## 3 Signal Processing
### Signal Processing Procedure
 \tag{45}\]  
where \(l=0,\cdots,MN-1\) is the ambiguity number, and \(\mathbf{1}_{MN}=\left[ \begin{array}{cccc}1&1&\cdots&1\end{array} \right]^{T}\) is the vector of \(MN\) ones. The \(MN\times MN\) channel phase delay matrix is a diagonal matrix whose elements can be defined as, \(\left(\mathbf{\Phi}\left(f_{a}\right)\right)_{k,k}=g_{k}\left(f_{a}\right)\), where \(g_{k}\left(f_{a}\right)\) is \(k\)-th element of the MIMO steering vector, \(\mathbf{g}\left(f_{a}\right)\). In Equation (45), we assume ideal and identical patterns for all antennas. This equation is included here for completeness of the signal processing procedure. The measured \(MN\times 1\) multi-channel signal vector is equal to:  
\[\mathbf{z}\left(r,f_{a}\right)=\mathbf{c}\left(r,f_{a}\right)+\mathbf{n} \left(r,f_{a}\right) \tag{46}\]  
where \(\mathbf{n}\left(r,f_{a}\right)\) is the noise vector. As mentioned in the introduction, there are many reconstruction algorithms. In the following, the matrix inversion method [9] is described due to the simplicity of its implementation. The reconstruction filters can be calculated through minimization as [9]:  
\[F_{M}=\min_{\mu_{p}\left(f_{a}\right)}\left\|z_{org}\left(r,t_{a}\right)-z_{ rec}\left(r,t_{a}\right)\right\| \tag{47}\]  
where \(\left\|\cdot\right\|\) denotes the L\({}_{2}\) norm and \(z_{org}\left(r,t_{a}\right)\) is the original signal, sampled at a frequency of \(MN\cdot f_{p}\). The reconstructed signal is obtained as:  
\[z_{rec}\left(r,f_{a}+lf_{p}\right)=\mu_{p}\left(f_{a}\right)\mathbf{z}\left(r,f_{a}\right) \tag{48}\]  
where \(\mathbf{z}\left(r,f_{a}\right)\) is the multi-channel data given by:  
\[\mathbf{z}\left(r,f_{a}\right)=\left[ \begin{array}{cccc}z_{0}\left(r,f_{a} \right)&z_{1}\left(r,f_{a}\right)&\cdots&z_{MN-1}\left(r,f_{a}\right)\end{array}  \right]^{T} \tag{49}\]  
and the reconstruction filter can be written as:  
\[\mu_{l}\left(f_{a}\right)=\mathbf{H}_{int}^{H}\cdot\mathbf{1}_{MN}^{l} \tag{50}\]  
where the superscript, \({}^{H}\), denotes the Hermitian transpose, and \(\mathbf{1}_{MN}^{l}=\left[ \begin{array}{cccc}0&0&\cdots&1&\cdots&0\end{array} \right]^ {T}\) is an \(MN\times 1\) vector of zeros except for the \(l+1\)th position which contains a one, and,  
\[\mathbf{H}_{int}\left(f_{a}\right)=\mathbf{H}\left(f_{a}\right)^{-1} \tag{51}\]  
\[\mathbf{H}\left(f_{a}\right)=\left[ \begin{array}{cccc}\mathbf{g}_{0}\left(f _{a}\right)&\mathbf{g}_{1}\left(f_{a}\right)&\cdots&\mathbf{g}_{MN-1}\left(f_{ a}\right)\end{array} \right] \tag{52}\]  
\[\mathbf{g}_{l}\left(f_{a}\right)=\mathrm{Vec}\left\{\mathbf{A}\left(f_{a}+lf_{p }\right)\right\} \tag{53}\]Note that, as mentioned in [7], in the case of a single-platform system or collocated antennas, the columns of the channel matrix, \(\mathbf{H}\)\((f_{a})\), consist of the steering vectors, \(\mathbf{g}_{l}\)\((f_{a})\). After reconstruction, SAR video frames are formed by the PFA, with \(z_{rec}\)\((r,f_{a}+lf_{p})\) as input data. In the circular path case, image rotation is implemented in the frequency domain using Equation (54), so that the image is in a fixed orientation. This orientation is called the cardinal direction. Image rotation is performed in the frequency domain to reduce distortion. Rotation proceeds as follows:  
\[\mathbf{k}_{CDU}=\mathbf{R}\cdot\mathbf{k}_{FRU} \tag{54}\]  
where \(\mathbf{k}_{CDU}\) is the spatial frequency vector in the cardinal direction up (CDU) coordinate system (global coordinate), \(\mathbf{k}_{FRU}\) is the spatial frequency vector in the far range up (FRU) coordinate system (local coordinate), and the rotation matrix is defined as follows:  
\[\mathbf{R}=\left[ \begin{array}{cc}\cos\theta_{as}&\sin\theta_{as}\\ -\sin\theta_{as}&\cos\theta_{as}\end{array} \right] \tag{55}\]  
where \(\theta_{as}\) is the aspect angle with respect to the cardinal direction, as shown in Figure 2.  
A block diagram of signal processing procedures in the proposed ViSAR system is shown in Figure 9. The first part of signal processing is BFD waveform demodulation, which includes Doppler compensation.