# mmSight: A Robust Millimeter-Wave Near-Field SAR Imaging Algorithm
## 3 System Design
### Target Imaging
#### 3.3.2 Imaging with Analytic Fourier Transform  
The 2D imaging algorithm at this stage is the Analytic Fourier Transform, which is derived from the Back Projection algorithm. The Back Projection is a spatial domain imaging algorithm often used for SAR imaging, which was inspired by the Computer Aided Tomography (CAT) technique. The idea of the BP (Back Projection) algorithm, which establishes a relationship between the antenna array position and the target grid position, is to calculate the pixel value, i.e., the reflection intensity of each grid pixel point, by back-projecting the radar echo data to each pixel in the imaging area and using the distance of the echo between the antenna and the image pixel to compensate for the time-delayed phase of the echo signal. The Back Projection algorithm has the following equation:  
\[f(x,y)=\iint s(x^{\prime},y^{\prime})e^{-j2k\sqrt{\left(x-x^{\prime}\right)^{2 }+\left(y-y^{\prime}\right)^{2}+2\phi^{2}}}dx^{\prime}dy^{\prime} \tag{6}\]  
where, \(f(x,y)\) denotes the reflection distribution of the target, which, in the BP algorithm, is the imaging grid to be divided. \(s(x^{\prime},y^{\prime})\) denotes the radar echo data after dealing with distance-FFT, pulse compression, and each coordinate point, corresponding to a spatial sampling point of the SISO monostatic virtual antenna array. Back Projection can adapt to different types of antenna arrays, and also has the best imaging results in theory because no approximation is made. However, due to the point-by-point traversal operation of this algorithm, it has a high time-complexity and is less efficient than other imaging algorithms. Due to the low efficiency of the Back Projection algorithm, the more commonly used, is the Analytic Fourier Transform imaging algorithm. It is a wavenumber domain imaging  
Figure 4: Procedure of windowing. algorithm with high efficiency, which can be derived from the Back Projection algorithm, and the derivation process is as follows. First, the spherical wave is approximated as a superposition of an infinite number of plane waves [5]:  
\[e^{-j2k\sqrt{\left(x-x^{\prime}\right)^{2}+\left(y-y^{\prime}\right)^{2}+z_{0}}}= e^{-j2k\sqrt{\left(x^{\prime}-x\right)^{2}+\left(y^{\prime}-y\right)^{2}+z_{0} ^{2}}}\approx\iint e^{jk_{x}(x-x^{\prime})+jk_{y}(y-y^{\prime})-jk_{z}z_{0}}dk_ {x}dk_{y} \tag{7}\]  
Substituting into the Back Projection equation:  
\[f(x,y)=\iint s(x^{\prime},y^{\prime})\iint e^{jk_{x}(x-x^{\prime})+jk_{y}(y-y^{ \prime})-jk_{z}z_{0}}dk_{x}dk_{y}dx^{\prime}dy^{\prime} \tag{8}\]  
After collating the equations, the following is shown:  
\[f(x,y)=\iint\Bigl{[}\iint s(x^{\prime},y^{\prime})e^{-jk_{x}x^{\prime}-jk_{y}y ^{\prime}}dx^{\prime}dy^{\prime}\Bigr{]}e^{-jk_{z}z_{0}}e^{jk_{x}x+jk_{y}y}dk_ {x}dk_{y} \tag{9}\]  
The equation for the Analytic Fourier Transform imaging algorithm is as follows [11]:  
\[f(x,y)=FT_{2D}^{-1}\Bigl{[}FT_{2D}[s(x^{\prime},y^{\prime})]e^{-jk_{z}z_{0}} \Bigr{]} \tag{10}\]  
where, \(FT_{2D}^{-1}\) and \(FT_{2D}\) denote the two-dimensional Inverse Fourier Transform and two-dimensional Fourier Transform, respectively, and \(k_{z}\) denote the wavenumber on the z-axis. According to the dispersion relation of plane waves in free space, the relationship between the wavenumber \(k\) and \(k_{x}\), \(k_{y}\), \(k_{z}\) is as follows [11]:  
\[{k_{x}}^{2}+{k_{y}}^{2}+{k_{z}}^{2}=(2k)^{2} \tag{11}\]  
Thus, the formula for the Analytic Fourier Transform imaging algorithm, in turn, can be expressed in the following form:  
\[f(x,y)=FT_{2D}^{-1}\Bigl{[}FT_{2D}[s(x^{\prime},y^{\prime})]e^{-j\sqrt{4k^{2} -k_{x}z^{2}-k_{y}^{2}}z_{0}}\Bigr{]} \tag{12}\]  
This is the formula that is referred to when the specific code is implemented.