## 2 MCC \(l_{1}\)-IWF Formulation  
In the channel estimation problem, we assume that at time instant n the observed signal \(y(n)\) is the result of the input signal \(x(k)\) sequence passing through the system \(\mathbf{w}_{o}~{}=~{}[w_{0},\cdots,w_{M-1}]^{T}\) in the M-dimensional finite impulse response (FIR) format. Especially, in the sparse channel estimation problem, we assume that the system response w is sparse. In the adaptive channel estimation, we apply an M dimensional channel \(\mathbf{w}(k)\) to the same dimensional signal vector \(\mathbf{x}(k)\), estimate an output \(\hat{y}(k)~{}=~{}\mathbf{x}^{T}(k)\mathbf{w}(k)\), and calculate the error signal \(e(k)~{}=~{}y(k)+n(k)-\hat{y}(k)~{}=~{}\overline{y}(k)-\hat{y}(k)\), where \(y(k)\) is the output of the actual system, \(\hat{y}(k)\) is the estimated output, and \(n(k)\) is the measurement noise. Especially, the measurement noise is nonGaussian. To estimate the channel in nonGaussian noise, we define an MCC cost function with exponential forgetting factor \(\lambda\) shown in (1) [20; 21] and minimize it adaptively. \[\underset{\hat{\mathbf{w}}(n)}{\text{minimize}}\left\{\sum_{m~{}=~{}0}^{n} \lambda^{n-m}\exp\left(-\frac{\left(y(m)-\hat{\mathbf{w}}(n)^{T}\mathbf{x}(m) \right)^{2}}{2\sigma^{2}}\right)\text{s.t.}\big{\|}\hat{\mathbf{w}}(n)\big{\|} _{1}\leq c\right\}, \tag{1}\]  
where \(\hat{\mathbf{w}}(n)~{}=~{}[\hat{w}_{0}(n),\cdots,\hat{w}_{M-1}(n)]^{T}\), \(\mathbf{x}(m)~{}=~{}[x(m),x(m-1),\cdots,x(m-M+1)]^{T}\), \(\lambda\) is a forgetting factor, and \(\big{\|}\hat{\mathbf{w}}(n)\big{\|}_{1}\triangleq\sum_{k~{}=~{}0}^{M-1}[\hat{ w}_{k}(n)\big{|}\). The Lagrangian for (1) becomes  
\[J(\hat{\mathbf{w}}(n),\gamma(n))~{}=~{}\zeta(\hat{\mathbf{w}}(n))+\gamma(n) \big{(}\big{\|}\hat{\mathbf{w}}(n)\big{\|}_{1}-c\big{)}, \tag{2}\]  
where \(\zeta(\hat{\mathbf{w}}(n))~{}=~{}\sum\limits_{m~{}=~{}0}^{n}\lambda^{n-m}\exp \left(-\frac{\left(y(m)-\hat{\mathbf{w}}(n)^{T}\mathbf{x}(m)\right)^{2}}{2 \sigma^{2}}\right)\), and \(\gamma(n)\) is a real-valued Lagrangian multiplier. We minimize the regularized cost function to find the optimal vector in the same way that IWF was derived [24]. The regularized cost function is convex and nondifferentiable; therefore, subgradient analysis replaces the gradient. When denoting a subgradient vector of \(f\) at \(\hat{\mathbf{w}}\) with \(\
abla^{s}f(\hat{\mathbf{w}})\), the subgradient vector of \(J(\hat{\mathbf{w}}(n),\gamma(n))\) with respect to \(\hat{\mathbf{w}}(n)\) can be written as follows:  
\[\
abla^{s}J(\hat{\mathbf{w}}(n),\gamma(n))~{}=~{}\
abla\zeta(\hat{\mathbf{w}} (n))+\gamma(n)\
abla^{s}\big{(}\big{\|}\hat{\mathbf{w}}(n)\big{\|}_{1}\big{)}. \tag{3}\]  
Hence, for the optimal \(\hat{\mathbf{w}}(n)\) minimizing \(J(\hat{\mathbf{w}}(n),\gamma(n))\), we set the subgradient of \(J(\hat{\mathbf{w}}(n),\gamma(n))\) to \(0\) at the optimal point. When evaluating the gradient \(\
abla\zeta(\hat{\mathbf{w}}(n))\), we can derive a gradient vector as (4). \[\
abla^{s}J(\hat{\mathbf{w}}(n),\gamma(n))~{}=~{}\frac{1}{\sigma^{2}}(\mathbf{ \Phi}(n)\hat{\mathbf{w}}(n)-\mathbf{r}(n))+\gamma(n)\text{sgn}(\hat{\mathbf{ w}}(n))~{}=~{}\mathbf{g}_{n}, \tag{4}\]where \(e(n)\ =\ y(n)-\hat{\mathbf{w}}(n)^{T}\mathbf{x}(n)\), \(\mathbf{\Phi}(n)\ =\ \sum\limits_{m\ =\ 0}^{n}\lambda^{n-m}\mathbf{x}(m) \mathbf{x}(m)^{T}\ =\ \lambda\mathbf{\Phi}(n-1)+\exp\biggl{(}-\frac{e(n)^{2}}{2\sigma^{2}} \biggr{)}\mathbf{x}(n)\mathbf{x}(n)^{T}\), \(\mathbf{r}(n)\ =\ \sum\limits_{m\ =\ 0}^{n}\lambda^{n-m}y(m)\mathbf{x}(m)\ =\ \lambda \mathbf{r}(n-1)+\exp\biggl{(}-\frac{e(n)^{2}}{2\sigma^{2}}\biggr{)}y(n) \mathbf{x}(n)\), and \(\
abla^{s}\Bigl{(}\bigl{\|}\hat{\mathbf{w}}(n)\bigr{\|}_{1}\Bigr{)}\ =\ \mathrm{sgn}(\hat{ \mathbf{w}}(n))\)[6]. Using Equation (4), we can obtain the update expression for \(\hat{\mathbf{w}}(n)\) as (5).