# Bistatic Forward ISAR with DVB-T Transmitter of Opportunity
## 4 BFISAR Image Reconstruction Algorithm  
The image extraction is the inverse operation to Equation (12). From the 2-D BFISAR signal \(S(k,p)\), a 2-D image can be extracted, i.e.,  
\[a(\hat{k},\hat{p})=\sum_{p=0,k=0}^{N-1,K-1}S(k,p)\exp\Big{\{}-j2\pi.\Big{[} \big{(}f_{0}+k.\Delta f.\big{)}\Big{(}t-t_{ijk}\Big{)}-k.\Delta f.\tilde{t} \Big{]}\Big{\}}, \tag{14}\]  
where \(\hat{k}=\overline{0,\ K-1},\hat{p}=\overline{0,\ N-1}\) are the new unknown coordinate indices of the \(ijk\)-th scattering point. To describe this process, the transformation of the 3-D image into a 2-D image, whose plane coincides with a signal plane, has to be considered. The 3-D coordinates of each scattering point are transformed into 2-D coordinates. The problem can be referred to a 3-D to 2-D projection, a design technique used to display the 3D object on the 2D surface. The exponential term in Equation (12) is a 3-D to 2-D projective operator. Consider the argument of the exponential term and denote as:  
\[\hat{\Phi}(k,p)=2\pi\Bigg{\{}.(f_{0}+k\Delta f)\Bigg{(}t-\frac{R_{ijk}(p)}{c} \Bigg{)}-k\Delta f.\tilde{t}\Bigg{\}}. \tag{15}\]  
Based on the 2-D Taylor expansion of \(\hat{\Phi}(k,p)\) in the vicinity of the imaging point defined by signal coordinates \(k=0,p=0\), the following expression can be written:  
\[\hat{\Phi}(k,p)=\hat{\Phi}(0,0)+\frac{\hat{\[PERSON]}^{(r)}(0,0)}{r!}(p.T_{p})^{r} +\frac{\hat{\[PERSON]}^{(q)}(0,0)}{q!}(k.\Delta T)^{q}, \tag{16}\]  
where \(\hat{\[PERSON]}(0,0)\) is the constant phase term that does not influence the imaging process; \(\hat{\[PERSON]}^{(r)}(0,0)\) is the \(r\)-th derivative of \(\hat{\[PERSON]}(k,p)\), where \(r=1,2,\ldots\), on the slow time \(\tilde{t};\hat{\[PERSON]}^{(q)}(0,0)\) is the \(q\)-th derivative of \(\hat{\[PERSON]}(k,p)\), where \(q=1,2,\ldots\), on the fast time \(\tau\) calculated for \(k=0,p=0\).  
In the case \(r=1\), the first derivative of \(\hat{\Phi}(k,p)\) for \(k=0,p=0\) is equal to \(2\pi.\frac{f_{0}}{c}.\frac{dR_{ijk}(0)}{dt}\), where \(\frac{dR_{ijk}(0)}{dt}=v_{rijk}(0)\) defines the radial velocity of the scattering point. The linear term can be modified as follows.  
\[\hat{\Phi}^{(1)}(0,0).(p.T_{p})=\frac{d\hat{\Phi}(00)}{d\hat{t}}(p.T_{p})=-2\pi. \frac{f_{0}}{c}\frac{dR_{ijk}(0)}{d\hat{t}}(p.T_{p})=-2\pi.\frac{c}{c.\lambda_{ 0}}\frac{dR_{ijk}(0)}{d\hat{t}}(p.T_{p}). \tag{17}\]  
Multiply and divide the expression with the period of the aperture synthesis and 2 (two), i.e.,  
\[\hat{\Phi}^{(1)}(0,0).(p.T_{p})=-2\pi.\frac{2}{\lambda_{0}}v_{rijk}(0).\frac{ \hat{T}}{2.\hat{T}}(p.T_{p}). \tag{18}\]  
Denote \(N=\frac{\hat{T}}{T_{p}}\) as the number of emitted segments for aperture synthesis, \(\Delta F_{D}=\frac{1}{\hat{T}}\) as the Doppler bandwidth, \(f_{Dijk}(0)=\frac{2}{\lambda_{0}}v_{rijk}(0)\) as the Doppler frequency of the scattering point, and \(\hat{p}=\frac{f_{Dijk}(0)}{2.\Delta F_{D}}\) as the Doppler index, the dimensionless coordinate of the \(ijk\)-th scattering point in the projection signal plane. In the case of \(r=1\), the first linear term can be presented as:  
\[\hat{\Phi}^{(1)}(0,0).(p.T_{p})=-2\pi\frac{\hat{p}.p}{N}. \tag{19}\]  
In the case of \(q=1\), the second linear term \(\hat{\Phi}^{(1)}(0,0). (k.\Delta T)=\frac{d\hat{\Phi}(0,0)}{dt}(k\Delta T)\) is modified as follows.