n)]1+εRZA|h^(n)|.By multiplyingU(n) on both sides of (29), the following equation can be obtained:
(30)U(n)h^(n+1)=U(n)h^(n)+12U(n)UT(n)λRZAU(n)h^(n+1)=-12γRZAU(n)sgn⁡[h^(n)]1+εRZA|h^(n)|.Taking (7) and (30) into consideration, we can get
(31)r(n)=U(n)h^(n)+12U(n)UT(n)λRZAr(n)=-12γRZAU(n)sgn⁡[h^(n)]1+εRZA|h^(n)|.Thus, the Lagrange multiplier vectorλRZA is obtained:
(32)λRZA=[U(n)UT(n)]-1λRZA=×{2e(n)+γRZAU(n)sgn⁡[h^(n)]1+εRZA|h^(n)|},
where e(n)=r(n)-U(n)h^(n). Substituting (32) into (29), we can get the update equation of the RZA-APA:
(33)h^(n+1)=h^(n)+U+(n)e(n)h^(n+1)=+12γRZAU+(n)U(n)sgn⁡[h^(n)]1+εRZA|h^(n)|h^(n+1)=-12γRZAsgn⁡[h^(n)]1+εRZA|h^(n)|.Similarly, a step-sizeμRZA 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 as
(34)h^(n+1)=h^(n)+μRZAU+(n)e(n)h^(n+1)=+12γRZAU+(n)U(n)sgn⁡[h^(n)]1+εRZA|h^(n)|h^(n+1)=-12γRZAsgn⁡[h^(n)]1+εRZA|h^(n)|.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 to1/εRZA while it has less shrinkage exerted on |h^(n)|≫1/εRZA. Thus, the RZA-APA can improve steady-state performance compared to the ZA-APA.


## 2.1. Affine Projection Algorithm (APA)

The channel estimation technique called the standard APA estimates the unknown sparse channelh using the input signal x(n) and the output signal y(n). In the standard APA, let us assume that we keep the last Q input signal x(n) to form the matrix U(n) as follows [28]:
(1)U(n)=[xT(n)xT(n-1)⋮xT(n-Q+1)]U(n)=[x(n)x(n-1)⋯x(n-N+1)x(n-1)x(n-2)⋯x(n-N)⋮⋮⋱⋮x(n-Q+1)x(n-Q)⋯x(n-N-Q+2)],
where Q denotes the projection order of the APA. Furthermore, we also define some vectors representing reusing results at a given instant n, such as the output y(n) of the channel, the output y^(n) of the filter, the received signal r(n), and the additive white Gaussian noise vector v(n) and these vectors are expressed as
(2)y(n)=U(n)h=[y(n)y(n-1)⋮y(n-Q+1)],(3)y^(n)=U(n)h^(n)=[y^(n)y^(n-1)⋮y^(n-Q+1)],(4)v(n)=[v(n)v(n-1)⋮v(n-Q+1)],(5)r(n)=[r(n)r(n-1)⋮r(n-Q+1)].From (1)–(5), the instantaneous error e(n) can be written as
(6)e(n)=[e(n)e(n-1)⋮e(n-Q+1)]e(n)=[r(n)-y^(n)r(n-1)-y^(n-1)⋮r(n-Q+1)-y^(n-Q+1)]e(n)=r(n)-y^(n).As for the channel estimation, the purpose of the APA is to minimize(7)∥h^(n+1)-h^(n)∥2subject  to:r(n)-U(n)h^(n+1)=0.The APA maintains the next coefficienth^(n+1) as close as possible to the current coefficient h^(n) 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 JAPA(n) of the APA:
(8)JAPA(n)=∥h^(n+1)-h^(n)∥2JAPA(n)=+[r(n)-U(n)h^(n+1)]TλAPA,
where λAPA is a Q×1 vector of Lagrange multiplier and λAPA=[λ0λ1⋯λQ-1]T. Equation (8) can be rewritten as
(9)JAPA(n)=[h^(n+1)-h^(n)]T[h^(n+1)-h^(n)]JAPA(n)=+[rT(n)-h^T(n+1)UT(n)]λAPA.Then, the gradient ofJAPA(n) with respect to h^(n+1) is given by
(10)∂JAPA(n)∂h^(n+1)=2h^(n+1)-2h^(n)-UT(n)λAPA.After setting the gradient ofJAPA(n) with respect to h^(n+1) equal to zero, we get
(11)h^(n+1)=h^(n)+12UT(n)λAPA.MultiplyingU(n) on both sides of (11), we have
(12)U(n)h^(n+1)=U(n)h^(n)+12U(n)UT(n)λAPA.By taking the constraint condition of (7) into consideration, we have
(13)r(n)=U(n)h^(n)+12U(n)UT(n)λAPA.Taking (3), (6), and (12) into account, we can get
(14)e(n)=12U(n)UT(n)λAPA.Then(15)λAPA=2[U(n)UT(n)]-1e(n).The update equation is now given by (11) with λAPA being the solution of (14) and is expressed as
(16)h^(n+1)=h^(n)+UT(n)[U(n)UT(n)]-1e(n)h^(n+1)=h^(n)+U+(n)e(n),
where U+(n)=UT(n)[U(n)UT(n)]-1. The above update equation corresponds to the conventional APA with unity convergence factor [28]. In the practical engineering applications, a convergence factor μAPA, 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 as
(17)h^(n+1)=h^(n)+μAPAUT(n)[U(n)UT(n)]-1e(n)h^(n+1)=h^(n)+μAPAU+(n)e(n).In general, the step-sizeμAPA should be chosen in the range 0<μAPA<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 Q=1.


## 2.2. 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, anl1-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 JAPA(n) of standard APA with l1-penalty of the channel estimator and is given by
(18)JZA(n)=∥h^(n+1)-h^(n)∥2+[r(n)-U(n)h^(n+1)]TλZAJZA(n)=+γZA∥h^(n+1)∥1,
where λZA is the vector of Lagrange multiplier with Q×1. γZA>0 is a regularization parameter to balance the estimation error and the sparse l1-penalty of h^(n+1). In order to minimize the cost function JZA(n), we use the Lagrange multiplier to calculate its gradient, which is expressed as
(19)∂JZA(n)∂h^(n+1)=2h^(n+1)-2h^(n)-UT(n)λZA∂JZA(n)∂h^(n+1)=+γZAsgn⁡[h^(n+1)],
where sgn⁡[·] is a component-wise sign function defined as
(20)sgn⁡[x]={x|x|,x≠00,x=0.As is known to us, the minimum is obtained by letting∂JZA(n)/∂h^(n+1)=0. Thus, we can get
(21)h^(n+1)=h^(n)+12UT(n)λZA-12γZAsgn⁡[h^(n+1)].Multiplying both sides byU(n), we can obtain
(22)U(n)h^(n+1)=U(n)h^(n)+12U(n)UT(n)λZAU(n)h^(n+1)=-12γZAU(n)sgn⁡[h^(n+1)].Considering the constraint condition of (7), we can get the following expression:
(23)r(n)=U(n)h^(n)+12U(n)UT(n)λZAr(n)=-12γZAU(n)sgn⁡[h^(n+1)].From the above discussion, we know thate(n)=r(n)-U(n)h^(n). Thus, the Lagrange multipliers vector λZA is obtained:
(24)λZA=[U(n)UT(n)]-1{2e(n)+γZAU(n)sgn⁡[h^(n+1)]}.Substituting (24) into (21) and assuming that sgn⁡[h^(n+1)]≈sgn⁡[h^(n)], we can obtain the update function of the ZA-APA:
(25)h^(n+1)=h^(n)+U+(n)e(n)h^(n+1)=+12γZAU+(n)U(n)sgn⁡[h^(n)]h^(n+1)=-12γZAsgn⁡[h^(n)].To balance the convergence speed and steady-state error, a step-sizeμZA is introduced and integrated into (25). Then, (25) can be rewritten as
(26)h^(n+1)=h^(n)+μZAU+(n)e(n)h^(n+1)=+12γZAU+(n)U(n)sgn⁡[h^(n)]h^(n+1)=-12γZAsgn⁡[h^(n)].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 γZA. 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.


