## 2 System Model and Problem Statement
### Problem Statement  
An LFM signal, generally used as the preamble signal, can be denoted as  
\[s_{\text{LFM}}(t)=\cos(2\pi f_{0}t+\pi kt^{2}),0\leq t\leq T_{\text{LFM}}, \tag{4}\]  
\[k=\frac{f_{1}-f_{0}}{T_{\text{LFM}}}, \tag{5}\]  
where \(f_{0}\) and \(f_{1}\) are the start frequency and stop frequency of the LFM signal, and \(T_{\text{LFM}}\) is the time duration of the LFM signal. If \(f_{0}<f_{1}\), the LFM signal is up-sweeping. If \(f_{0}>f_{1}\), the LFM signal is down-sweeping. By differentiation, the instantaneous frequency of the LFM signal is  
\[f(t)=\frac{1}{2\pi}\cdot\frac{d}{dt}(2\pi f_{0}+\pi kt^{2})=f_{0}+kt, \tag{6}\]  
which is a linear function of time. As an LFM signal is transmitted, the received LFM signal at the receiver can be written as  
\[y_{\text{LFM}}(t)=\sum_{p=0}^{P-1}A_{p}\cos(2\pi f_{0}((1+\alpha)t-\tau_{p})+ \pi k((1+\alpha)t-\tau_{p})^{2})+w(t). \tag{7}\]  
If the transmitter and the receiver are relatively stationary, a matched filter can correlate the transmitted and received waveforms to determine when the signal is detected. However, if relative motion exists, a mismatch will occur. As a Doppler-insensitive waveform, the main ridge of the ambiguity function for the LFM signal is approximately a sloping curve. It is indicated that the peak position of the matched filter will deviate from the real-time delay position in the case of relative motion, which is called range-Doppler coupling [29]. For a realistic Doppler factor level (\(v<15\) m/s), the offset almost linearly varies with theDoppler factor. It is shown in [30] that the relationship between the offset and the Doppler factor can be approximately expressed as  
\[\Delta t(\alpha)=-\frac{\alpha(f_{0}+f_{1})T_{\text{LFM}}}{2(f_{1}-f_{0})}. \tag{8}\]  
From (8), one finds that the Doppler factor can be estimated by the offset of the peak position of the matched filter. The estimated Doppler factor can be used to perform Doppler compensation to improve the performance of UWA communication. Moreover, the relative velocity of the transmitter/receiver can be indirectly calculated based on the estimated Doppler factor. If the transmitter or receiver is stationary, the radial velocity of the other can be obtained. For existing Doppler estimation algorithms using the LFM signal, two LFM signals are inserted before and after the data frame, respectively. Hence, these algorithms cannot achieve real-time estimation. Moreover, for existing Doppler estimation algorithms using the HFM signal, a longer time duration of the preamble signal is needed. So, these algorithms may lead to the increase in overhead. Inspired by (8), a preamble signal should be designed to achieve real-time Doppler estimation and without increasing the time duration. The preamble signal can also serve as the training sequence for the channel estimation. In a discrete-time system, the received preamble signal after Doppler compensation can be expressed as  
\[y_{\text{P}}(n)=\sum_{l=0}^{L-1}s_{\text{P}}(n-l)h(l)+w(n),0\leq n\leq N-1, \tag{9}\]  
where \(n\) is the index, \(y_{\text{P}}(n)\) is the sampled received preamble signal, \(s_{\text{P}}(n)\) is the sampled preamble signal, \(h(l)\) is the discrete CIR, \(L\) is the length of the discrete channel, \(N\) is the total number of sampled preamble signals, and \(w(n)\) is the sampled noise. Under the assumption that the channel is time-invariant within the time duration of the data frame, (9) can be presented in a matrix-vector form as  
\[\mathbf{y}_{\text{P}}=\mathbf{\Lambda}\mathbf{h}+\mathbf{w}, \tag{10}\]  
where \(\mathbf{\Lambda}\) is the dictionary matrix, which is composed of the sampled preamble signal with a size of \(N\times L\), as well as a Toeplitz matrix. Matrix \(\mathbf{\Lambda}\) can be defined as  
\[\mathbf{\Lambda}= \begin{pmatrix}s_{\text{P}}(0)&s_{\text{P}}(-1)&\cdots&s_{\text{ P}}(-(L-1))\\ s_{\text{P}}(1)&s_{\text{P}}(0)&\cdots&s_{\text{P}}(-(L-2))\\ \vdots&\vdots&\ddots&\vdots\\ s_{\text{P}}(N-1)&s_{\text{P}}(N-2)&\cdots&s_{\text{P}}(N-L)\\ \end{pmatrix} .