## 1 Introduction
Inspired by this, the hypersonic platform SAR adopts the displaced phase center antenna multichannel (DPCAM) technology to meet the HRWS imaging requirement in this paper. Although the DPCAM technique can effectively widen the range swath for hypersonic platform, there are strict restrictions on SAR system parameters such as the velocity of the platform, PRF and the phase center spacing of sub-apertures. Any deviation of parameters will lead to the non-uniform sampling in azimuth [16; 17]. However, due to the curved trajectory of the hypersonic platform SAR, each sub-aperture is linearly configured along the tangential direction of the flight trajectory, thus the azimuth data is difficult to meet the uniform sampling condition. Moreover, the PRF selection should be diverse since it is required to satisfy different beam position design. In addition, the velocity of this platform cannot hold as a constant. All of these factors would induce a non-uniform sampling in azimuth. Thus, based on the multichannel signal reconstruction theory [18], [PERSON] et al. [19; 20] proposed the spectrum reconstruction methods for SAR azimuth non-uniform sampling to alleviate the limitation of PRF selection. In this paper, we present a more efficient spectrum reconstruction model using the discrete Fourier transform and multiplication of complex data, which can pre-eliminate the effects of azimuth non-uniform sampling. According to the curved trajectory of the hypersonic platform SAR, the range-azimuth coupling will show a strong space-varying property, which further increases the difficulty for SAR imaging. Similar to the imaging geometry of the hypersonic platform, [PERSON] [21] studied the truck-mounted arc-scanning SAR, where the measured results with different imaging modes mainly focused on the monitoring application of the terrain deformation. Moreover, to improve the azimuth resolution of the rotating synthetic aperture radar (ROSAR) with circular trajectory, [PERSON] et al. [22] derived the approximate linear sampling model based on resampling and rotational transformation processing, and the extended Omega-K algorithm is used to obtain the ideal imaging results. In addition, since the second-order Taylor series expansion of the slant range will limit the effective synthetic aperture length and the range of reconstruction angle for ROSAT, [PERSON] et al. [23; 24] put forward the modified Omega-K algorithm to improve the imaging results by keeping higher order terms of Taylor series expansion. Although the traditional Omega-K algorithm adopting the hyperbolic range model can accurately correct the range cell migration, and obtain a well-focused SAR image for hypersonic platform SAR, this algorithm can only accurately focus the targets located at the reference range, and the residual phase at the non-reference range cannot be compensated by the Stolt interpolation operation. Meanwhile, it is difficult to deduce the imaging algorithm for hypersonic platform SAR due to the complexity of the slant range model. In this paper, an equivalent range model based on the minimax criterion is proposed, where the optimal second-order fitting coefficients of cosine function are derived using the two-population coevolutionary algorithm. On this basis, an Exact Transfer Function (ETF) imaging algorithm is then designed for hypersonic platform SAR with curved trajectory.