## 2.3. 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, ∑i=1Nlog⁡(1+εRZA|h^i(n)|) is adopted instead of ∥h^(n)∥1. Thus, the cost function of the RZA-APA can be written as
(27)JRZA(n)=∥h^(n+1)-h^(n)∥2JRZA(n)=+[r(n)-U(n)h^(n+1)]TλRZAJRZA(n)=+γRZA∑i=1N‍log⁡(1+εRZA|h^i(n+1)|),
where γRZA>0 is a regularization parameter, εRZA>0 is a positive threshold, and λRZA is the vector of the Lagrange multiplier with size of Q×1. The Lagrange multiplier is used for calculating the minimization of JRZA(n) and the gradient of JRZA(n) can be expressed as
(28)∂JRZA(n)∂h^(n+1)=2h^(n+1)-2h^(n)-UT(n)λRZA∂JRZA(n)∂h^(n+1)=+γRZAsgn⁡[h^(n+1)]1+εRZA|h^(n+1)|.Let∂JRZA(n)/∂h^(n+1)=0 and assume sgn⁡[h^(n+1)]/(1+εRZA|h^(n+1)|)≈sgn⁡[h^(n)]/(1+εRZA|h^(n)|), and then we can get
(29)h^(n+1)=h^(n)+12UT(n)λRZA-12γRZAsgn⁡[h^(n)]1+εRZA|h^(n)|.By multiplyingU(n) on both sides of (29), the following equation can be obtained:
(30)U(n)h^(n+1)=U(n)h^(n)+12U(n)UT(n)λRZAU(n)h^(n+1)=-12γRZAU(n)sgn⁡[h^(n)]1+εRZA|h^(n)|.Taking (7) and (30) into consideration, we can get
(31)r(n)=U(n)h^(n)+12U(n)UT(n)λRZAr(n)=-12γRZAU(n)sgn⁡[h^(n)]1+εRZA|h^(n)|.Thus, the Lagrange multiplier vectorλRZA is obtained:
(32)λRZA=[U(n)UT(n)]-1λRZA=×{2e(n)+γRZAU(n)sgn⁡[h^(n)]1+εRZA|h^(n)|},
where e(n)=r(n)-U(n)h^(n). Substituting (32) into (29), we can get the update equation of the RZA-APA:
(33)h^(n+1)=h^(n)+U+(n)e(n)h^(n+1)=+12γRZAU+(n)U(n)sgn⁡[h^(n)]1+εRZA|h^(n)|h^(n+1)=-12γRZAsgn⁡[h^(n)]1+εRZA|h^(n)|.Similarly, a step-sizeμRZA 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 as
(34)h^(n+1)=h^(n)+μRZAU+(n)e(n)h^(n+1)=+12γRZAU+(n)U(n)sgn⁡[h^(n)]1+εRZA|h^(n)|h^(n+1)=-12γRZAsgn⁡[h^(n)]1+εRZA|h^(n)|.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 to1/εRZA while it has less shrinkage exerted on |h^(n)|≫1/εRZA. Thus, the RZA-APA can improve steady-state performance compared to the ZA-APA.


## 3. Proposed Smooth Approximationl0-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∑i=1Nlog⁡(1+εRZA|h^i(n+1)|) is more similar to l0-norm [22, 32, 33]. On the other hand, solving l0-norm ∥h^(n+1)∥0 is a NP-hard problem [18]. Fortunately, smooth approximation l0-norm (SL0) with low complexity has been proposed as an accurate approximation of ∥h^(n+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 l0-norm 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.


### 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(35)JSL0(n)=∥h^(n+1)-h^(n)∥2JSL0(n)=+[r(n)-U(n)h^(n+1)]TλSL0JSL0(n)=+γSL0∥h^(n+1)∥0,
where λSL0 is the vector of the Lagrange multiplier with size of Q×1 and γSL0>0 is a regularization parameter to tradeoff the estimation error and the sparse l0-penalty of h^(n+1). Here, the smooth approximation of l0-norm ∥h^(n+1)∥0 is a continuous function defined as follows:
(36)∥h^(n+1)∥0=∑i=1N-1‍|h^i(n+1)||h^i(n+1)|+δ=|h^(n+1)||h^(n+1)|+δ,
where δ is a small positive constant which is used for avoiding division by zero, and the gradient of this continuous functions for SL0 is obtained:
(37)∂∥h^(n+1)∥0∂h^(n+1)=δsgn⁡(h^(n+1))(|h^(n+1)|+δ)2.To obtain the minimum of theJSL0(n), we use Lagrange multiplier to calculate the gradient of JSL0(n). Then the gradient of the cost function of the SL0-APA is written as
(38)∂JSL0(n)∂h^(n+1)=2h^(n+1)-2h^(n)-UT(n)λSL0∂JSL0(n)∂h^(n+1)=+γSL0δsgn⁡(h^(n+1))(|h^(n+1)|+δ)2.Let the left-hand side of (38) be equal to zero. We can get the following equation:
(39)h^(n+1)=h^(n)+12UT(n)λSL0-12γSL0δsgn⁡(h^(n+1))(|h^(n+1)|+δ)2.MultiplyingU(n) on both sides of (39), we can get
(40)U(n)h^(n+1)=U(n)h^(n)+12U(n)UT(n)λSL0U(n)h^(n+1)=-12γSL0U(n)δsgn⁡(h^(n+1))(|h^(n+1)|+δ)2.By taking (7) into consideration, (40) can be rewritten as
(41)r(n)=U(n)h^(n)+12U(n)UT(n)λSL0r(n)=-12γSL0U(n)δsgn⁡(h^(n+1))(|h^(n+1)|+δ)2.From the discussion of the ZA-APA and RZA-APA, we can get the Lagrange multiplier vectorλSL0 from (41) by taking e(n)=r(n)-U(n)h^(n) into account:
(42)λSL0=[U(n)UT(n)]-1λSL0=×{2e(n)+γSL0U(n)δsgn⁡(h^(n+1))(|h^(n+1)|+δ)2}.Substituting (42) into (39) and assuming that δsgn⁡(h^(n+1))/(|h^(n+1)|+δ)2≈δsgn⁡(h^(n))/(|h^(n)|+δ)2, the update function of the SL0-APA can be achieved:
(43)h^(n+1)=h^(n)+U+(n)e(n)h^(n+1)=+12γSL0U+(n)U(n)δsgn⁡(h^(n))(|h^(n)|+δ)2h^(n+1)=-12γSL0δsgn⁡(h^(n))(|h^(n)|+δ)2h^(n+1)=h^(n)+U+(n)e(n)h^(n+1)=+12γSL0U+(n)U(n)T(n)-12γSL0T(n),
where T(n)=δsgn⁡(h^(n))/(|h^(n)|+δ)2. Similar to the ZA-APA and RZA-APA, a step-size μSL0 is introduced into (43) to create a balance between the convergence speed and steady-state error of the SL0-APA:
(44)h^(n+1)=h^(n)+μSL0U+(n)e(n)h^(n+1)=+12γSL0U+(n)U(n)T(n)-12γSL0T(n).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∥h^(n+1)∥0, which is proved to be an approximate and near-accurate approximation of l0-norm in comparison with the sum-log function ∑i=1Nlog⁡(1+εRZA|h^i(n+1)|) in the RZA-APA. Moreover, it is easy to calculate the gradient, as we can easily find a continuous gradient for this smoothed l0-norm function.


