## 2 MCC \(l_{1}\)-IWF Formulation
\[\hat{\mathbf{w}}(n+1)\ =\ \hat{\mathbf{w}}(n)-\mu_{n}\
abla^{s}J(\hat{ \mathbf{w}}(n),\gamma(n))\ =\ \hat{\mathbf{w}}(n)-\mu_{n}\mathbf{g}_{n} \tag{5}\]  
To get the step size \(\mu_{n}\), we find the \(\mu_{n}\) that minimizes exponentially averaged _a posteriori_ error energy, \(J(\hat{\mathbf{w}}(n+1),\gamma(n))\), where _a posteriori_ error is \(e(n)\ =\ y(n)-\hat{\mathbf{w}}(n+1)^{T}\mathbf{x}(n)\). \[ \begin{array}{ll}\
abla_{\mu}J(\hat{\mathbf{w}}(n+1),\gamma(n))&=-\frac{1}{ \sigma^{2}}\hat{\mathbf{w}}(n+1)^{T}\mathbf{\Phi}(n)\mathbf{g}_{n}+\frac{1}{ \sigma^{2}}\mathbf{r}(n)^{T}\mathbf{g}_{n}-\gamma(n)\
abla^{s}\Bigl{(}\bigl{\|} \hat{\mathbf{w}}(n+1)\bigr{\|}_{1}\Bigr{)}^{T}\mathbf{g}_{n}\\ &\cong-\frac{1}{\sigma^{2}}\hat{\mathbf{w}}(n+1)^{T}\mathbf{\Phi}(n)\mathbf{g} _{n}+\frac{1}{\sigma^{2}}\mathbf{r}(n)^{T}\mathbf{g}_{n}-\gamma(n)\
abla^{s} \Bigl{(}\bigl{\|}\hat{\mathbf{w}}(n)\bigr{\|}_{1}\Bigr{)}^{T}\mathbf{g}_{n}\\ &=-\frac{1}{\sigma^{2}}\hat{\mathbf{w}}(n+1)^{T}\mathbf{\Phi}(n)\mathbf{g}_{n} +\frac{1}{\sigma^{2}}\mathbf{r}(n)^{T}\mathbf{g}_{n}-\gamma(n)\mathrm{sgn}( \hat{\mathbf{w}}(n))^{T}\mathbf{g}_{n}.\end{array}  \tag{6}\]  
Substituting Equation (5) into Equation (6), we get  
\[\
abla_{\mu}J(\hat{\mathbf{w}}(n+1),\gamma(n))\ =\ -\frac{1}{\sigma^{2}}\hat{ \mathbf{w}}(n)^{T}\mathbf{\Phi}(n)\mathbf{g}_{n}+\frac{1}{\sigma^{2}}\mu_{n} \mathbf{g}_{n}^{T}\mathbf{\Phi}(n)\mathbf{g}_{n}+\frac{1}{\sigma^{2}}\mathbf{ r}(n)^{T}\mathbf{g}_{n}-\gamma(n)\mathrm{sgn}(\hat{\mathbf{w}}(n))^{T} \mathbf{g}_{n}. \tag{7}\]  
To find \(\mu_{n}\), we set \(\
abla_{\mu}J(\hat{\mathbf{w}}(n),\gamma(n))\ =\ 0\), and  
\[\mu_{n}\ =\ \frac{\frac{1}{\sigma^{2}}\Bigl{(}\hat{\mathbf{w}}(n)^{T}\mathbf{ \Phi}(n)-\mathbf{r}(n)^{T}\Bigr{)}\mathbf{g}_{n}+\gamma(n)\mathrm{sgn}(\hat{ \mathbf{w}}(n))^{T}\mathbf{g}_{n}}{\frac{1}{\sigma^{2}}\mathbf{g}_{n}^{T} \mathbf{\Phi}(n)\mathbf{g}_{n}}\ =\ \sigma^{2}\frac{\mathbf{g}_{n}^{T}\mathbf{g}_{n}}{ \mathbf{g}_{n}^{T}\mathbf{\Phi}(n)\mathbf{g}_{n}} \tag{8}\]  
We have to derive regularization coefficient \(\hat{\gamma}(n)\) such that \(\bigl{\|}\hat{\mathbf{w}}(n+1)\bigr{\|}_{1}\ =\ c\), i.e., the \(l_{1}\)-norm of vector \(\hat{\mathbf{w}}(n+1)\) is preserved at all time steps of \(n\). This can be represented by a flow equation in continuous time-domain in [25]. \[\frac{\partial\bigl{\|}\hat{\mathbf{w}}(t)\bigr{\|}_{1}}{\partial t}\ =\ \biggl{(}\frac{\partial\bigl{\|}\hat{ \mathbf{w}}(t)\bigr{\|}_{1}}{\partial\hat{\mathbf{w}}}\biggr{)}^{T}\frac{ \partial\hat{\mathbf{w}}}{\partial t}\ =\ \Bigl{(}\
abla^{s}\bigl{\|}\hat{\mathbf{w}}(t)\bigr{\|}_{1} \Bigr{)}^{T}\frac{\partial\hat{\mathbf{w}}}{\partial t}\ =\ 0. \tag{9}\]  
Using a sufficiently small interval \(\delta\), the time derivative in (9) can be approximated as  
\[\Bigl{(}\
abla^{s}\bigl{\|}\hat{\mathbf{w}}(t)\bigr{\|}_{1}\Bigr{)}^{T}\frac{ \partial\hat{\mathbf{w}}}{\partial t}\cong\Bigl{(}\
abla^{s}\bigl{\|}\hat{ \mathbf{w}}(n)\bigr{\|}_{1}\Bigr{)}^{T}\frac{(\hat{\mathbf{w}}(n+1)-\hat{ \mathbf{w}}(n))}{\delta}\ =\ 0. \tag{10}\]  
Using (5) and (6), (10) becomes  
\[\mathrm{sgn}(\hat{\mathbf{w}}(n))^{T}(\hat{\mathbf{w}}(n+1)-\hat{\mathbf{w}}(n) )\ =\ \mathrm{sgn}(\hat{\mathbf{w}}(n))^{T}(-\mu_{n}\mathbf{g}_{n})\ =\ 0. \tag{11}\]  
and  
\[\mathrm{sgn}(\hat{\mathbf{w}}(n))^{T}\biggl{(}\frac{1}{\sigma^{2}}\mathbf{\Phi} (n)\mathbf{w}(n)-\frac{1}{\sigma^{2}}\mathbf{r}(n)+\gamma(n)\mathrm{sgn}(\hat{ \mathbf{w}}(n))\biggr{)}\ =\ 0. \tag{12}\]  
The regularization coefficient \(\hat{\gamma}(n)\) obtained from Equation (12) is as follows. \[\gamma(n)\ =\ -\frac{\mathrm{sgn}(\hat{\mathbf{w}}(n))^{T}(\mathbf{\Phi}(n) \mathbf{w}(n)-\mathbf{r}(n))}{\sigma^{2}\mathrm{sgn}(\hat{\mathbf{w}}(n))^{T} \mathrm{sgn}(\hat{\mathbf{w}}(n))}. \tag{13}\]On the contrary, CR-RMC algorithm in [20] uses the same regularization coefficient as that in [6].