## 3 Proposed Sensing Waveform
### Signal Detection and Estimation
As for the IAA algorithm, it transforms the solving of the Doppler frequency of the target in \(\hat{\mathbf{y}}_{m}\) into solving the following optimization problem:  
\[\min||\hat{\mathbf{y}}_{m}-\gamma_{k}\mathbf{a}(f_{k})||_{\mathbf{Q}_{k}^{-1}}^{2} \tag{18}\]  
where \(||\mathbf{x}||_{\mathbf{Q}_{k}^{-1}}^{2}=\mathbf{x}^{H}\mathbf{Q}_{k}^{-1}\mathbf{x}\), \(\mathbf{Q}_{k}\) denotes the covariance matrix of the clutter and noise, and \(|\gamma_{k}|^{2}\) is the power of the Doppler spectrum with frequency \(f_{k}\). The IAA algorithm operates in an iterative way and the power of the Doppler frequency points \(\mathbf{P}_{k,k}\) is calculated for each range of interest in each iteration, and then the IAA covariance matrix \(\mathbf{R}_{\text{IAA}}\) is updated. The derivation details can be seen in reference [38]. A summary of the IAA algorithm is shown in Algorithm 2. As for the OMP algorithm, it operates in an iterative way and computes the maximum inner product via the matrix-vector multiplication \(\mathbf{Ar}_{p}\) for each iteration, where \(r_{p}\) is the residual vector at the \(p\)th iteration. Moreover, the solution estimated at each iteration requires the solution of a least-squares problem, given by  
\[\min\Bigl{\|}\mathbf{A}_{\mathbf{\Omega}(p+1)}\mathbf{y}_{\mathbf{\Omega}(p+1)}-\hat{\mathbf{y}}_ {m}\Bigr{\|}_{2}^{2} \tag{19}\]  
where \(\mathbf{\Omega}(p+1)=\{b(p_{i}+1)|p_{i}=1,2,\ldots,p\}\) denotes the set consisting of \(b(p+1)\) for each iteration. The derivation details can be seen in reference [39]. A summary of the OMP algorithm is shown in Algorithm 3. Figure 5: Maximum and minimum eigenvalues of the Gram matrix against sparsity k.  
```
Initialization: \(\mathbf{A}=[\mathbf{a}(f_{1}),...,\mathbf{a}(f_{K})]\),  IAA covariance matrix \(R_{\text{IAA}}=\mathbf{I}\), \(\mathbf{P}_{k,k}=\frac{1}{L^{2}}|\mathbf{a}^{H}(f_{k})\hat{\mathbf{y}}_{m}|^{2}\), \(k=1,...,K\). Repeat: \(\mathbf{R}_{\text{IAA}}=\mathbf{A}\mathbf{P}\mathbf{A}^{H}\),  For\(k=1,...,K\), \(\mathbf{P}_{k,k}=|\gamma_{k}|^{2}=|\frac{\mathbf{a}^{H}(f_{k})\mathbf{R}_{\text{IAA}}^{-1 }\hat{\mathbf{y}}_{m}}{\mathbf{a}^{H}(f_{k})\mathbf{R}_{\text{IAA}}^{-1}\mathbf{a}(f_{k})}|^{2}\), End Until a certain number of iterations is reached. Output:\(\hat{\mathbf{y}}_{m}=diag(\mathbf{P})\). ```  
**Algorithm 2** The IAA algorithm. ```
Initialization: \(\hat{\mathbf{y}}_{m}\), \(\mathbf{A}=[\mathbf{a}(f_{1}),...,\mathbf{a}(f_{K})]\), \(p=0\),  Residual vector \(\mathbf{r}_{0}=\hat{\mathbf{y}}_{m}\),  Stopping parameter \(\epsilon\),  Selected index subset \(\mathbf{\Omega}(0)=\emptyset\). While\(||\mathbf{r}_{p}||_{2}^{2}>\epsilon\) \(b(p+1)=\operatorname*{argmax}_{b}|\mathbf{a}^{H}(f_{b})\mathbf{r}(p)|\), \(\mathbf{\Omega}(p+1)=\mathbf{\Omega}(p)\cup b(p+1)\), \(\mathbf{G}=(\mathbf{A}_{\mathbf{\Omega}(p+1)}^{H}\mathbf{\Lambda}_{\mathbf{\Omega}(p+1)})^{-1}\mathbf{ A}_{\mathbf{\Omega}(p+1)}^{H}\), \(\mathbf{y}_{\mathbf{\Omega}(p+1)}=\mathbf{G}\hat{\mathbf{y}}_{m}\),  Update the residual vector \(\mathbf{r}_{p+1}=\hat{\mathbf{y}}_{m}-A_{\mathbf{\Omega}(p+1)}\mathbf{y}_{\mathbf{\Omega}(p+1)}\), \(p=p+1\), End Compute the Doppler profile \(\hat{\mathbf{y}}_{m}=B_{s}\mathbf{y}_{\mathbf{\Omega}(p+1)}\). ```  
**Algorithm 3** The OMP algorithm. After applying the sparse recovery algorithm to \(\hat{\mathbf{y}}_{m}\), we obtained the Doppler dimension data \(\hat{\mathbf{y}}_{m}\) for the \(m\)th range cell. Then, the location of the target is detected on the range-Doppler map by the OS-CFAR [40] detector. The specific process of OS-CFAR is as follows: First, for the detection cell \(z_{m,k}\) in the range-Doppler map, we select some cells around the detection cell as the protection cell. We then select some cells that are not in the scope of the protection cells as reference cells. Moreover, we sort the reference cells according to their power, and then select the middle \(N_{\text{CFAR}}\) cells of the reference cells to estimate the power of clutter and noise \(P_{\text{CN}}\).