# Clutter Cancellation and Long Time Integration for GNSS-Based Passive Bistatic Radar
## 3 The Clutter Cancellation and Proposed Long Time Integration Algorithm
### The Proposed Long Time Integration Algorithm  
It is worth noticing that DC and DRM that are related to the motion of the air target are constant during the CPI. The ratio of DRM and DC are defined as relative acceleration (RA), which can be expressed as  
\[\mu=\frac{f_{dmr}}{f_{dc}} \tag{19}\]  
Subsequently, (9) can be rewritten as  
\[ \begin{array}{ll}s(\hat{t}_{m})&=A\exp(-j\frac{2\pi}{\lambda}(R_{0}+\lambda f _{dc}\hat{t}_{m}+\frac{1}{2}\lambda\mu f_{dc}\hat{t}_{m}^{2}))\\ &=A\exp(-j\frac{2\pi}{\lambda}(R_{0}+\lambda f_{dc}\hat{t}_{m}(1+\frac{1}{2} \mu\hat{t}_{m}^{2})))\end{array}  \tag{20}\]  
If the Doppler is constant during the CPI, FT can be applied to obtain the coherent integration processing gain. The target echo after cross-correlation and KT is denoted as \(S(m,n)\), where \(m\) is the slow time and \(n\) is the fast time. We denote the range-Doppler map as \(I\), the FT matrix as \(F\), and the process of the coherent integration can be expressed as  
\[I=F\cdot S \tag{21}\]  
For a coherent integration containing \(M\) pulses, the FT matrix can be written as  
\[F=W^{mm^{\prime}} \tag{22}\]  
where \(W=\exp(-j\frac{2\pi}{M})\), \(m,m^{\prime}=0,1,2,\ldots,M-1\). Each \(m^{\prime}\) value corresponds to a row vector in the FT matrix. Subsequently, the integration component for quadratic phase terms in (20) can be expressed as  
\[F_{c}=W^{(m-M/2)m^{\prime}\mu/2/PRF} \tag{23}\]  
According to (23), the compensation component can be constructed with the variation of \(m\) and \(m^{\prime}\). The size of the matrix \(F_{c}\) is the same as \(F\). The FT matrix containing the Doppler shift compensation is  
\[F^{\prime}=F\otimes F_{c} \tag{24}\]  
where \(\otimes\) denotes Hadamard product and \(F^{\prime}\) is named as the long-time integration Fourier transform (LIFT) matrix. Subsequently, the coherent integration process can be expressed as  
\[I=F^{\prime}\cdot S \tag{25}\]  
The above analysis is based on the fact that the RA is already known. However, the ratio is unknown for the target and it needs to be estimated in advance. Equation (9) indicates that the Doppler frequency can be described by a linear frequency modulated signal. Many parameter estimation methods can be applied to estimate the RA, such as fractional Fourier transform (FrFT) and Lv's distribution [33; 34]. Here, FrFT is used for parameter estimation. The definition of the FrFT for signal \(x(t)\) is  
\[X_{p}(u)=\int_{-\infty}^{\infty}x(t)K_{p}(u,t)dt \tag{26}\]\[K_{p}(u,t)= \begin{cases}c\delta(u-t)&p=(2n\pm 1)\pi\\ \delta(u-t)&p=2n\pi\\ \sqrt{1-j\cot p}\exp(j\pi(u^{2}+t^{2})\cot p-2 ut\csc p)&p\
eq n\pi\end{cases}  \tag{27}\]  
where \(K_{p}(u,t)\) is the transform kernel that is defined as (27) and \(n=1,2\ldots\). The result of the transformation concentrates the chirp signal energy on the specific point in the FrFT plane. Additionally, the parameters of the LFM signal can be obtained by  
\[ \begin{array}{l}\widetilde{f}_{dc}=u\csc p/T\\ \widetilde{f}_{dmr}=-(cotp).\frac{PRF}{T}\end{array}  \tag{28}\]  
Subsequently, the estimated RA can be calculated by  
\[\widetilde{\mu}=\widetilde{f}_{dmr}/\widetilde{f}_{dc}=PRF\cos p/u \tag{29}\]  
However, the estimated value is inaccurate due to the interfere and noise. Here, we define a new variable to evaluate the coherent integration performance. The definition of energy integration ratio (EIR) is  
\[\rho=\frac{E(\widetilde{\mu})}{E} \tag{30}\]  
where \(E(\widetilde{\mu})\) is the signal power of the Doppler cell that target located after coherent integration and \(E\) is the total signal power of the target echo after clutter cancellation. It can be noticed that \(\rho<1\) and a larger \(\rho\) indicates a better integration effect. Hence, the accurate \(\widetilde{\mu}\) can be obtained by maximizing the \(\rho\).