## 3 Proposed Methods
### The Enhanced OMP Channel Estimation Algorithm  
In this work, a correlation-peak-search-based improved OMP (CPS-IOMP) algorithm is proposed to estimate the channel using the designed preamble signal. The block diagram of the proposed channel estimation algorithm, which consists of the channel sparsity estimation module and the OMP module, is illustrated in Figure 4. The input signal of the proposed channel estimation algorithm is the received preamble signal after the Doppler compensation with the Doppler factor estimated in (21). Figure 4: The block diagram of proposed enhanced channel estimation algorithm. Figure 3: The structure for Doppler estimation. The channel sparsity is estimated according to the excellent autocorrelation characteristic of the designed preamble signal. In the proposed correlation-peak-search-based channel sparsity estimation module, correlation processing is first performed between the Doppler-compensated received preamble signal and the locally generated preamble signal. And then, the correlation peaks are searched. These correlation peaks are mainly caused by the multipath, and the positions of the correlation peaks correspond to the multipath delay. Hence, the positions of the correlation peaks can be used to determine the channel sparsity. The correlation peaks larger than the threshold \(\eta\) are selected. Let the number of selected correlation peaks be \(L^{\prime}\). Obviously, \(L^{\prime}\) is much smaller than \(L\). The corresponding normalized delay of each correlation peak is denoted as \(c_{i}\), \(i=1\), \(2\), \(\ldots\), \(L^{\prime}\). Hence, dictionary matrix \(\mathbf{A}\) can be simplified as  
\[\mathbf{\Lambda}=[\mathbf{a}_{c_{1}},\mathbf{a}_{c_{2}},\cdots,\mathbf{a}_{c_{L^{\prime}}}] = \begin{bmatrix}s_{\text{P}}(0-c_{1})&s_{\text{P}}(0-c_{2})&\cdots&s_{ \text{P}}(0-c_{L^{\prime}})\\ s_{\text{P}}(1-c_{1})&s_{\text{P}}(1-c_{2})&\cdots&s_{\text{P}}(1-c_{L^{\prime }})\\ \vdots&\vdots&\ddots&\vdots\\ s_{\text{P}}(N-1-c_{1})&s_{\text{P}}(N-1-c_{2})&\cdots&s_{\text{P}}(N-1-c_{L^ {\prime}})\end{bmatrix} , \tag{23}\]  
where \(\mathbf{a}_{i}\) is the \((i+1)\)th column of matrix \(\mathbf{A}\). Since the simplified dictionary matrix eliminates some irrelevant paths, the delay corresponding to each column of \(\mathbf{\Lambda}\) no longer changes continuously. The jump at the delay corresponding to the multipath is a stair-like change trend with the increase in the index, as shown in Figure 5a. The number of jumps is the estimated channel sparsity. Owing to the noise and large sidelobes in the correlation processing, the actual obtained jump at the delay may be as illustrated in Figure 5b. Comparing Figure 5b with Figure 5a, we observe that the overall trend of the change is basically consistent and the columns corresponding to multipath delay are almost included, although more columns are selected in the actual obtained result. Therefore, with an appropriate threshold, the channel sparsity can be estimated and the simplified dictionary matrix can be constructed by filtering out the columns corresponding to multipath delay. After obtaining the channel sparsity and the simplified dictionary matrix, the channel state information is estimated with the OMP algorithm in the proposed CPS-IOMP algorithm. The OMP algorithm is an iterative procedure for estimating the channel taps and its coefficients sequentially. The number of iterations is set as the estimated channel sparsity. Figure 5: The change in corresponding delay along with the increase in the column index of simplified dictionary matrix. (**a**) Ideal result; (**b**) actual result. At each iteration, the column of the dictionary matrix which has the largest absolute value of correlation with the residual is selected.