### 3.2. 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 [36–38] is employed to obtain the theoretical expressions for the MSE of the SL0-APA. Let us consider the received signal r(n) that is derived from the following linear model:
(45)r(n)=U(n)h+v(n),
where h is the sparse channel vector of the multipath communication system that we wish to estimate and v(n) is the additive Gaussian noise at instant n. Our objective is to evaluate the steady-state MSE performance of the proposed SL0-APA. The steady-state MSE is defined as
(46)MSE≜lim⁡n→∞E[|e(n)|2],
where E[·] denotes the expectation and
(47)e(n)=r(n)-U(n)h^(n)
is the estimated error at time n. Taking (45) and (47) into account, we obtain
(48)e(n)=U(n)h+v(n)-U(n)h^(n)e(n)=U(n)[h-h^(n)]+v(n).Subtractingh from both sides of the SL0-APA update function (44), we get the misalignment vector:
(49)Δ(n+1)=h-h^(n+1)Δ(n+1)=h-{12h^(n)+μSL0U+(n)e(n)11111111111111+12γSL0U+(n)U(n)T(n)-12γSL0T(n)}.Substituting (48) into (49), we can get
(50)Δ(n+1)=h-h^(n)-μSL0U+(n)Δ(n+1)=×{U(n)[h-h^(n)]+v(n)}Δ(n+1)=-12γSL0U+(n)U(n)T(n)+12γSL0T(n)Δ(n+1)=[IN-μSL0U+(n)U(n)]Δ(n)-μSL0U+(n)v(n)Δ(n+1)=-12γSL0U+(n)U(n)T(n)+12γSL0T(n).Taking expectations on both sides of (50), we get
(51)E[Δ(n+1)]=E[IN-μSL0U+(n)U(n)]E[Δ(n)]E[Δ(n+1)]=-μSL0E[U+(n)v(n)]E[Δ(n+1)]=-12γSL0E[U+(n)U(n)T(n)]E[Δ(n+1)]=+12γSL0E[T(n)].We assume that the additive noisev(n) is statistically independent of the input signal x(n), and hence we have E[U+(n)v(n)]=0. Therefore, (51) can be simplified as
(52)E[Δ(n+1)]=E[IN-μSL0U+(n)U(n)]E[Δ(n)]E[Δ(n+1)]=-12γSL0E[U+(n)U(n)T(n)]E[Δ(n+1)]=+12γSL0E[T(n)].From previous studies on sparse LMS algorithms [22, 39], in the steady state, we have
(53)E{sgn⁡[h^(n)]}≈sgn⁡(h^).Thus, theE[T(n)] in (52) can be written as
(54)E[T(n)]=E{δsgn⁡(h^(n))(|h^(n)|+δ)2}=δsgn⁡(h^(n))(|h^(n)|+δ)2.In addition, when the channel length is far larger than 1,N≫1, the E[U+(n)U(n)] can be written as [37, 40, 41]
(55)E[U+(n)U(n)]=E{UT(n)[U(n)UT(n)]-1U(n)}E[U+(n)U(n)]≈E{UT(n){E[U(n)UT(n)]}-1U(n)}.SinceE[xT(n)x(n-1)]=0 for sparse channel estimation, the inner expectation reduces to(56)E[U(n)UT(n)]=E{[xT(n)xT(n-1)⋮xT(n-Q-1)][x(n)x(n-1)⋯x(n-Q-1)]}E[U(n)UT(n)]=E{[∥x(n)∥2xT(n)x(n-1)⋯xT(n)x(n-Q-1)xT(n-1)x(n)∥x(n-1)∥2⋯⋮⋮⋮⋱⋮xT(n-Q-1)x(n)xT(n-Q-1)x(n-1)⋯∥x(n-Q-1)∥2]}E[U(n)UT(n)]=E{[∥x(n)∥20⋯00∥x(n-1)∥2⋯0⋮⋮⋱⋮00⋯∥x(n-Q-1)∥2]}.Here, we define(57)R=δx2IN,
where δx2 is the power of the input signal. Thus,
(58)E[U(n)UT(n)]≈Tr⁡(R)IQ=Nδx2IQ,
where Tr⁡(·) is the trace of matrix and IQ is the Q×Q identity matrix. Moreover, we can obtain
(59)E{[U(n)UT(n)]-1}≈{E[U(n)UT(n)]}-1=1Nδx2IQ.Then we can approximateE{UT(n)[U(n)UT(n)]-1U(n)} by
(60)E{UT(n)[U(n)UT(n)]-1U(n)}≈E{UT(n)1Nδx2IQU(n)}E{UT(n)[U(n)UT(n)]-1U(n)}≈QRNδx2.Therefore, (52) can be rewritten as
(61)E[Δ(n+1)]=E[IN-μSL0QRNδx2]E[Δ(n)]E[Δ(n+1)]=-12γSL0QRNδx2E[T(n)]+12γSL0E[T(n)].It is found that the matrixT(n) is approximately bounded between -δIN and δIN. Therefore, we see that such convergence is guaranteed only if (IN-μSL0QR/Nδx2) is less than 1 [28], which is given by
(62)0<μSL0<Nδx2Qλmax⁡,
where λmax⁡ is the maximum eigenvalue of the autocorrelation matrix R of x(n). We can observe that the stability condition of the SL0-APA is independent of the parameter γSL0. We assume that the estimated vector h^(n) converges when n→∞. Then, (61) can be rewritten as
(63)E[Δ(∞)]=[IN-μSL0QRNδx2]E[Δ(∞)]E[Δ(∞)]=-12γSL0QRNδx2δsgn⁡(h)(h+δ)2+12γSL0δsgn⁡(h)(h+δ)2.From (63), we can obtain
(64)E[Δ(∞)]=-γSL02μSL0δsgn⁡(h)(h+δ)2+γSL02μSL0Nδx2QRδsgn⁡(h)(h+δ)2,
which can be regarded as
(65)E[h^(∞)]=h-γSL02μSL0δsgn⁡(h)(h+δ)2+γSL02μSL0Nδx2QRδsgn⁡(h)(h+δ)2.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(n) from the left, we can get
(66)U(n)h^(n+1)=U(n)h^(n)+μSL0U(n)U+(n)e(n)U(n)h^(n+1)=+12γSL0U(n)U+(n)U(n)T(n)U(n)h^(n+1)=-12γSL0U(n)T(n).Furthermore,(67)U(n)h^(n+1)=U(n)h^(n)+μSL0e(n).Additionally, we define the a posteriori error vectorep(n) and the a priori error vector ea(n) as
(68)ep(n)=U(n)h-U(n)h^(n+1)ea(n)=U(n)h-U(n)h^(n).Combining (67) and (68), we have
(69)ep(n)=ea(n)-μSL0e(n).In addition,(70)e(n)=r(n)-U(n)h^(n)e(n)=U(n)h+v(n)-U(n)h^(n)e(n)=ea(n)+v(n).By substituting (70) into (69), we have
(71)ep(n)=(I-μSL0)e(n)-v(n).From (69), we can also write the e(n) as follows:
(72)e(n)=1μSL0[ea(n)-ep(n)].Substituting (72) to (44), we have
(73)h^(n+1)=h^(n)+U+(n)[ea(n)-ep(n)]h^(n+1)=+12γSL0U+(n)U(n)T(n)-12γSL0T(n).On the basis of the discussion mentioned above, we notice thatU(n)U+(n)=U(n)UT(n)[U(n)UT(n)]-1=I. By considering the power of both sides of (73), using the steady-state condition E[∥h^(n+1)∥2]≈E[∥h^(n)∥2] when n→∞, and assuming that ea(n), ep(n), and h^(n) are independent of x(n) in the steady state, we get
(74)E[epT(n)[U(n)UT(n)]-1ep(n)]=E[eaT(n)[U(n)UT(n)]-1ea(n)]-γSL024E{TT(n)U+(n)U(n)T(n)}+γSL024E[TT(n)T(n)].Substituting (71) into the left-hand side (LHS) of (74), we have
(75)LHS=(1-μSL0)2E{eT(n)[U(n)UT(n)]-1e(n)}LHS=-(1-μSL0)E{eT(n)[U(n)UT(n)]-1v(n)}LHS=-(1-μSL0)E{vT(n)[U(n)UT(n)]-1e(n)}LHS=+E{vT(n)[U(n)UT(n)]-1v(n)}.Moreover, substituting (70) into the right-hand side (RHS) of (74), we have
(76)RHS=E{eT(n)[U(n)UT(n)]-1e(n)}RHS=-E{eT(n)[U(n)UT(n)]-1v(n)}RHS=-E{vT(n)[U(n)UT(n)]-1e(n)}RHS=+E{vT(n)[U(n)UT(n)]-1v(n)}RHS=-γSL024E{TT(n)U+(n)U(n)T(n)}RHS=+γSL024E[TT(n)T(n)].By combining (75) and (76), we get
(77)(2μSL0-μSL02)E{eT(n)[U(n)UT(n)]-1e(n)}=μSL0E{eT(n)[U(n)UT(n)]-1v(n)}+μSL0E{vT(n)[U(n)UT(n)]-1e(n)}+γSL024E{TT(n)U+(n)U(n)T(n)}-γSL024E[TT(n)T(n)].We also assume that the additive Gaussian noisev(n) is statistically independent of the input signal x(n). Thus (77) can be simplified as
(78)E{eT(n)[U(n)UT(n)]-1e(n)}=12-μSL0E{vT(n)[U(n)UT(n)]-1v(n)}+γSL024(2μSL0-μSL02)E{TT(n)U+(n)U(n)T(n)}-γSL024(2μSL0-μSL02)E[TT(n)T(n)].Here, we also assume that theU(n) is statistically independent of e(n) at the steady state. Moreover, we use the definition of E[eT(n)e(n)]=E|et(n)|2S [36], where
(79)S≈{I,μSL0issmall1·1T,μSL0islarge,
where 1T=[10⋯0] and et(n) is the top entry of e(n) [36]. Then, the LHS of (78) can be rewritten as
(80)E{eT(n)[U(n)UT(n)]-1e(n)}≈Tr⁡{E[eT(n)e(n)[U(n)UT(n)]-1]}≈E|et(n)|2Tr⁡{S·E[[U(n)UT(n)]-1]}.Similar to the calculation of (80), the first term in the RHS of (78) can be written as
(81)E{vT(n)[U(n)UT(n)]-1v(n)}=Tr⁡{E[vT(n)v(n)[U(n)UT(n)]-1]}=Qδv2Tr⁡{E[[U(n)UT(n)]-1]}.In addition, the second term of RHS of (78) can be rewritten as
(82)γSL024(2μSL0-μSL02)E{TT(n)U+(n)U(n)T(n)}≈γSL024(2μSL0-μSL02)δsgn⁡T(h)(hT+δ)2E{U+(n)U(n)}δsgn⁡(h)(h+δ)2.Then the last term of the right-hand side of (78) can be expressed as
(83)γSL024(2μSL0-μSL02)E[T(n)TT(n)]=γSL024(2μSL0-μSL02)δsgn⁡T(h)(hT+δ)2δsgn⁡(h)(h+δ)2.When theμSL0 is small, we can get
(84)Tr⁡{S·E[[U(n)UT(n)]-1]}=Tr⁡{I·E[[U(n)UT(n)]-1]}=QNδx2.Therefore, the MSE of the proposed SL0-APA with small step-sizeμSL0 can be written as
(85)MSEsmall=12-μSL0δv2MSEsmall=+γSL024(2μSL0-μSL02)δsgn⁡T(h)(hT+δ)2Nδx2QQRNδx2δsgn⁡(h)(h+δ)2MSEsmall=-γSL024(2μSL0-μSL02)δsgn⁡T(h)(hT+δ)2Nδx2Qδsgn⁡(h)(h+δ)2MSEsmall=12-μSL0δv2+γSL024(2μSL0-μSL02)δsgn⁡T(h)(hT+δ)2MSEsmall=×(R-Nδx2QI)δsgn⁡(h)(h+δ)2.When the step-sizeμSL0 is large, S≈1·1T [36]. In this case,
(86)Tr⁡{S·E[[U(n)UT(n)]-1]}=1Nδx2.Thus, the MSE of the proposed SL0-APA with large step-sizeμSL0 can be written as
(87)MSElarge=12-μSL0δv2QMSElarge=+γSL024(2μSL0-μSL02)δsgn⁡T(h)(hT+δ)2Nδx2QRNδx2δsgn⁡(h)(h+δ)2MSElarge=-γSL024(2μSL0-μSL02)δsgn⁡T(h)(hT(n)+δ)2Nδx2δsgn⁡(h)(h+δ)2MSElarge=12-μSL0δv2Q+γSL024(2μSL0-μSL02)δsgn⁡T(h)(hT+δ)2MSElarge=×(QR-Nδx2I)δsgn⁡(h)(h+δ)2.


