Algorithm 1 The Iterative Off-grid Turbo Channel Estimation Algorithm

Input: Received signal $\mathit{y}\in {\mathbb{C}}^{RK\times 1}$; Sampling matrix $\mathit{W}\in {\mathbb{C}}^{RK\times N}$, noise power ${\sigma }^2$.

Output: Estimate of $\mathit{g}$, ${\mathit{g}}^t$; Estimation of δ, ${\delta }^t$.

Iteration: Step 1: Initialization $$\begin{array}{c}\hfill t=0,{\delta }^0=0\end{array}$$ Step 2: Estimation of g (1) Fixing δ, $$\begin{array}{cc}\hfill {\mathit{d}}^t& ={\left[1e^{j2\pi {\delta }^t/N}{e}^{j2\pi 2{\delta }^t/N}\cdots e^{j2\pi (N-1){\delta }^t/N}\right]}^T\hfill \\ \hfill {\mathit{g}}^{t+1}& =\text{ITCCS}({\mathit{W}}^T,\text{Diag}({\mathit{d}}^t),\frac{1}{N},{\sigma }^2)\hfill \end{array}$$ Step 3: Estimation of δ (1) Fixing $\mathit{g}$, $$\begin{array}{c}\hfill {[{\mathit{d}}^{t+1}]}_{\mathcal{R}}=\mathit{y}\oslash {[{\mathit{g}}^{t+1}]}_{\mathcal{R}}\end{array}$$ where ⊘ denotes the element-wise division operation. (2) Calculating ${\delta }^{t+1}$ by linear fitting after phase extraction from the equation, $$\begin{array}{c}\hfill {[{\mathit{d}}^{t+1}]}_{\mathcal{R}}={\left[e^{j2\pi (n_1-1){\delta }^{t+1}/N}{e}^{j2\pi (n_2-1){\delta }^{t+1}/N}\cdots e^{j2\pi (n_K-1){\delta }^{t+1}/N}\right]}^T\end{array}$$ Step 4: Iteration of loop IF iteration stopping condition is satisfied output the estimated value. ELSE $t=t+1$. Then, and turn to Step 2.