## 3 Methodology
### Two-Dimensional Adaptive Receiving Module  
In this subsection, we introduce the first module in our proposed imaging framework, the 2D adaptive receiving module. From the perspective of hardware resource consumption, this module is designed to reduce the consumption of receiving resources. In general, receiving resource consumption is mainly determined by two factors, which are sampling frequency and storage occupation. For the sampling frequency, according to  
Figure 5: The proposed target-oriented imaging concept and its realization framework. the well-known Nyquist sampling theorem [33], it should be at least the bandwidth of the receiving echo. Thus, the requirement for the sampling device is expressed as  
\[F_{s}\geq B_{r} \tag{5}\]  
where \(F_{s}\) is the sampling rate and \(B_{r}\) is the signal bandwidth. For the storage occupation, it is determined by the sampling length and the sampling frequency, which is expressed as  
\[D_{s}\geq (F_{s}T_{s}) \tag{6}\]  
where \(D_{s}\) is the storage occupation and \(T_{s}\) is the sampling time length. As seen from the above two formulas, to reduce the consumption of receiving resources, we need to reduce the sampling frequency and the sampling length. In our 2D adaptive receiving module, necessary bandwidth and time-length echoes from targets are received by dechirping and subaperture decomposition in the range and the azimuth direction, respectively, as illustrated in Figure 6.  
Unlike the normal receiving in the range direction, which is received by mixing the echoes with the local oscillator, the module adapts dechirping by mixing them with the time-delayed transmitting signal. It can be expressed as  
\[s_{rc}(t,\eta,i) =A_{0}rect\bigg{[}\frac{t-t_{0}(\eta)}{T_{r}}\bigg{]}\text{exp} \Big{\{}j\pi K_{r}[t-t_{0}(\eta)]^{2}\Big{\}}\text{exp}\{j2\pi f_{0}[t-t_{0}( \eta)]\} \tag{7}\] \[s_{t}(t) =rect\bigg{[}\frac{t}{T_{r}}\bigg{]}\text{exp}\Big{\{}j\pi K_{r} t^{2}\Big{\}}\text{exp}\{j2\pi f_{0}t\}\] \[s_{r2}(t,\eta,i) =s_{rc}(t,\eta,i)s_{*}^{*}(t)\] \[s_{r2}(t,\eta,i) \approx A_{0}rect\bigg{[}\frac{t-t_{0}(\eta)}{T_{r}}\bigg{]}\text{ exp}\{-j2\pi K_{r}t_{0}(\eta)t\}\text{exp}\{-j2\pi f_{0}t_{0}(\eta)\}\]  
where \(*\) is the conjugate operator. This formula shows that for a point target, its echo becomes a single-frequency signal, and the frequency is related to its slant range. Compared with the signal form in Formula (1), the bandwidth is reduced to zero. In practice, for a typical target such as a ship, it cannot be taken as a point target but as a distributed target, as shown in Figure 7.  
Figure 6: Two-dimensional adaptive receiving by dechirping and subaperture decomposition in the range and azimuth direction, respectively. Only necessary bandwidth and time-length echoes from targets are received. For a distributed target, the dechirped echo can be expressed as  
\[F_{s}\geq B_{r} \tag{5}\]0  
where \(l=1,2,\cdots,N_{T}\) is the target point index and \(N_{T}\) is the number of points, \(A_{l}\) is the intensity of the \(l\) point of the target, and \(t_{0l}(\eta)\) is the instantaneous range time delay of the \(l\) th point of the target. From this formula, we can see that its echo consists of multiple single-frequency signals for a distributed target, and its bandwidth is determined by the size of the target as follows:  
\[F_{s}\geq B_{r} \tag{5}\]1  
where \(W(\eta)=max[t_{0l}(\eta)]-\min[t_{0l}(\eta)]\) is the time length of the target echo in the range direction. As the maximum length equals the receiving window's length, the largest bandwidth is \(B^{\prime}_{max}=K_{r}T_{R}\). In general, the size of the ship does not exceed the order of 100 m, corresponding to the order of 1 \(\mu\)s time length. In comparison with the length of the receiving window, the order of 10 \(\mu\)s, it is much less. Benefitting from this, the needed sampling frequency is reduced a lot.