## 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(35)JSL0(n)=∥h^(n+1)-h^(n)∥2JSL0(n)=+[r(n)-U(n)h^(n+1)]TλSL0JSL0(n)=+γSL0∥h^(n+1)∥0,
where λSL0 is the vector of the Lagrange multiplier with size of Q×1 and γSL0>0 is a regularization parameter to tradeoff the estimation error and the sparse l0-penalty of h^(n+1). Here, the smooth approximation of l0-norm ∥h^(n+1)∥0 is a continuous function defined as follows:
(36)∥h^(n+1)∥0=∑i=1N-1‍|h^i(n+1)||h^i(n+1)|+δ=|h^(n+1)||h^(n+1)|+δ,
where δ is a small positive constant which is used for avoiding division by zero, and the gradient of this continuous functions for SL0 is obtained:
(37)∂∥h^(n+1)∥0∂h^(n+1)=δsgn⁡(h^(n+1))(|h^(n+1)|+δ)2.To obtain the minimum of theJSL0(n), we use Lagrange multiplier to calculate the gradient of JSL0(n). Then the gradient of the cost function of the SL0-APA is written as
(38)∂JSL0(n)∂h^(n+1)=2h^(n+1)-2h^(n)-UT(n)λSL0∂JSL0(n)∂h^(n+1)=+γSL0δsgn⁡(h^(n+1))(|h^(n+1)|+δ)2.Let the left-hand side of (38) be equal to zero. We can get the following equation:
(39)h^(n+1)=h^(n)+12UT(n)λSL0-12γSL0δsgn⁡(h^(n+1))(|h^(n+1)|+δ)2.MultiplyingU(n) on both sides of (39), we can get
(40)U(n)h^(n+1)=U(n)h^(n)+12U(n)UT(n)λSL0U(n)h^(n+1)=-12γSL0U(n)δsgn⁡(h^(n+1))(|h^(n+1)|+δ)2.By taking (7) into consideration, (40) can be rewritten as
(41)r(n)=U(n)h^(n)+12U(n)UT(n)λSL0r(n)=-12γSL0U(n)δsgn⁡(h^(n+1))(|h^(n+1)|+δ)2.From the discussion of the ZA-APA and RZA-APA, we can get the Lagrange multiplier vectorλSL0 from (41) by taking e(n)=r(n)-U(n)h^(n) into account:
(42)λSL0=[U(n)UT(n)]-1λSL0=×{2e(n)+γSL0U(n)δsgn⁡(h^(n+1))(|h^(n+1)|+δ)2}.Substituting (42) into (39) and assuming that δsgn⁡(h^(n+1))/(|h^(n+1)|+δ)2≈δsgn⁡(h^(n))/(|h^(n)|+δ)2, the update function of the SL0-APA can be achieved:
(43)h^(n+1)=h^(n)+U+(n)e(n)h^(n+1)=+12γSL0U+(n)U(n)δsgn⁡(h^(n))(|h^(n)|+δ)2h^(n+1)=-12γSL0δsgn⁡(h^(n))(|h^(n)|+δ)2h^(n+1)=h^(n)+U+(n)e(n)h^(n+1)=+12γSL0U+(n)U(n)T(n)-12γSL0T(n),
where T(n)=δsgn⁡(h^(n))/(|h^(n)|+δ)2. Similar to the ZA-APA and RZA-APA, a step-size μSL0 is introduced into (43) to create a balance between the convergence speed and steady-state error of the SL0-APA:
(44)h^(n+1)=h^(n)+μSL0U+(n)e(n)h^(n+1)=+12γSL0U+(n)U(n)T(n)-12γSL0T(n).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∥h^(n+1)∥0, which is proved to be an approximate and near-accurate approximation of l0-norm in comparison with the sum-log function ∑i=1Nlog⁡(1+εRZA|h^i(n+1)|) in the RZA-APA. Moreover, it is easy to calculate the gradient, as we can easily find a continuous gradient for this smoothed l0-norm function.


