## 2 Signal Model  
In this paper, the scatterer model of the wideband LFM signal is used. It is assumed that the target is in the far field and the incident wave is a plane wave. The scatterers are located on the line of sight. The range profile of target in wideband radar is illustrated in Figure 1. In stretching processing, the wideband echo signal reflected by the scatterers is mixed with the local wideband LFM signal. For single scatterer, the mixing result is a single frequency signal, whose frequency is determined by the round trip time of the scatterer. For multiple scatterers, the mixing result is composed of multiple signals with different frequencies. Therefore, the wideband echo signal is converted to a narrowband signal, of which the bandwidth is determined by the length of the target. After performing FFT operation, this stretched narrowband signal can be transformed to a range profile along the radar line of sight (RLOS), as shown in the bottom of Figure 1. The parameters that will be frequently used in later derivation are listed in Table 1.  
The transmitted signal of the wideband LFM radar can be modeled as:  
\[s_{tr}(t)=\text{rect}\left(\frac{t}{T}\right)e^{i\pi\gamma t^{2}}e^{i2\pi f_{c }t}, \tag{1}\]  
where \(\text{rect}(u)=\left\{ \begin{array}{ll}1,|u|\leq 1/2\\ 0,\text{otherwise}\end{array} \right.\) denotes the rectangle pulse shape, \(t\) the fast time, \(T\) the pulse width and \(f_{c}\) the radar center frequency. \(\gamma=B/T\) is the chirp rate of the LFM signal, where \(B\) is the swept bandwidth of the LFM signal. \begin{table}
\begin{tabular}{c c l} \hline \hline
**Notation** & **Type** & **Meaning** \\ \hline \(ur\) & Subscript & The received signal is uncompressed. \\ \(st\) & Subscript & The signal is stretched. \\ \(ac\) & Subscript & The signal is obtained by multiplying a echo signal with the conjugate of another echo signal. \\ \(se\) & Subscript & The signal is the multiplication result of the different echo signal from the same scatterers. \\ \(cr\) & Subscript & The signal is the multiplication result of the different echo signal from the different scatterers. \\ \(s_{t}(t)\) & Signal & The transmitted signal. \\ \(s_{tD}(t)\) & Signal & The local reference signal. \\ \(s_{ur}(t,t_{m})\) & Signal & The uncompressed received signal. \\ \(s_{st}(t,t_{m})\) & Signal & The stretched signal. \\ \(R_{p}(t_{m})\) & Signal & The distance from radar to the \(p\)-th scatterer. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations. Figure 1: Target’s range profile of wideband radar. The UR signal of the wideband LFM radar can be represented as:  
\[ \begin{split} s_{ur}(t,t_{m})&=\sum_{p=0}^{P-1}\text{ rect}\left(\frac{t-\tau_{p}(t_{m})}{T}\right)\\ &\times A_{p}e^{j2\pi f_{c}(t-\tau_{p}(t_{m}))}\\ &\times e^{j\gamma\pi(t-\tau_{p}(t_{m}))^{2}}+\omega_{0}(t,t_{m} )\\ &=s_{ur,s}(t,t_{m})+\omega_{0}(t,t_{m}),\end{split}  \tag{2}\]  
where \(P\) denotes the number of scatterers, \(A_{p}\) the scattering coefficient of the \(p\)-th scatterer and \(\tau_{p}(t_{m})\) the time delay from the radar to the \(p\)-th scatterer. \(t_{m}=mT_{pr}\) is the slow time, where \(m\) is the pulse number, and \(T_{pr}\) is the pulse repetition interval (PRI). \(s_{ur,s}(t,t_{m})\) denotes the signal term of \(s_{ur}(t,t_{m})\), \(\omega_{0}(t,t_{m})\) is the Gaussian noise with mean zero and variance \(\sigma^{2}\). The time delay can be expressed as:  
\[\tau_{p}(t_{m})=\frac{2R_{p}(t_{m})}{c}, \tag{3}\]  
where \(R_{p}(t_{m})\) is the distance from the radar to the \(p\)-th scatterer at slow time \(t_{m}\). Due to the wide bandwidth of the signal, the data size of the uncompressed signal is considerably large, which leads to a high computational complexity. Therefore, in wideband radar systems, the stretching operation is often used to reduce the data size.