# An Improved Set-Membership Proportionate Adaptive Algorithm for a Block-Sparse System
## 3 The New BS-SMPNLMS and BS-SMPNLMS Algorithms  
Although the SM-PNLMS algorithm can both utilize the sparsity and simplify the computational complexity, its performance still needs to be improved when the response is block-sparse, which is common in real-world echo cancellation systems. Time-domain segmentation is an effective approach for block-sparse processing, and is adopted by some existing algorithms mentioned in Section 1. In addition, using a mixed norm such as the \(l_{2,1}\) norm, \(l_{2,0}\) norm, or \(l_{q,1}\) norm is also an effective method of block-sparse system identification [43; 44; 45; 46; 47]. To further take advantage of the block-sparsity, we propose the block-sparse set-membership PNLMS (BS-SMPNLMS) algorithm to obtain better convergence behavior and a lower level of misadjustment. We introduce the \(l_{2,1}\) norm into the cost function, and then the optimization problem turns to  
\[\min\|\hat{\mathbf{w}}(n)-\hat{\mathbf{w}}(n-1)\|_{2,1\mathbf{Q}^{-1}(n-1)}^{2 }\ \ \text{s.t.}\ \ d(n)-\mathbf{x}^{T}(n)\hat{\mathbf{w}}(n)=\gamma. \tag{12}\]  
Using the Lagrange multiplier, we can get the cost function of the devised BS-SMPNLMS algorithm  
Substituting Equation (23) into (22), we can get  
\[d(n)-\mathbf{x}^{T}\left(n\right)\mathbf{\hat{w}}\left(n-1\right)=\gamma+\lambda \left\|\mathbf{\hat{w}}_{\left[t\right]}\right\|_{2}\mathbf{x}^{T}\left(n \right)\mathbf{Q}\left(n-1\right)\mathbf{x}\left(n\right), \tag{24}\]  
and we combine the following equation  
\[d(n)-\mathbf{x}^{T}\left(n\right)\mathbf{\hat{w}}\left(n-1\right)=e\left(n \right). \tag{25}\]  
Therefore, we can get  
\[\lambda=\frac{e\left(n\right)-\gamma}{\left\|\mathbf{\hat{w}}_{\left[t\right]} \right\|_{2}\mathbf{x}^{T}\left(n\right)\mathbf{Q}\left(n-1\right)\mathbf{x} \left(n\right)}. \tag{26}\]  
Substituting (26) into (22), we can obtain  
\[\mathbf{\hat{w}}\left(n\right)=\mathbf{\hat{w}}\left(n-1\right)+\frac{e\left( n\right)-\gamma}{\mathbf{x}^{T}\left(n\right)\mathbf{Q}\left(n-1\right)\mathbf{x} \left(n\right)}\mathbf{Q}\left(n-1\right)\mathbf{x}\left(n\right). \tag{27}\]  
The updating function is  
\[\mathbf{\hat{w}}\left(n\right)=\mathbf{\hat{w}}\left(n-1\right)+\mu_{\text{SM }}\frac{\mathbf{Q}\left(n-1\right)\mathbf{x}\left(n\right)e\left(n\right)}{ \mathbf{x}^{T}\left(n\right)\mathbf{Q}\left(n-1\right)\mathbf{x}\left(n\right) +\varepsilon_{\text{BS}}}, \tag{28}\]  
where \(\varepsilon_{\text{BS}}\) is a positive constant whose value is very small, and it is used to avoid the denominator being zero. The \(\mu_{\text{SM}}\) in (28) is the same as that in (11). Herein, the step size assignment matrix \(\mathbf{Q}\left(n-1\right)\) is described as  
\[\mathbf{Q}\left(n-1\right)=\text{diag}\left\{q_{0}\left(n-1\right)\mathbf{1}_ {L},q_{1}\left(n-1\right)\mathbf{1}_{L},\cdots,q_{B-1}\left(n-1\right)\mathbf{ 1}_{L}\right\}, \tag{29}\]  
where  
\[q_{t}(n-1)=\frac{\alpha_{t}(n-1)}{\sum\limits_{s=0}^{B-1}\alpha_{s}(n-1)},0 \leq t\leq B-1, \tag{30}\]  
and  
\[\alpha_{t}(n-1)=\max\left\{\rho\max\left\{\delta,\left\|\mathbf{\hat{w}}_{ \left[1\right]}\right\|_{2^{\prime}}\left\|\mathbf{\hat{w}}_{\left[2\right]} \right\|_{2^{\prime}},\cdots,\left\|\mathbf{\hat{w}}_{\left[B\right]}\right\|_ {2}\right\},\left\|\mathbf{\hat{w}}_{\left[t\right]}\right\|_{2}\right\}. \tag{31}\]  
The parameter \(\mathbf{1}_{L}\) in Equation (29) denotes the row vector with \(L\) length and all of its elements are ones. In order to expand the BS-SMPNLMS to make it more useful for the dispersive application field, we propose an improved BS-SMPNLMS (BS-SMIPNLMS) algorithm. The relationship between the BS-SMPNLMS and the BS-SMIPNLMS is similar to that between the PNLMS and the IPNLMS. Hence, the updating equation of the BS-SMIPNLMS is the same as that of the BS-SMPNLMS.