## 3.2. 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 [36–38] is employed to obtain the theoretical expressions for the MSE of the SL0-APA. Let us consider the received signal r(n) that is derived from the following linear model:
(45)r(n)=U(n)h+v(n),
where h is the sparse channel vector of the multipath communication system that we wish to estimate and v(n) is the additive Gaussian noise at instant n. Our objective is to evaluate the steady-state MSE performance of the proposed SL0-APA. The steady-state MSE is defined as
(46)MSE≜lim⁡n→∞E[|e(n)|2],
where E[·] denotes the expectation and
(47)e(n)=r(n)-U(n)h^(n)
is the estimated error at time n. Taking (45) and (47) into account, we obtain
(48)e(n)=U(n)h+v(n)-U(n)h^(n)e(n)=U(n)[h-h^(n)]+v(n).Subtractingh from both sides of the SL0-APA update function (44), we get the misalignment vector:
(49)Δ(n+1)=h-h^(n+1)Δ(n+1)=h-{12h^(n)+μSL0U+(n)e(n)11111111111111+12γSL0U+(n)U(n)T(n)-12γSL0T(n)}.Substituting (48) into (49), we can get
(50)Δ(n+1)=h-h^(n)-μSL0U+(n)Δ(n+1)=×{U(n)[h-h^(n)]+v(n)}Δ(n+1)=-12γSL0U+(n)U(n)T(n)+12γSL0T(n)Δ(n+1)=[IN-μSL0U+(n)U(n)]Δ(n)-μSL0U+(n)v(n)Δ(n+1)=-12γSL0U+(n)U(n)T(n)+12γSL0T(n).Taking expectations on both sides of (50), we get
(51)E[Δ(n+1)]=E[IN-μSL0U+(n)U(n)]E[Δ(n)]E[Δ(n+1)]=-μSL0E[U+(n)v(n)]E[Δ(n+1)]=-12γSL0E[U+(n)U(n)T(n)]E[Δ(n+1)]=+12γSL0E[T(n)].We assume that the additive noisev(n) is statistically independent of the input signal x(n), and hence we have E[U+(n)v(n)]=0. Therefore, (51) can be simplified as
(52)E[Δ(n+1)]=E[IN-μSL0U+(n)U(n)]E[Δ(n)]E[Δ(n+1)]=-12γSL0E[U+(n)U(n)T(n)]E[Δ(n+1)]=+12γSL0E[T(n)].From previous studies on sparse LMS algorithms [22, 39], in the steady state, we have
(53)E{sgn⁡[h^(n)]}≈sgn⁡(h^).Thus, theE[T(n)] in (52) can be written as
(54)E[T(n)]=E{δsgn⁡(h^(n))(|h^(n)|+δ)2}=δsgn⁡(h^(n))(|h^(n)|+δ)2.In addition, when the channel length is far larger than 1,N≫1, the E[U+(n)U(n)] can be written as [37, 40, 41]
(55)E[U+(n)U(n)]=E{UT(n)[U(n)UT(n)]-1U(n)}E[U+(n)U(n)]≈E{UT(n){E[U(n)UT(n)]}-1U(n)}.SinceE[xT(n)x(n-1)]=0 for sparse channel estimation, the inner expectation reduces to(56)E[U(n)UT(n)]=E{[xT(n)xT(n-1)⋮xT(n-Q-1)][x(n)x(n-1)⋯x(n-Q-1)]}E[U(n)UT(n)]=E{[∥x(n)∥2xT(n)x(n-1)⋯xT(n)x(n-Q-1)xT(n-1)x(n)∥x(n-1)∥2⋯⋮⋮⋮⋱⋮xT(n-Q-1)x(n)xT(n-Q-1)x(n-1)⋯∥x(n-Q-1)∥2]}E[U(n)UT(n)]=E{[∥x(n)∥20⋯00∥x(n-1)∥2⋯0⋮⋮⋱⋮00⋯∥x(n-Q-1)∥2]}.Here, we define(57)R=δx2IN,
where δx2 is the power of the input signal. Thus,
(58)E[U(n)UT(n)]≈Tr⁡(R)IQ=Nδx2IQ,
where Tr⁡(·) is the trace of matrix and IQ is the Q×Q identity matrix. Moreover, we can obtain
(59)E{[U(n)UT(n)]-1}≈{E[U(n)UT(n)]}-1=1Nδx2IQ.Then we can approximateE{UT(n)[U(n)UT(n)]-1U(n)} by
(60)E{UT(n)[U(n)UT(n)]-1U(n)}≈E{UT(n)1Nδx2IQU(n)}E{UT(n)[U(n)UT(n)]-1U(n)}≈QRNδx2.Therefore, (52) can be rewritten as
(61)E[Δ(n+1)]=E[IN-μSL0QRNδx2]E[Δ(n)]E[Δ(n+1)]=-12γSL0QRNδx2E[T(n)]+12γSL0E[T(n)].It is found that the matrixT(n) is approximately bounded between -δIN and δIN. Therefore, we see that such convergence is guaranteed only if (IN-μSL0QR/Nδx2) is less than 1 [28], which is given by
(62)0<μSL0<Nδx2Qλmax⁡,
where λmax⁡ is the maximum eigenvalue of the autocorrelation matrix R of x(n). We can observe that the stability condition of the SL0-APA is independent of the parameter γSL0. We assume that the estimated vector h^(n) converges when n→∞. Then, (61) can be rewritten as
(63)E[Δ(∞)]=[IN-μSL0QRNδx2]E[Δ(∞)]E[Δ(∞)]=-12γSL0QRNδx2δsgn⁡(h)(h+δ)2+12γSL0δsgn⁡(h)(h+δ)2.From (63), we can obtain
(64)E[Δ(∞)]=-γSL02μSL0δsgn⁡(h)(h+δ)2+γSL02μSL0Nδx2QRδsgn⁡(h)(h+δ)2,
which can be regarded as
(65)E[h^(∞)]=h-γSL02μSL0δsgn⁡(h)(h+δ)2+γSL02μSL0Nδx2QRδsgn⁡(h)(h+δ)2.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(n) from the left, we can get
