## 3 Virtual Space-Time Diversity Turbo Equalization
### Cluster Sparse \(\ell_{p,q}\)-PRLS Adaptive Filtering Algorithm  
The update of equalizer coefficients \(\mathbf{w}_{n}\) can be regarded as an adaptive filtering process. Under the condition of sparsity and zero-attracting principle, a penalty item should be added into the RLS cost function to satisfy the cluster sparse UWA channel. The modified cost function is given as [32]:  
\[J[\mathbf{w}_{n}]=J_{\text{RLS}}[\mathbf{w}_{n}]+J_{p,q}[\mathbf{w}_{n}] \tag{15}\]  
Figure 4: Result comparisons of phase detectors.  
Figure 3: Step updating diagram.  
where the \(\ell_{p,q}\)-norm penalty item is defined as [44]:  
\[J_{p,q}[\mathbf{w}_{n}]\triangleq\gamma\Bigg{(}\sum_{j=1}^{C}\|\mathbf{w}_{ \mathbf{w}_{j,n}}\|_{p}^{q}\Bigg{)}^{q^{-1}} \tag{16}\]  
\(\gamma\in\mathbb{R}^{+}\) is the regularization factor. Therefore, the \(\ell_{p,q}\)-RLS algorithm is derived by minimizing the modified cost function \(J[\mathbf{w}_{n}]\) in Equation (15). Item \(J_{p,q}[\mathbf{w}_{n}]\) may be a non-differentiable convex function or a non-convex function; therefore, a subgradient analysis is introduced to replace the gradient of the objective function. The subgradient of \(J_{p,q}[\mathbf{w}_{n}]\), with respect to \(\mathbf{w}_{n}\), is denoted by \(\
abla^{s}J_{p,q}[\mathbf{w}_{n}]\).  
We can derive the \(\ell_{p,q}\)-PRLS algorithm, which utilizes the proportionate updating matrix, to control the adjustments caused by the Kalman gain vector of the \(\ell_{p,q}\)-RLS. The proposed \(\ell_{p,q}\)-PRLS algorithm is expressed as follows:  
\[\mathbf{w}_{n}=\mathbf{w}_{n-1}+\mathbf{G}_{n-1}\mathbf{k}_{n}e_{n}^{*}-\hat{ \gamma}\
abla^{s}J_{p,q}[\mathbf{w}_{n-1}] \tag{17}\]  
where \(\mathbf{G}_{n}\) is the \(N_{eq}\times N_{eq}\) diagonal proportionate matrix, and its \(i\)-th diagonal entry \(g_{n,i}\) is calculated as [28; 33]:  
\[g_{n,i}=\begin{cases}\frac{\tau_{1}(1-\beta_{1})}{2N}+\frac{\tau_{1}(1+\beta_{ 1})|w_{n,i}|}{2\|\mathbf{w}_{n}\|_{1}+\varepsilon_{g}}&i\in[1,K_{s}(N_{f,1}+N_ {f,2}+1)]\\ \frac{\tau_{2}(1-\beta_{2})}{2N}+\frac{\tau_{2}(1+\beta_{2})|w_{n,i}|}{2\| \mathbf{w}_{n}\|_{1}+\varepsilon_{g}}&i\in[K_{s}(N_{f,1}+N_{f,2}+1)+1,N_{eq}] \end{cases} \tag{18}\]  
where \(\{\beta_{m}\}_{m=1}^{2}\in[-1,1]\), \(\{\tau_{m}\}_{m=1}^{2}\in(0,N_{eq})\) is a trading off parameter and \(\varepsilon_{g}\) is a small positive constant for computational stability. Parameter \(\hat{\gamma}\) is a simplified form of \(\hat{\gamma}\leftarrow\gamma(1-\lambda)\mathbf{P}_{n}\), where \(\beta\) is an adjustment factor of sparse regularization, \(\mathbf{P}_{n}\) is the inverse of the input correlation matrix. The Kalman gain vector \(\mathbf{k}_{n}\), \(\mathbf{P}_{n}\) and a priori error \(e_{n}\) are respectively given by  
\[\mathbf{k}_{n}=\frac{\mathbf{P}_{n-1}\mathbf{u}_{n}}{\lambda+\mathbf{u}_{n}^{ H}\mathbf{P}_{n-1}\mathbf{u}_{n}} \tag{19}\]  
\[\mathbf{P}_{n}=\frac{1}{\lambda}\Big{(}\mathbf{P}_{n-1}-\mathbf{k}_{n}\mathbf{ u}_{n}^{H}\mathbf{P}_{n-1}\Big{)} \tag{20}\]  
\[e_{n}=\begin{cases}x_{n}-\hat{d}_{n}&\text{Training Stage}\\ Q[\hat{d}_{n}]-\hat{d}_{n}&\text{Tracking Stage}\end{cases} \tag{21}\]  
where \(\lambda\) is the forgetting factor, \(Q[\cdot]\) is the decision function. Several gradients or subgradients of \(\ell_{p,q}\)-norm penalty regularization items are listed in Table 1.  
\begin{table}
\begin{tabular}{c c} \hline \(\{p,q\}\) & \(\
abla^{s}J_{p,q}[w_{n,i}]\) \\ \hline \(1\)1  \\ \(0\)2  \\ \(\{2,1\}\) & \(\sum_{j=1}^{C}\frac{\alpha w_{\mathbf{w}_{j,n,i}}}{\|\mathbf{w}_{\mathbf{w}_{j, n}}\|_{2}+\varepsilon_{p}}{\|\mathbf{w}_{\mathbf{w}_{j,n}}\|_{2}+\varepsilon_{p}}\) \\ \(\{2,0\}\) & \(\sum_{j=1}^{C}\frac{\alpha w_{\mathbf{w}_{j,n,i}}}{\|\mathbf{w}_{\mathbf{w}_{j, n}}\|_{2}+