## 3 Imaging Algorithm Based on Modified Chirp Fourier Transform  
### Azimuth Fast Compression Based on MCFT  
The chirp Fourier transform is a polar coordinate representation of the matched Fourier transform [27; 28]. Since the transform kernel is a chirp function, it is a linear transform for the chirp signal. The chirp Fourier transform (CFT) of the signal \(s(t)\) is:  
\[F(f,\gamma)=\int s(t)\exp\left(j2\pi\left(ft+\frac{1}{2}\gamma t^{2}\right) \right)dt \tag{12}\]  
Figure 2: Example of time-frequency diagram of a range cell. (**a**) Range profiles after alignment and (**b**) smooth pseudo Wigner-Ville distribution (SPWVD) of a range cell. When the signal is a chirp signal as shown in Equation (9), its frequency-modulated Fourier transform is:  
\[F(f,\gamma)=\int\exp\left(j2\pi[(f-f_{i})t+\frac{1}{2}(\gamma-k_{i})t^{2}\right) dt=\delta(f-f_{i})\delta(\gamma-k_{i}) \tag{13}\]  
When \((f,\gamma)=(f_{i},k_{i})\), the signal is accumulatively added to reach the maximum value. On the contrary, when \((f,\gamma)\
eq(f_{i},k_{i})\), there is a mutual cancellation of the signal integration and the corresponding amplitude is small. CFT is a linear transformation, so there is no cross-term problem. Therefore, the chirp Fourier transform is performed on the azimuth multi-component LFM signal, and a spike occurs at the corresponding \((f_{i},k_{i})\) position to achieve compression. However, CFT requires a complex two-dimensional parameter search, resulting in a large amount of computation, which greatly reduces the engineering practicality of the algorithm [29; 30]. From the analysis in the previous section, the rotation ratio \(K_{\Omega\omega}\) for the maneuvering target is a fixed value. If \(K_{\Omega\omega}\) is known, we can redefine the modified chirp Fourier transform (MCFT) as:  
\[F(f)=\int s(t)\exp\left(j2\pi ft\left(1+\frac{1}{2}K_{\Omega\omega}t\right) \right)dt \tag{14}\]  
Combining Equations (11) and (14) produces:  
\[ \begin{array}{ll}S_{a}(f)&=\int\sum\limits_{i=1}^{N}\sigma_{i}\exp(-j2\pi f _{i}t_{m}(1+\frac{1}{2}K_{\Omega\omega}t_{m}))\exp\left(j2\pi ft_{m}(1+\frac{1 }{2}K_{\Omega\omega}t_{m})\right)dt_{m}\\ &=\sum\limits_{i=1}^{N}\sigma_{i}f\exp(-j2\pi(f-f_{i})t_{m}(1+\frac{1}{2}K_{ \Omega\omega}t_{m}))dt_{m}\\ &=\sum\limits_{i=1}^{N}\sigma_{i}\delta(f-f_{i})\end{array}  \tag{15}\]  
when \(f=f_{i}\), Equation (15) provides the maximum value. That is, if the rotation ratio \(K_{\Omega\omega}\) is known, MCFT is only needed once to compress the azimuth echo signal, and no two-dimensional search transformation is needed. This greatly reduces the computational complexity of the algorithm and improves the applicability of the algorithm. The discrete form of MCFT is:  
\[F(k)=\frac{1}{\sqrt{N}}\sum\limits_{n=0}^{N-1}s(n)W_{N}^{kn(1+\frac{K_{\Omega \omega}}{2N}n)} \tag{16}\]  
where \(n\) is the sampling point, \(N\) is the number of sampling points, \(s(n)\) is the discrete sequence of signals, \(W_{N}\) is the transform rotation factor and \(W_{N}=\exp(j2\pi/N)\), and \(k\) is the digital frequency. Because the phase factor of Equation (16) contains \(n^{2}\) terms, it can only be calculated by discrete Fourier transform. In order to improve its speed, Equation (16) can be written in matrix form:  
\[F=\frac{1}{\sqrt{N}}\mathbf{s}\mathbf{W}_{FN}=\frac{1}{\sqrt{N}}\mathbf{s}\mathbf{W}_{N}^{k( \mathbf{n+\frac{K_{\Omega\omega}}{N}n\otimes\mathbf{n})}} \tag{17}\]  
where \(\otimes\) is the Hadamard product and:  
\[\mathbf{n}=[0,1,2,\cdots,N] \tag{18}\]  
\[\mathbf{k}=[0,1,2,\cdots,N]^{T} \tag{19}\]  
\[\mathbf{s}=[s(0),s(1),s(2)\cdots,s(N-1)]^{T} \tag{20}\]  
\[\mathbf{F}=[F(0),F(1),F(2),\cdots,F(N-1)]^{T} \tag{21}\]\[\mathbf{W}_{FN}=\left[