## 2 Problem Formulation  
### Review of SLMS-RL1 Algorithm  
Consider an additive noise interference channel, which is modeled by the unknown \(N\)-length finite impulse response (FIR) vector \(\mathbf{w}=[w_{0},w_{1},...,w_{N-1}]^{T}\) at discrete time index \(n\). The ideal received signal is expressed as\[d(n)=\mathbf{x}^{T}(n)\mathbf{w}+z(n) \tag{1}\]  
where \(\mathbf{x}(n)=[x(n),x(n-1),...,x(n-N+1)]^{T}\) is the input signal vector of the \(N\) most recent input samples; \(\mathbf{w}\) is an \(N\)-dimensional column vector of the unknown system that we wish to estimate, and \(z(n)\) is impulsive noise which can be described by Gaussian mixture model (GMM) [13] distribution as  
\[p\big{(}z(n)\big{)}=(1-\phi)\cdot\mathcal{CN}(0,\sigma_{n}^{2})+\phi\cdot \mathcal{CN}(0,T\sigma_{n}^{2}) \tag{2}\]  
where \(T\gg 1\) denotes impulsive-noise strength and \(\mathcal{CN}(0,\sigma_{n}^{2})\) denotes the Gaussian distributions with zero mean and variance \(\sigma_{n}^{2}\), and the \(\phi\) is the mixture parameter to control the impulsive noise level. According to Equation (2), one can find that stronger impulsive noises can be described by larger noise variance \(T\sigma_{n}^{2}\) as well as larger mixture parameter \(\phi\). According to Equation (2), variance of GMM is obtained. \[\sigma_{z}^{2}=E\{z^{2}(n)\}=(1-\phi)\sigma_{n}^{2}+\phi T\sigma_{n}^{2} \tag{3}\]  
Note that \(z(n)\) will reduce to Gaussian noise model if \(\phi=0\). The objective of the adaptive channel estimation is to perform adaptive estimate of \(\mathbf{w}(n)\) with limited complexity and memory given sequential observation \(\{d(n),\mathbf{x}(n)\}\) in the presence of additive GMM noise \(z(n)\). According to Equation (1), instantaneous estimation error \(e(n)\) can be written as  
\[e(n)=d(n)-\mathbf{w}^{T}(n)\mathbf{x}(n) \tag{4}\]  
where \(\mathbf{w}(n)\) is the estimator of \(\mathbf{w}\) at iteration \(n\). To obtain the optimal channel estimation, one can construct the \(\ell_{0}\)-norm minimization problem as  
\[G(n)=\left\|e(n)\right\|_{1}+\lambda\left\|\mathbf{w}(n)\right\|_{0} \tag{5}\]  
where \(\left\|\mathbf{w}(n)\right\|_{0}\) denotes \(\ell_{0}\)-norm operator which is defined as \(\left\|\mathbf{w}(n)\right\|_{0}=\#\{i|w_{i}(n)\
eq 0\}\). That is to say, the main function of \(\left\|\mathbf{w}(n)\right\|_{0}\) is to find the total number of nonzero coefficients. However, solving the \(\ell_{0}\)-norm minimization is a Non-Polynomial (NP) hard problem. Hence, it is necessary to introduce an approximate \(\ell_{0}\)-norm minimization function so that Equation (5) is solvable. On the adaptive sparse channel estimation, reweighted \(\ell_{1}\)-norm (RL1) minimization has a better performance than \(\ell_{1}\)-minimization that is usually employed in compressive sensing [24]. It is due to the fact that a properly RL1 can approximate the \(\ell_{0}\)-norm more accurate than \(\ell_{1}\)-norm. Hence, one approach to enforce the sparsity of the solution for the sparse SLMS algorithm is to introduce the RL1 penalty term in thee cost function as RL1-LAE which considers a penalty term proportional to the RL1 of the coefficient vector. Hence, the cost function Equation (5) can be revised as  
\[G(n)=\left\|e(n)\right\|_{1}+\lambda\left\|\mathbf{F}(n)\mathbf{w}(n)\right\| _{1} \tag{6}\]  
where \(\lambda\) is the weight associated with the penalty term and elements of the \(N\times N\) diagonal reweighted matrix \(\mathbf{F}(n)\) are devised as  
\[[\mathbf{F}(n)]_{ii}=\frac{1}{\varepsilon+\left|[\mathbf{w}(n-1)]_{i}\right|} \,,i=0,1,...,N-1 \tag{7}\]where \(\varepsilon\) being some positive number and hence \([\mathbf{F}(n)]_{li}>0\) for \(i=0,1,...,N-1\). The update equation can be derived by differentiating (6) with respect to the FIR channel vector \(\mathbf{w}(n)\). Then, the resulting update equation is  
\[