## 2 System Model
### Adaptive Channel Estimation  
For convenience of presentation, we introduce an intermediate quantity \[U_{t^{{}^{\prime},k^{{}^{\prime}}}}=X_{[l-l^{{}^{\prime}}]_{M};[k-k^{{}^{\prime}}]_{ N}}e^{\varphi(l^{{}^{\prime}}-l,k^{{}^{\prime}}-k)} \tag{11}\]  
where \(\varphi(l^{{}^{\prime}}-l,k^{{}^{\prime}}-k)\) can be listed as  
\[\varphi(l^{{}^{\prime}}-l,k^{{}^{\prime}}-k)=\] \[\left\{ \begin{array}{cc}\frac{-\beta\pi(k^{{}^{\prime}}-k+\frac{ N}{2})((M-1)+(l^{{}^{\prime}}-l))}{(M+N_{\rm CP})N},&(k-k^{{}^{\prime}})\geq \frac{N}{2}\\ \frac{-\beta\pi(k^{{}^{\prime}}-k)((M-1)+(l^{{}^{\prime}}-l))}{(M+N_{\rm CP}) N},&(k-k^{{}^{\prime}})<\frac{N}{2}\end{array} \right. \tag{12}\]  
Thus, the Equation (8) can be rewritten as  
\[Y_{l,k} =\sum_{l^{{}^{\prime}}=0}^{M-1}\sum_{k^{{}^{\prime}}=0}^{N-1}H_{l ^{{}^{\prime}},k^{{}^{\prime}}}X_{[l-l^{{}^{\prime}}]_{M},[k-k^{{}^{\prime}}] _{N}}e^{\varphi(l^{{}^{\prime}}-l,k^{{}^{\prime}}-k)}+V_{l,k}\] \[=\sum_{l^{{}^{\prime}}=0}^{M-1}\sum_{k^{{}^{\prime}}=0}^{N-1}H_{l ^{{}^{\prime}},k^{{}^{\prime}}}U_{l^{{}^{\prime}},k^{{}^{\prime}}}+V_{l,k} \tag{13}\]  
To note this, the training signal \(X_{l,k}\) and compensated phase \(e^{\varphi(l^{{}^{\prime}}-l,k^{{}^{\prime}}-k)}\) are both known for the receiver. For convenience, we transform Equation (13) into a one-dimensional vector form, which can be described as follows:  
\[Y_{l,k}=\mathbf{\Lambda}^{H}\mathbf{u}+\mathbf{v} \tag{14}\]  
where \(\mathbf{\Lambda}=\text{vec}\{\mathbf{H}^{*}\}\), \(\mathbf{u}=\text{vec}\{\mathbf{U}\}\), \(\mathbf{v}=\text{vec}\{\mathbf{V}\}\). Let \(\vec{H}_{l,k}\) denote the estimated channel. Then, the error between the received signal and the system output based on the estimated channel can be written as follows:  
\[E_{l,k} =Y_{l,k}-\sum_{l^{{}^{\prime}}=0}^{M-1}\sum_{k^{{}^{\prime}}=0}^{ N-1}\hat{H}_{l^{{}^{\prime}},k^{{}^{\prime}}}U_{l^{{}^{\prime}},k^{{}^{\prime}}}\] \[=Y_{l,k}-\hat{\mathbf{\Lambda}}^{H}\mathbf{u} \tag{15}\]  
Since the channels in the DD domain are sparse, the cost function is given by  
\[J_{l,k}=\|E_{l,k}\|_{2}^{2}+2\gamma\|\hat{\mathbf{\Lambda}}\|_{1} \tag{16}\]  
where the operation \(\|\cdot\|_{1}\) and \(\|\cdot\|_{2}\) denote the \(l_{1}\) norm and \(l_{2}\) norm, respectively. In Equation (16), an \(l_{1}\) norm penalty is introduced, which could improve the convergence and accuracy of the adaptive algorithm by exploiting the sparsity of the channel. We adopt the IPNLMS algorithm to estimate the parameters. The update equation is given by  
\[(\hat{\mathbf{\Lambda}})^{\dagger}=\hat{\mathbf{\Lambda}}+\mu\frac{(E_{l,k})^ {*}\mathbf{G}}{\mathbf{u}^{H}\mathbf{G}\mathbf{u}+\delta}\mathbf{u} \tag{17}\]  
where \((\hat{\mathbf{\Lambda}})^{\dagger}\) is the updated parameters, \(\mu\) is the step, \(\mathbf{G}\) denotes a diagonal proportionate matrix with \(p\)-th diagonal element  
\[G_{p,p}=\frac{1-\alpha}{2L_{h}}+(1+\alpha)\frac{|\hat{\mathbf{\Lambda}}|}{2|| \hat{\mathbf{\Lambda}}||_{1}+\epsilon} \tag{18}\]  
\(\delta\) and \(\epsilon\) are small positive numbers, called regularizers. \(L_{h}\) is the size of \(\hat{\mathbf{\Lambda}}\), and \(|\cdot|\) is the absolute value operation. \(-1\leq\alpha<1\) determines the filter sparsity. Compared with the normalized least mean square (NLMS) algorithm, the updating Equation (17) of the IPNLMS algorithm has very fast convergence and tracking when the impulse response path is sparse. By using an average coefficient, the updating Equation (17) even performs better than the proportionate normalized least mean square (PNLMS) algorithm when the impulse response is dispersive. If \(\alpha=-1\), this algorithm reduces to the NLMS, and if \(\alpha=1\), the algorithm performs more like the PNLMS.