## 3 Proposed Sensing Waveform
### Signal Detection and Estimation  
In this section, we detail the sensing signal detection and estimation using the designed waveform. The specific workflow is illustrated in Figure 4. After preprocessing the received signal to obtain \(\mathbf{Y}\) as shown in (14), we perform the corresponding signal processing on \(\mathbf{Y}\) to achieve target detection and parameter estimation. First, we perform an inverse discrete Fourier transform (IDFT) on the fast-time dimension of \(\mathbf{Y}\) to obtain range data. We then apply static clutter suppression on the slow-time dimension. Sparse recovery or a non-uniform discrete Fourier transform (NUDFT) is performed on the slow-time dimension to obtain Doppler data. Subsequently, the ordered-statistic constant false-alarm rate (OS-CFAR) is applied to each range-Doppler cell. Once a target is detected, frequency-time phase regression (FTPR) is conducted to obtain the target's range and velocity. Specific details are as follows. We calculate the range spectrum by IDFT along the fast-time dimension of \(\mathbf{Y}\). \[\tilde{\mathbf{Y}}=\mathbf{Y}\mathbf{W}, \tag{15}\]  
where \(\mathbf{W}\) is the inverse discrete Fourier transform matrix, given by  
\[\mathbf{W}= \begin{bmatrix}1&1&\cdots&1\\ 1&e^{j2\pi 1/N_{c}}&\cdots&e^{j2\pi(N_{c}-1)/N_{c}}\\ \vdots&\vdots&\ddots&\vdots\\ 1&e^{j2\pi(N_{c}-1)/N_{c}}&\cdots&e^{j2\pi(N_{c}-1)(N_{c}-1)/N_{c}}\\ \end{bmatrix}  \tag{16}\]  
Let \(\tilde{\mathbf{Y}}=[\tilde{\mathbf{y}}_{0},\tilde{\mathbf{y}}_{1},\ldots,\tilde{\mathbf{y}}_{ N_{c}-1}]\), where \(\tilde{\mathbf{y}}_{m}\) is the received data of the \(m\)th range cell. We then perform the static clutter suppression on \(\tilde{\mathbf{y}}_{m}\) along the slow-time dimension, as given by  
\[\hat{\mathbf{y}}_{m}=\tilde{\mathbf{y}}_{m}-\tilde{g}_{m}\mathbf{v}_{L}, \tag{17}\]  
where \(\tilde{g}_{m}=1/L\sum_{l=0}^{L-1}\tilde{\mathbf{y}}_{m}(l)\), and \(\mathbf{v}_{L}\) is the \(L\times 1\) vector, with all elements equal to one. It is clear that \(\hat{\mathbf{y}}_{m}\) is sparsely sampled in the Doppler dimension. According to compressed sensing (CS) theory, if a signal is sparse or compressible in a certain transform domain, it may be reconstructed using nonlinear methods by solving an optimization problem with  
Figure 4: Signal processing flowchart. high probability. Furthermore, the problem can be abstracted as recovering the vector \(\hat{\mathbf{y}}_{m}\) from data \(\hat{\mathbf{y}}_{m}=A\hat{\mathbf{y}}_{m}\), where \(\mathbf{A}\) is required to satisfy the restricted isometry property (RIP) [34]. Specifically, \(\mathbf{A}=[\mathbf{a}(f_{1}),\mathbf{a}(f_{2}),\ldots,\mathbf{a}(f_{K})]\) is the temporal steering vector dictionary, where \(\mathbf{a}(f_{k})=[e^{2\pi f_{k}n_{0}T_{s}},\ldots,e^{2\pi f_{k}n_{L-1}T_{s}}]^{T}\) is the steering vector. From [35; 36], it is known that \(\mathbf{A}\) satisfies RIP with high probability if the eigenvalues of the Gram matrix \(\mathbf{A}_{\Gamma}^{H}\mathbf{A}_{\Gamma}\) are bounded near \(1\) for every \(\Gamma\) (\(\Gamma\) is a subset of \(\{1\ldots K\}\)). Since it is difficult to prove the temporal steering vector dictionary \(\mathbf{A}\) satisfying the RIP, we compare the eigenvalues of its Gram matrices to that of a matrix with Gaussian entries of zero mean and variance \(1/L\) after scaling \(\mathbf{A}\), so that its columns have unit norm [34]. According to the CS theory, the Gaussian random matrix satisfies the RIP with high probability [37]. As shown in Figure 5, the maximum and minimum eigenvalues of the Gram matrix \(\mathbf{A}_{\Gamma}^{H}\mathbf{A}_{\Gamma}\) are close to \(1\), and are similar to the eigenvalues of the Gaussian random matrix. Therefore, \(\mathbf{A}\) satisfies the RIP with high probability, and \(\hat{\mathbf{y}}_{m}\) can be reconstructed with high probability in the dictionary \(\mathbf{A}\). Here, we introduce two sparse recovery algorithms, namely the iterative adaptive approach (IAA) and the orthogonal matching pursuit (OMP) to reconstruct \(\hat{\mathbf{y}}_{m}\).