## 3 The Non-Uniform Hybrid Sampling Technology
### Analysis of the Non-Uniform Hybrid Sampling Technology for High-Resolution SAR Imaging  
Due to the high relative velocity between the space-borne X-band SAR and space targets, the phase and time of the SAR echoes will be disturbed when the proposed non-uniform hybrid sampling technology is applied to receive wideband SAR echoes. The beam angle in the azimuth direction is narrow, so the maximum time disturbance appears at the edge of the antenna beam. As shown in Figure 4, due to the relative motion between the space-borne X-band SAR and space targets, the maximum wave-path difference can be expressed as [19]  
\[\Delta R_{\rm i}\approx v_{\rm st}\cdot N\cdot t_{\rm hs}\cdot\sin(\theta_{\rm A }/2)\approx\frac{v_{\rm st}\cdot N\cdot t_{\rm hs}\cdot\lambda_{\rm c}}{2D_{\rm A}} \tag{4}\]  
where \(v_{\rm st}\), \(\lambda_{\rm c}\), and \(\theta_{\rm A}\) are the relative velocity between the space-borne X-band SAR and space targets, the wavelength of the center frequency, and the beam angle, respectively. In order to reduce the side lobes of the SAR echoes in the range direction, the maximum time disturbance should be smaller than the equivalent-time sampling interval of the non-uniform hybrid sampling, which is estimated as  
\[\frac{v_{\rm st}\cdot N\cdot t_{\rm hs}\cdot\lambda_{\rm c}}{D_{\rm A}\cdot c }\leq\frac{t_{\rm int}}{P}\leq\frac{1}{P\cdot\eta_{\rm r}\cdot B_{\rm w}},\ P \geq 2,\eta_{\rm r}\geq 2 \tag{5}\]  
where \(B_{\rm w}\), \(\eta_{\rm r}\) and \(P\) are the bandwidth of SAR signal, the oversampling rate in the range direction, and the limited factor for the time disturbance, respectively. Then, Equation (5) can be organized as  
\[\frac{D_{\rm A}\cdot f_{\rm c}}{v_{\rm st}\cdot N\cdot t_{\rm hs}\cdot\eta_{ \rm r}\cdot B_{\rm w}}\geq P \tag{6}\]  
where \(f_{\rm c}\) is the center frequency of the X-band SAR signals. Obviously, Equation (6) can be satisfied easily. If the relative velocity is not too large, such as 1 km/s, Equation (6) is easy to meet in engineering applications.  
On the other hand, to reduce the side-lobes of the SAR echoes in azimuth direction, the maximum phase perturbation needs to be limited as  
\[\frac{2\pi\cdot\Delta R_{\rm i}}{\lambda_{\rm c}}=\frac{\pi\cdot v_{\rm st} \cdot N\cdot t_{\rm hs}}{D_{\rm A}}\leq\frac{\pi}{Q},Q\geq 2 \tag{7}\]  
where \(Q\) is the qualification factor for the phase disturbance of SAR echoes in a complete synthetic aperture time. Equation (7) is further simplified to  
\[\frac{D_{\rm A}}{v_{\rm st}\cdot N\cdot t_{\rm hs}}\geq Q \tag{8}\]  
Figure 4: Analysis of the non-uniform hybrid sampling technology for the mini-SAR.  
Likewise, Equation (8) is easy to meet when the relative velocity is not too high. It is worth emphasizing that the sampling requirement of Equation (8) is more difficult to meet than that of the Equation (6) when the space-borne SAR is a typical narrowband system. However, the space-borne X-band SAR proposed in this paper is a typical wideband system, which can easily meet the sampling requirement of Equation (8). According to Equations (3) and (8), the interval between two adjacent hybrid sampling (\(t_{\text{hs}}\)) should meet the following condition:  
\[\frac{2W_{\text{swath}}\cdot\sin\theta_{\text{in}}}{c}+t_{\text{P}}<t_{\text{ hs}}\leq\frac{D_{\text{A}}}{v_{\text{st}\cdot N\cdot Q}} \tag{9}\]  
It can be concluded that the interval (\(t_{\text{hs}}\)) should be small enough to reduce the time and phase disturbance. Meanwhile, the interval (\(t_{\text{hs}}\)) should also be large enough to receive the backscattered signals of all the targets located in the entire imaging swath.