(66)U(n)h^(n+1)=U(n)h^(n)+μSL0U(n)U+(n)e(n)U(n)h^(n+1)=+12γSL0U(n)U+(n)U(n)T(n)U(n)h^(n+1)=-12γSL0U(n)T(n).Furthermore,(67)U(n)h^(n+1)=U(n)h^(n)+μSL0e(n).Additionally, we define the a posteriori error vectorep(n) and the a priori error vector ea(n) as
(68)ep(n)=U(n)h-U(n)h^(n+1)ea(n)=U(n)h-U(n)h^(n).Combining (67) and (68), we have
(69)ep(n)=ea(n)-μSL0e(n).In addition,(70)e(n)=r(n)-U(n)h^(n)e(n)=U(n)h+v(n)-U(n)h^(n)e(n)=ea(n)+v(n).By substituting (70) into (69), we have
(71)ep(n)=(I-μSL0)e(n)-v(n).From (69), we can also write the e(n) as follows:
(72)e(n)=1μSL0[ea(n)-ep(n)].Substituting (72) to (44), we have
(73)h^(n+1)=h^(n)+U+(n)[ea(n)-ep(n)]h^(n+1)=+12γSL0U+(n)U(n)T(n)-12γSL0T(n).On the basis of the discussion mentioned above, we notice thatU(n)U+(n)=U(n)UT(n)[U(n)UT(n)]-1=I. By considering the power of both sides of (73), using the steady-state condition E[∥h^(n+1)∥2]≈E[∥h^(n)∥2] when n→∞, and assuming that ea(n), ep(n), and h^(n) are independent of x(n) in the steady state, we get
(74)E[epT(n)[U(n)UT(n)]-1ep(n)]=E[eaT(n)[U(n)UT(n)]-1ea(n)]-γSL024E{TT(n)U+(n)U(n)T(n)}+γSL024E[TT(n)T(n)].Substituting (71) into the left-hand side (LHS) of (74), we have
(75)LHS=(1-μSL0)2E{eT(n)[U(n)UT(n)]-1e(n)}LHS=-(1-μSL0)E{eT(n)[U(n)UT(n)]-1v(n)}LHS=-(1-μSL0)E{vT(n)[U(n)UT(n)]-1e(n)}LHS=+E{vT(n)[U(n)UT(n)]-1v(n)}.Moreover, substituting (70) into the right-hand side (RHS) of (74), we have
(76)RHS=E{eT(n)[U(n)UT(n)]-1e(n)}RHS=-E{eT(n)[U(n)UT(n)]-1v(n)}RHS=-E{vT(n)[U(n)UT(n)]-1e(n)}RHS=+E{vT(n)[U(n)UT(n)]-1v(n)}RHS=-γSL024E{TT(n)U+(n)U(n)T(n)}RHS=+γSL024E[TT(n)T(n)].By combining (75) and (76), we get
(77)(2μSL0-μSL02)E{eT(n)[U(n)UT(n)]-1e(n)}=μSL0E{eT(n)[U(n)UT(n)]-1v(n)}+μSL0E{vT(n)[U(n)UT(n)]-1e(n)}+γSL024E{TT(n)U+(n)U(n)T(n)}-γSL024E[TT(n)T(n)].We also assume that the additive Gaussian noisev(n) is statistically independent of the input signal x(n). Thus (77) can be simplified as
(78)E{eT(n)[U(n)UT(n)]-1e(n)}=12-μSL0E{vT(n)[U(n)UT(n)]-1v(n)}+γSL024(2μSL0-μSL02)E{TT(n)U+(n)U(n)T(n)}-γSL024(2μSL0-μSL02)E[TT(n)T(n)].Here, we also assume that theU(n) is statistically independent of e(n) at the steady state. Moreover, we use the definition of E[eT(n)e(n)]=E|et(n)|2S [36], where
(79)S≈{I,μSL0issmall1·1T,μSL0islarge,
where 1T=[10⋯0] and et(n) is the top entry of e(n) [36]. Then, the LHS of (78) can be rewritten as
(80)E{eT(n)[U(n)UT(n)]-1e(n)}≈Tr⁡{E[eT(n)e(n)[U(n)UT(n)]-1]}≈E|et(n)|2Tr⁡{S·E[[U(n)UT(n)]-1]}.Similar to the calculation of (80), the first term in the RHS of (78) can be written as
(81)E{vT(n)[U(n)UT(n)]-1v(n)}=Tr⁡{E[vT(n)v(n)[U(n)UT(n)]-1]}=Qδv2Tr⁡{E[[U(n)UT(n)]-1]}.In addition, the second term of RHS of (78) can be rewritten as
(82)γSL024(2μSL0-μSL02)E{TT(n)U+(n)U(n)T(n)}≈γSL024(2μSL0-μSL02)δsgn⁡T(h)(hT+δ)2E{U+(n)U(n)}δsgn⁡(h)(h+δ)2.Then the last term of the right-hand side of (78) can be expressed as
(83)γSL024(2μSL0-μSL02)E[T(n)TT(n)]=γSL024(2μSL0-μSL02)δsgn⁡T(h)(hT+δ)2δsgn⁡(h)(h+δ)2.When theμSL0 is small, we can get
(84)Tr⁡{S·E[[U(n)UT(n)]-1]}=Tr⁡{I·E[[U(n)UT(n)]-1]}=QNδx2.Therefore, the MSE of the proposed SL0-APA with small step-sizeμSL0 can be written as
(85)MSEsmall=12-μSL0δv2MSEsmall=+γSL024(2μSL0-μSL02)δsgn⁡T(h)(hT+δ)2Nδx2QQRNδx2δsgn⁡(h)(h+δ)2MSEsmall=-γSL024(2μSL0-μSL02)δsgn⁡T(h)(hT+δ)2Nδx2Qδsgn⁡(h)(h+δ)2MSEsmall=12-μSL0δv2+γSL024(2μSL0-μSL02)δsgn⁡T(h)(hT+δ)2MSEsmall=×(R-Nδx2QI)δsgn⁡(h)(h+δ)2.When the step-sizeμSL0 is large, S≈1·1T [36]. In this case,
(86)Tr⁡{S·E[[U(n)UT(n)]-1]}=1Nδx2.Thus, the MSE of the proposed SL0-APA with large step-sizeμSL0 can be written as
(87)MSElarge=12-μSL0δv2QMSElarge=+γSL024(2μSL0-μSL02)δsgn⁡T(h)(hT+δ)2Nδx2QRNδx2δsgn⁡(h)(h+δ)2MSElarge=-γSL024(2μSL0-μSL02)δsgn⁡T(h)(hT(n)+δ)2Nδx2δsgn⁡(h)(h+δ)2MSElarge=12-μSL0δv2Q+γSL024(2μSL0-μSL02)δsgn⁡T(h)(hT+δ)2MSElarge=×(QR-Nδx2I)δsgn⁡(h)(h+δ)2.


