## 2 Traditional Channel Estimation Technique  
An input-output wireless system under the S\(\alpha\)S noise environment is considered. The wireless channel vector is described by \(N\)-length FIR sparse vector \(\mathbf{w}=\left[w_{0},w_{1},\ldots,w_{N-1}\right]^{T}\) at discrete time-index \(n\). The received signal is obtained as  
\[d(n)=\mathbf{w}^{T}\mathbf{x}(n)+z(n) \tag{1}\]  
where \(\mathbf{x}(n)=\left[x(n),x(n-1),\cdots,x(n-N+1)\right]^{T}\) is the input signal vector of the \(N\) most recent input samples with distribution of \(\mathcal{CN}\left(0,\sigma_{x}^{2}\right)\) and \(z(n)\sim\phi\left(\alpha,\beta,\gamma,\delta\right)\) denotes a S\(\alpha\)S noise variable. To understand the characteristic function of S\(\alpha\)S noise, here we define it as  
\[p(t)=\exp\left\{j\delta t-\gamma^{\alpha}\left|t\right|^{\alpha}\left[1-j \beta\text{sgn}(t)\phi\left(t,\alpha\right)\right]\right\} \tag{2}\]  
where  
\[0<\alpha\leq 2,\ -1\leq\beta\leq 1,\ \gamma>0,\ -\infty<\delta<\infty \tag{3}\]  
\[\text{sgn}(t)= \begin{cases}1,&t>0\\ 0,&t=0\\ -1,&t<0\end{cases}  \tag{4}\]  
\[\phi\left(t,\alpha\right)= \begin{cases}\tan(\alpha\pi/2),&\alpha\
eq 1\\ -\log(\left|t\right|),&\alpha=1\end{cases}  \tag{5}\]  
In Equation (2), \(\alpha\in(0,2]\) controls the tail heaviness of S\(\alpha\)S noise. Since when \(\alpha<1\) is rare to happen S\(\alpha\)S noise in practical systems, \(\alpha\in(1,2]\) is considered throughout this paper [11]. \(\gamma>0\) denotes the dispersive parameter, which can perform a similar role to Gaussian distribution; \(\beta\in[-1,1]\) stands for the symmetrical parameter. To have a better understanding of the alpha-stable noise, its probability density function (PDF) curves are depicted in Figures 1 and 2 as examples. Figure 1a shows that the PDF curve of symmetric alpha-stable noise changes with the parameter \(\alpha\), i.e., a smaller \(\alpha\) produces a larger PDF of the alpha-stable noise model and vice versa. In other words, \(\alpha\) controls the strength of the impulsive noise. Similarly, Figure 1b shows that the PDF curve of skewed \(\alpha\)-stable noise model also changes simultaneously with \(\alpha\) and \(\beta\). Since the skewed noise model may not exist in practical wireless communication systems, the symmetrical \(\alpha\)-stable noise model is used in this paper; the characteristic function of \(\alpha\)-stable process reduces as  
\[p(t)=\exp\left(-\gamma\left|t\right|^{\alpha}\right) \tag{6}\]  
For convenience, symmetric \(\alpha\)-stable noise variance is defined by  
\[\sigma_{z}^{2}=\gamma^{2/\alpha} \tag{7}\]  
and the generalized received signal noise ratio (GSNR) is defined by  
\[E_{s}/N_{0}\left(\mathrm{dB}\right)\triangleq 10\times\log_{10}\left\{P_{0}/ \gamma^{2/\alpha}\right\} \tag{8}\]  
where \(P_{0}\) denotes the received signal power while \(\sigma_{z}^{2}=\gamma^{2/\alpha}\) plays the same role as the noise variance. The objective of adaptive channel estimation is to perform adaptive estimate of \(\mathbf{w}(n)\) with limited complexity and memory given sequential observation \(\left\{d(n),\mathbf{x}(n)\right\}\) in the presence of additive noise \(z(n)\). That is to say, the estimate observation signal \(y(n)\) is given as  
\[y(n)=\mathbf{w}^{T}(n)\mathbf{x}(n) \tag{9}\]  
where \(\mathbf{w}(n)\) denotes channel estimator. By combining (1) and (4), the estimation error \(e(n)\) is  
\[e(n)=d(n)-y(n)=z(n)-\mathbf{x}^{T}(n)\mathbf{v}(n) \tag{10}\]  
where \(\mathbf{v}(n)=\mathbf{w}(n)-\mathbf{w}\) is the updating error of \(\mathbf{w}(n)\) at iteration \(n\). The cost function of standard LMS was written as  
\[G_{\text{LMS}}\left(\mathbf{w}(n)\right)=\left(1/2\right)e^{2}(n) \tag{11}\]  
Using Equation (11), the standard LMS algorithm was derived as  
\[ \begin{array}{ll}\mathbf{w}(n+1)&=\mathbf{w}(n)+\upmu\frac{\partial G(\mathbf{w}(n))}{ \partial\mathbf{w}(n)}\\ &=\mathbf{w}(n)+\upmu e(n)\mathbf{x}(n)\end{array}  \tag{12}\]  
where \(\upmu\) denotes a step-size that controls the gradient descend speed of the LMS.