## 4 HRWS Imaging Processing  
The hypersonic platform SAR adopting the DPCAM technique can obtain the uniform azimuth echo signal after the spectrum reconstruction; then, the SAR imaging algorithm can be further designed to process the echo data. Due to the curved trajectory of this platform, when the traditional Omega-K algorithm is utilized, the precise focus of the targets located at the reference range will be achieved after the bulk compression processing. However, the residual phase of targets at the non-reference range is difficult to be compensated because the residual phase is not only related to the range frequency and the azimuth frequency, but the range variable \(r\) is also involved. Due to the fact that the Stolt mapping along the range axis cannot be implemented, the traditional Omega-K imaging algorithm cannot fully achieve the focus of the targets for hypersonic platform SAR with curved trajectory. To solve this problem, the ETF algorithm is designed in this section, and the SAR imaging for the wide swath is further considered by using the method of range division.  
As given in Equation (2), the SAR slant range of the hypersonic platform is relatively complex, thus it is difficult to derive the two-dimensional spectrum of the SAR echo signal due to the existence of the cosine function. Meanwhile, it is not easy for the mutual coupled range-azimuth frequency to carry out the subsequent imaging processing. For this problem, the approximate processing of the SAR slant range is executed to reduce the complexity of the ETF imaging algorithm. Here,  
Figure 3: SAR spectrum reconstruction for azimuth multichannel non-uniform sampling.  
the derived hyperbolic range model using the second-order Tylor series expansion is beneficial for deducing the approximate spectrum of SAR echo signal due to its relatively simple expression. However, under the circumstance of the enhanced space-variant feature resulted from the curved flight trajectory, the ignored high-order terms will further increase the phase error in azimuth. Furthermore, the fourth-order or higher-order term of the Taylor series expansion is considered to approximate the slant range, then the focusing performance of the targets in azimuth can be effectively improved, but the computational complexity will be significantly increased. Therefore, from the point of the lower complexity and wider applicability, the cosine function existed in the SAR slant range form is directly processed using a second-order polynomial fitting. The equivalent range model without any series expansion is helpful to improve the azimuth phase precision. Furthermore, the difference of the approximate cosine function relative to the original cosine function is constructed as the object function, where the analytic form is a function of the azimuth angle and polynomial fitting coefficients. Then the optimal fitting coefficients can be obtained to minimize the maximum approximation error under different azimuth angle variables when the objective function is approximated. It is worth noting that the solution of the polynomial coefficients can be further transformed into a minimax optimization problem, thus the fitting coefficients are further solved by using the two-population coevolutionary algorithm. Then, the approximate processing of SAR slant range is completed.  
Based on the approximate model of SAR slant range, we further design the ETF imaging algorithm. Here, the implementation of the imaging algorithm mainly includes the derivation of the two-dimensional spectrum, and the design of the exact transfer function and the compensation transfer function. Meanwhile, the ETF algorithm generally restricts the SAR imaging swath to avoid the defocus of targets near swath edge. Therefore, the wide swath will be divided into limited sub-swaths under the constraint of the ETF imaging algorithm. Then, the independent imaging processing on each sub-swath and the subsequent image mosaic operation are executed to obtain an integrate SAR image.