## 4. 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 Section3.2. 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.


### 4.1. Performance of the Proposed SL0-APA

Firstly, we set up a simulation example to discuss the convergence speed of 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 channelh whose length N is equal to 16 and whose number of dominant taps K is set to two different sparsity levels, namely, K=1,K=4, similarly to [6, 22, 25, 26]. The dominant channel taps are obtained from a Gaussian distribution subjected to ∥h∥22=1, and the positions of the dominant channel taps are random within the length of the channel. The input signal x(n) of the channel is a Gaussian random signal, while the output of the channel is corrupted by an independent white Gaussian noise v(n). An example of a typical sparse multipath channel with a channel length of N=16 and a sparsity level of K=3 is shown in Figure 2. In the simulations, the power of the received signal is Eb=1, while the noise power is given by δv2. 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:
(88)MSE(n)=10log⁡10E{∥h-h^(n)∥22}(dB).Figure 2
A typical sparse multipath channel.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:μNLMS=0.25, μAPA=0.125, μZA=0.165, μRZA=0.18, μSL0=0.21, γZA=5×10-5, γRZA=8×10-5, γSL0=3×10-6, εRZA=10, δSL0=0.001, Q=2, and δv2=10-3, where μNLMS 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 Q. Thus, we discuss the effects of the affine projection order Q for SL0-APA and compare it with the APA and NLMS algorithms. The computer simulation results with different values of Q are shown in Figure 4. It reveals that the convergence speed is improved by the increment of the affine projection order Q. However, the steady-state performance has deteriorated from Q=2 to Q=8. Thus, in our proposed SL0-APA, the affine projection Q, the step-size μSL0, the regularization parameter γSL0, and δSL0 should be take into account to balance the convergence speed and the steady-state behavior.Figure 3
Convergence of the proposed SL0-APA.Figure 4
Affine projection order effects on the SL0-APA.Next, we show the effects of the sparsity levels on the steady-state performance of the proposed SL0-APA atK=1 and K=4. To obtain the same convergence speed, the simulation parameters used in this experiment are listed as follows: μNLMS=0.095, μAPA=μZA=μRZA=μSL0=0.05, γZA=5×10-5, γRZA=8×10-5, γSL0=4×10-6, εRZA=10, δ=0.01, and δv2=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 K=1 and Q=2 shown in Figure 5(a). When the sparsity level K 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 K=1. When the affine projection order increases to Q=3, we can see from Figure 6 that the convergence speed is significantly improved compared to that of Q=2 shown in Figure 5. However, the steady-state error is also slightly increased when the Q increases. Furthermore, our proposed SL0-APA still has the best convergence speed and lowest steady-state error.Performances of the SL0-APA with different sparsity levels forQ=2.
(a)
K
=
1
(b)
K
=
4Performances of the SL0-APA with different sparsity levels forQ=3.
(a)
K
=
1
(b)
K
=
4Finally, we use the theoretical expressions obtained in Section3.2 to predict the mean-square-error (MSE) of the proposed SL0-APA with different μSL0 and compare the theoretical results with the simulation ones. The MSE comparisons of the SL0-APA as a function of the step-size μSL0 for the designated sparse multipath communication channel with the simulation parameters of γSL0=4×10-6, δ=0.01, δv2=10-3, Q=3, and K=1 are shown in Figure 7. The theoretical results are obtained from (85) to (87) for small values of μSL0 and large values of μSL0, 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 μSL0. 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 μSL0 for small μSL0 shown in Figure 7(a), but the steady-state error is becoming lower. For the large μSL0, both the steady-state error and the convergence speed are deteriorated by the increment of the step-size μSL0. Generally speaking, as μSL0 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 μSL0 carefully in order to balance convergence speed and steady-state performance.Steady-state MSE performance of the SL0-APA with different step-sizeμSL0 for K=4.
(a)
SmallμSL0
(b)
LargeμSL0


