## 4 Convergence Analysis of the Proposed Algorithms  
Unlike the standard SLMS algorithm, the proposed sparse SLMS algorithms can further improve estimation accuracy by exploiting channel sparsity. For convenience of theoretical analysis without loss of generality, the above proposed sparse SLMS algorithms are generalized as  
\[\mathbf{w}(n+1)=\mathbf{w}(n)+\mu\text{sgn}\left(e(n)\right)\mathbf{x}(n)-\rho f\left(\bm {w}(n)\right) \tag{38}\]  
where \(f\left(\mathbf{w}(n)\right)\) denotes sparsity constraint function and \(\rho\) denotes the regularization parameter. Throughout this paper, our analysis is based on independent assumptions as below:  
\[E\left(z(n)\mathbf{x}(n)\right)=0 \tag{39}\]  
\[E\left[z(n)\mathbf{x}(n)\mathbf{v}^{T}(n)\right]=0 \tag{40}\]  
\[E\left[z(n)\mathbf{v}(n)\mathbf{x}^{T}(n)\right]=0 \tag{41}\]  
\[E\left[z(n)f\left(\mathbf{w}(n)\right)\mathbf{x}^{T}(n)\right]=0 \tag{42}\]  
\[E\left[z(n)\mathbf{x}(n)f^{T}\left(\mathbf{w}(n)\right)\right]=0 \tag{43}\]  
**Theorem 1**. : _If \(\mu\) satisfies (15), the mean coefficient vector \(E\left\{\mathbf{w}(n)\right\}\) approaches_  
\[\mathbf{w}(\infty)=\mathbf{w}-\sqrt{\pi/2}\mu^{-1}\rho R^{-1}\gamma^{1/\alpha}f\left( \mathbf{w}(\infty)\right) \tag{44}\]  
**Proof. ** By subtracting \(\mathbf{w}\) from both sides of (35), the mean estimation error \(E\left\{\mathbf{v}(n+1)\right\}\) is derived as \[ \begin{array}{ll}E\left\{\mathbf{v}(n+1)\right\}&=E\left\{\mathbf{v}(n)\right\}+\mu E \left\{\text{sgn}(e(n))\mathbf{x}(n)\right\}-\rho f\left(\mathbf{w}(n)\right)\\ &\approx E\left\{\mathbf{v}(n)\right\}+\mu E\left\{\text{sgn}\left(e(n)\right)\mathbf{x }(n)\left|\mathbf{v}(n)\right.\right\}-\rho f\left(\mathbf{w}(n)\right)\\ &=E\left\{\mathbf{v}(n)\right\}+\sqrt{2/\pi}\mu\sigma_{e}^{-1}(n)E\left\{e(n)\mathbf{ x}(n)\right\}-\rho f\left(\mathbf{w}(n)\right)\\ &=\left\{\mathbf{I}-\sqrt{2/\pi}\mu\sigma_{e}^{-1}(n)\mathbf{R}\right\}E\left\{\mathbf{v}(n )\right\}-\rho f\left(\mathbf{w}(n)\right)\end{array}  \tag{45}\]  
It is worth mentioning that vector \(\rho f\left(\mathbf{w}(n)\right)\) is bounded for all sparse constraints. For example, if \(f(\mathbf{w}(n))=\text{sgn}(\mathbf{w}(n))\), then the bound is between \(-\rho I_{N}\) and \(\rho I_{N}\), where \(I_{N}\) is an \(N\)-length identity vector. For \(n\to\infty\), Equation (45) can be rewritten as  
\[ \begin{array}{ll}E\left\{\mathbf{v}(\infty)\right\}&=\left\{\mathbf{I}-\sqrt{2/\pi} \mu\sigma_{e}^{-1}(\infty)\mathbf{R}\right\}E\left\{\mathbf{v}(\infty)\right\}-\rho f \left(\mathbf{w}(\infty)\right)\\ &\approx\left\{\mathbf{I}-\sqrt{2/\pi}\mu\gamma^{-1/\alpha}\mathbf{R}\right\}E\left\{ \mathbf{v}(\infty)\right\}-\rho f\left(\mathbf{w}(\infty)\right)\end{array}  \tag{46}\]  
where \(\text{Tr}\left(\mathbf{RC}(\infty)\right)=\sigma_{x}^{2}c(\infty)\ll\gamma^{1/\alpha}\) and  
\[\sigma_{e}^{2}(\infty)=\lim_{n\to\infty}\sigma_{e}^{2}(n)=\gamma^{2/\alpha}+ \text{Tr}\left(\mathbf{RC}(\infty)\right)\approx\gamma^{2/\alpha} \tag{47}\]  
are utilized in the above equation. Since \(E\left\{\mathbf{w}(\infty)\right\}=\mathbf{w}-E\left\{\mathbf{v}(\infty)\right\}\), according to Equation (47), one can easily get Theorem 1. Theorem 2: _Let \(\Omega\) denotes the index set of nonzero taps, i.e., \(w_{i}\
eq 0\) for \(i\in\Omega\).