## 3 Azimuth Spectrum Reconstruction
### Azimuth Spectrum Reconstruction  
As shown in Figure 2, to achieve HRWS imaging, the hypersonic platform SAR adopting the DPCAM technique has the access to permit a low PRF in azimuth sampling. Based on which multiple sub-apertures are linearly configured in the along-track dimensional and each of them deviates from the transceiver, then azimuth sampling signal of the multiple sub-apertures can be obtained in one pulse repetition time (PRT). However, since the DPCAM equivalent azimuth echo is a non-uniform sampling sequence, which cannot be directly processed as an input signal for SAR imaging. Thus, in this section, the non-uniform sampling signal is pre-processed using a more efficient spectrum reconstruction method, and then the azimuth uniform signal of the equivalent single channel SAR system with low PRF can be obtained. Generally, based on the multichannel signal reconstruction theory [18], the multi-filter banks reconstruction method for SAR non-uniform sampling mainly includes the linear filter banks and reconstruction filter banks. Specially, the azimuth baseband signal with a limited bandwidth \(B_{a}\) is passed through the \(Q\) band-limited filter banks, and the filtered signal is further sampled at the under-sampling rate of \(B_{a}/Q\). Then, the multichannel reconstruction filter banks should be designed to restore the azimuth uniform signal. Based on the azimuth response of the single channel SAR system, [PERSON] et al. [20] derived the transfer function of the linear filter banks by introducing a constant phase shift and a time delay. Furthermore, the reconstruction filter banks are derived from the \(Q\) transfer functions. In this paper, we present a new reconstruction method with only fast Fourier transform (FFT) and multiplication of complex data, thus the high complexity of implementing the general interpolation operation can be effectively avoided. When the DPCAM uniform sampling condition is not satisfied, although the azimuth signal is shown as a non-uniform sampling sequence, the adjacent \(Q\) sampling points are uniformly spaced in azimuth time. Therefore, we consider that the non-uniform azimuth signal is generated after sampling the filtered signal of \(Q\) channels uniformly. Here, the impulse response of the \(q\)th linear filter is recorded as \(\left\{\delta(\eta-\Delta x_{q}/2L\omega_{s})\right\}\). Then, the transfer function of each pre-filter bank can be expressed as  
\[H_{q}\big{(}f_{\eta}\big{)}=\exp\left(-\beta 2\pi\frac{\triangle x_{q}}{2L \omega_{s}}f_{\eta}\right). \tag{8}\]  
The reconstruction filter banks can be further derived based on the linear filter banks, and each reconstruction filter is composed of \(Q\) band-limited filter banks, based on which the frequency range of the azimuth baseband signal can be further divided. Here, consider an odd number of receiving channels. Then, the frequency of the \(q\)th sub-band can be obtained by  
\[E_{q}=\left(-\frac{Q\cdot PRF}{2}+(q-1)\cdot PRF,-\frac{Q\cdot PRF}{2}+q\cdot PRF \right). \tag{9}\]  
Referring to Equation (8), the independent pre-filter bank of each receiving channel can be further expressed as matrix \(H(f_{\eta})\), which is  
\[H\big{(}f_{\eta}\big{)}=\left[ \begin{array}{ccc}H_{1}\big{(}f_{\eta}\big{)} &\cdots&H_{Q}\big{(}f_{\eta}\big{)}\\ H_{1}\big{(}f_{\eta}+PRF\big{)}&\cdots&H_{Q}\big{(}f_{\eta}+PRF\big{)}\\ \vdots&\ddots&\vdots\\ H_{1}\big{(}f_{\eta}+(Q-1)PRF\big{)}&\cdots&H_{Q}\big{(}f_{\eta}+(Q-1)PRF\big{)} \end{array} \right], \tag{10}\]  
where \(f_{\eta}\in E_{1}\), and the \(q\)th row corresponds to the pre-filter function of the \(E_{q}\) frequency range, and the \(q\)th column corresponds to the pre-filter function of the \(q\)th receiving channel. Then, the inverse transformation of Equation (10) is performed to derive the reconstruction filters, where the corresponding matrix expression is expressed as follows:  
\[G\big{(}f_{\eta}\big{)}=H^{-1}\big{(}f_{\eta}\big{)}=\left[