### 4.2. Computational Complexity

In this subsection, we present the computational complexity of the proposed SL0-APA and compare it with the conventional sparsity-aware channel estimation algorithms, including the APA, ZA-APA, and RZA-APA. It is worth noting that when the affine projection orderQ 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 K nonzero taps in a sparse channel model as an FIR filter with N coefficients, and the order of these affine projection algorithms is Q. The computational complexity of the proposed SL0-APA and the relevant sparsity-aware algorithms are shown in Table 1.Table 1
Computational complexity.
Algorithms
Additions
Multiplications
Divisions
NLMS
3
N
3
N
+
1
1
ZA-NLMS
N
+
3
K
N
+
3
K
+
1
1
RZA-NLMS
N
+
4
K
N
+
4
K
+
1
N
+
1
APA
N
Q
2
+
N
Q
+
K
-
N
+
O
(
Q
3
)
N
Q
2
+
Q
2
+
N
Q
+
Q
+
O
(
Q
3
)
Q
ZA-APA
N
Q
2
+
Q
2
+
3
N
Q
-
2
Q
-
2
N
+
3
K
+
O
(
Q
3
)
N
Q
2
+
2
Q
2
+
3
N
Q
+
2
Q
+
K
+
O
(
Q
3
)
Q
RZA-APA
N
Q
2
+
Q
2
+
3
N
Q
-
2
Q
-
2
N
+
4
K
+
O
(
Q
3
)
N
Q
2
+
2
Q
2
+
3
N
Q
+
Q
+
N
+
2
K
+
O
(
Q
3
)
N
+
Q
SL0-APA
N
Q
2
+
Q
2
+
3
N
Q
-
2
Q
-
2
N
+
4
K
+
O
(
Q
3
)
N
Q
2
+
2
Q
2
+
3
N
Q
+
Q
+
N
2
-
N
+
2
K
+
O
(
Q
3
)
N
+
QFrom Table1, 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.


## 4.1. Performance of the Proposed SL0-APA

Firstly, we set up a simulation example to discuss the convergence speed of 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 channelh whose length N is equal to 16 and whose number of dominant taps K is set to two different sparsity levels, namely, K=1,K=4, similarly to [6, 22, 25, 26]. The dominant channel taps are obtained from a Gaussian distribution subjected to ∥h∥22=1, and the positions of the dominant channel taps are random within the length of the channel. The input signal x(n) of the channel is a Gaussian random signal, while the output of the channel is corrupted by an independent white Gaussian noise v(n). An example of a typical sparse multipath channel with a channel length of N=16 and a sparsity level of K=3 is shown in Figure 2. In the simulations, the power of the received signal is Eb=1, while the noise power is given by δv2. 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:
(88)MSE(n)=10log⁡10E{∥h-h^(n)∥22}(dB).Figure 2
A typical sparse multipath channel.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:μNLMS=0.25, μAPA=0.125, μZA=0.165, μRZA=0.18, μSL0=0.21, γZA=5×10-5, γRZA=8×10-5, γSL0=3×10-6, εRZA=10, δSL0=0.001, Q=2, and δv2=10-3, where μNLMS 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 Q. Thus, we discuss the effects of the affine projection order Q for SL0-APA and compare it with the APA and NLMS algorithms. The computer simulation results with different values of Q are shown in Figure 4. It reveals that the convergence speed is improved by the increment of the affine projection order Q. However, the steady-state performance has deteriorated from Q=2 to Q=8. Thus, in our proposed SL0-APA, the affine projection Q, the step-size μSL0, the regularization parameter γSL0, and δSL0 should be take into account to balance the convergence speed and the steady-state behavior.Figure 3
Convergence of the proposed SL0-APA.Figure 4
Affine projection order effects on the SL0-APA.Next, we show the effects of the sparsity levels on the steady-state performance of the proposed SL0-APA atK=1 and K=4. To obtain the same convergence speed, the simulation parameters used in this experiment are listed as follows: μNLMS=0.095, μAPA=μZA=μRZA=μSL0=0.05, γZA=5×10-5, γRZA=8×10-5, γSL0=4×10-6, εRZA=10, δ=0.01, and δv2=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 K=1 and Q=2 shown in Figure 5(a). When the sparsity level K 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 K=1. When the affine projection order increases to Q=3, we can see from Figure 6 that the convergence speed is significantly improved compared to that of Q=2 shown in Figure 5. However, the steady-state error is also slightly increased when the Q increases. Furthermore, our proposed SL0-APA still has the best convergence speed and lowest steady-state error.Performances of the SL0-APA with different sparsity levels forQ=2.
(a)
K
=
1
(b)
K
=
4Performances of the SL0-APA with different sparsity levels forQ=3.
(a)
K
=
1
(b)
K
=
4Finally, we use the theoretical expressions obtained in Section3.2 to predict the mean-square-error (MSE) of the proposed SL0-APA with different μSL0 and compare the theoretical results with the simulation ones. The MSE comparisons of the SL0-APA as a function of the step-size μSL0 for the designated sparse multipath communication channel with the simulation parameters of γSL0=4×10-6, δ=0.01, δv2=10-3, Q=3, and K=1 are shown in Figure 7. The theoretical results are obtained from (85) to (87) for small values of μSL0 and large values of μSL0, 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 μSL0. 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 μSL0 for small μSL0 shown in Figure 7(a), but the steady-state error is becoming lower. For the large μSL0, both the steady-state error and the convergence speed are deteriorated by the increment of the step-size μSL0. Generally speaking, as μSL0 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 μSL0 carefully in order to balance convergence speed and steady-state performance.Steady-state MSE performance of the SL0-APA with different step-sizeμSL0 for K=4.
(a)
SmallμSL0
(b)
LargeμSL0


## 4.2. Computational Complexity

In this subsection, we present the computational complexity of the proposed SL0-APA and compare it with the conventional sparsity-aware channel estimation algorithms, including the APA, ZA-APA, and RZA-APA. It is worth noting that when the affine projection orderQ 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 K nonzero taps in a sparse channel model as an FIR filter with N coefficients, and the order of these affine projection algorithms is Q. The computational complexity of the proposed SL0-APA and the relevant sparsity-aware algorithms are shown in Table 1.Table 1
Computational complexity.
Algorithms
Additions
Multiplications
Divisions
NLMS
3
N
3
N
+
1
1
ZA-NLMS
N
+
3
K
N
+
3
K
+
1
1
RZA-NLMS
N
+
4
K
N
+
4
K
+
1
N
+
1
APA
N
Q
2
+
N
Q
+
K
-
N
+
O
(
Q
3
)
N
Q
2
+
Q
2
+
N
Q
+
Q
+
O
(
Q
3
)
Q
ZA-APA
N
Q
2
+
Q
2
+
3
N
Q
-
2
Q
-
2
N
+
3
K
+
O
(
Q
3
)
N
Q
2
+
2
Q
2
+
3
N
Q
+
2
Q
+
K
+
O
(
Q
3
)
Q
RZA-APA
N
Q
2
+
Q
2
+
3
N
Q
-
2
Q
-
2
N
+
4
K
+
O
(
Q
3
)
N
Q
2
+
2
Q
2
+
3
N
Q
+
Q
+
N
+
2
K
+
O
(
Q
3
)
N
+
Q
SL0-APA
N
Q
2
+
Q
2
+
3
N
Q
-
2
Q
-
2
N
+
4
K
+
O
(
Q
3
)
N
Q
2
+
2
Q
2
+
3
N
Q
+
Q
+
N
2
-
N
+
2
K
+
O
(
Q
3
)
N
+
QFrom Table1, 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.


## 5. Conclusion

In this paper, we proposed an SL0-APA to exploit the sparsity of sparse channel and to improve the performance on both the convergence speed and steady-state error of the APA, ZA-APA, and RZA-APA. This algorithm is mainly developed by introducing a smooth approximationl0-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 sparsity-aware 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.



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*Source: 937252-2014-03-26.xml*