# Feasibility Study of Passive Bistatic Radar Based on Phased Array Radar Signals
## 4 Coherent Integration Method Based on Radon-NUFFT
### Radon-NulFFT Method  
Since the irregular RCM \(2\left(R_{0}-v_{0}t_{m}\right)/c\) is correlated with \(R_{0}\), \(v0\), and \(t_{m}\) and the irregular range-Doppler coupling effect \(f_{d}/\mu_{m}=2v_{0}B/\lambda\tau_{m}\) is correlated with \(v_{0}\) and \(\tau_{m}\), we can compensate them by searching through the motion parameters \(R_{0}\) and \(v_{0}\) for each pulse respectively.  
As for the nonuniform phase fluctuations \(\exp\left(j2\pi f_{d}t_{m}\right)\) among different pulses, which can be converted to the problem of spectral analysis of nonuniformly sampled complex-valued data, we use the nonuniform discrete Fourier transform (NUDFT) of nonuniform sampling data, which is given as follows:  
\[y(f_{k})=\sum_{i=0}^{N-1}x_{i}e^{-j\frac{2\pi}{N}f_{k}t_{i}} \tag{14}\]  
where \(x_{i}\) is the nonuniformly sampled data at nonuniform sampling time \(t_{i}\), \(N\) is the total number of sampling points, and \(f_{k}\) is the frequency point we want to analyze. However, the NUDFT algorithm requires a large amount of computation, so we use a NUFFT method based on a class of regular Fourier matrices [27], which is much faster than NUDFT. For the nonuniformly sampled data sequence \(\{x(t_{i})\}_{i=1}^{N}\), the spectral estimation at \(f_{k}\) can be denoted as \(y(f_{k})=\text{NUFFT}[x(t_{i}),f_{k}]\).  
Figure 7 shows the flowchart of the proposed method. The coherent integration time \(T_{n}\), the number of pulses \(N_{p}\), the range search scope \(\left[r_{min},r_{max}\right]\), and interval \(\Delta r\) are predetermined according to the dwell time of antenna and radar parameters. The searching scope of the initial velocity and interval are preset based on prior information such as moving status of targets to be detected. Therefore, the number of range searching parameters is \(N_{r}=\left[(r_{max}-r_{min})/\Delta r\right]\), where \(\left[\cdot\right]\) denotes the round up to an integer operation and the number of velocity searching parameters is \(N_{v}=\left[2v_{max}/\Delta v\right]\). Therefore, the moving trajectories of the target determined by the searching parameters can be given as follows:  
\[r(t_{m})=r_{i}-v_{j}t_{m} \tag{15}\]  
where \(m=1,2,\cdots,N_{p}\), \(r_{i}=r_{min}+(i-1)\Delta r\), \(i=1,2,\cdots,N_{r}\), \(v_{j}=-v_{max}+(j-1)\Delta v\), and \(j=1,2,\cdots,N_{p}\). Then, compensate the irregular range-Doppler coupling \(2v_{j}\tau_{m}/B\lambda\) for each pulse.  
For the signal shown in Equation (13), extract the \(N_{p}\) dimension data vector for coherent integration.  
\[\mathbf{X}_{N_{p}}=s_{PC}\left[t_{m},\frac{2(r_{i}-v_{j}t_{m})}{c}-\frac{2v_{ j}\tau_{m}}{B\lambda}\right] \tag{16}\]  
At last, calculate the spectral estimate of \(\mathbf{X}_{N_{p}}\) at frequency \(f_{j}=2v_{j}/\lambda\) to compensate the irregular RCM and range-Doppler coupling and nonuniform phase fluctuations among different pulses simultaneously, which can be expressed as  
\[s_{p}(r_{i},v_{j})=\text{NUFFT}\left\{s_{PC}\left[t_{m},\frac{2(r_{i}-v_{j}t_{m })}{c}-\frac{2v_{j}\tau_{m}}{B\lambda}\right],\frac{2v_{j}}{\lambda}\right\} \tag{17}\]  
where \(s_{p}(r_{i},v_{j})\) is the coherent integration result of the target with the initial bistatic range sum \(r_{i}\) and constant radial velocity \(v_{j}\). Go through all the searching parameters of range and velocity, then a two-dimensional result defined in the \((r,v)\) plane can be formed, which is the result of coherent integration based on the proposed method.  
According to the abovementioned analysis, the sampled data are accumulated coherently along the slow time dimension via a general Doppler filter bank, so when the searching parameters are equal to the real motion parameters, the irregular RCM and range-Doppler coupling can be compensated and the energy of target can be accumulated completely.