# A Novel Channel Calibration Method for Bistatic ISAR Imaging System
## 2. Bistatic ISAR Signal Model with Non-Ideal Channel  
Assuming that the transmitting station and receiving station are ideal synchronously, the transmitted LFM signal with pulse repetition period \(T_{\text{PRT}}\) is shown as:  
\[s_{t}\big{(}\hat{t},t_{m}\big{)}=\text{rect}\bigg{(}\frac{\hat{t}}{T_{p}} \bigg{)}\exp\bigg{[}j2\pi\bigg{(}f_{c}t+\frac{1}{2}\mu\hat{t}^{2}\bigg{)} \bigg{]} \tag{1}\]  
where \(\text{rect}(x)\) is the rectangular window function defined as \(\text{rect}(x)=1\), if \(|x|\leq 0.5\), otherwise \(\text{rect}(x)=0\). \(\hat{t}\) is the fast time, \(t_{m}=mT_{\text{PRT}},\ (m=0,1,2,\ldots)\) is the slow time. \(t\) is the total time and \(\hat{t}=t-t_{m}\). \(T_{p}\) is the pulse width. \(f_{c}\) is the carrier frequency. \(\mu\) is the chirp rate. The baseband signal can be expressed as:  
\[s_{b}\big{(}\hat{t},t_{m}\big{)}=\text{rect}\bigg{(}\frac{\hat{t}}{T_{p}} \bigg{)}\exp\bigg{(}j\pi\mu\hat{t}^{2}\bigg{)} \tag{2}\]  
Assuming that delay time of the target is \(\tau\), and the RCS (Radar Cross Section) is constant and can be ignored, the echo signal is given as:  
\[s_{r}\big{(}\hat{t},t_{m}\big{)}=\text{rect}\bigg{(}\frac{\hat{t}-\tau}{T_{p} }\bigg{)}\exp\bigg{\{}j2\pi\bigg{[}f_{c}(t-\tau)+\frac{1}{2}\mu\big{(}\hat{t} -\tau\big{)}^{2}\bigg{]}\bigg{\}} \tag{3}\]The baseband signal of the echo is obtained by converting the echo signal to zero intermediate frequency (zero-IF) through a coherent local oscillator. It can be written as:  
\[s_{if}\big{(}\hat{t},t_{m}\big{)}=s_{b}\big{(}\hat{t}-\tau,t_{m}\big{)}\exp(-j2 \pi f_{c}\tau) \tag{4}\]  
The corresponding frequency domain signal can be expressed as:  
\[S_{if}(f,t_{m})=S_{b}(f)\exp[-j2\pi(f+f_{c})\tau] \tag{5}\]  
where \(S_{b}(f)\) is the spectrum of \(s_{b}\big{(}\hat{t},t_{m}\big{)}\). The high resolution range profile (HRRP) is obtained through pulse compression with a matched filter. The matched filter \(H(f)\) is the frequency domain conjugate of baseband signal \(s_{b}\big{(}\hat{t},t_{m}\big{)}\), that is \(H(f)=S_{b}^{*}(f)\). The HRRP and its spectrum can be written as:  
\[s_{if\_c}(\hat{t},t_{m}\big{)}\approx\sqrt{\mu}T_{p}\cdot\sin\c[\mu T_{p}\big{(} \hat{t}-\tau\big{)}]\exp(-j2\pi f_{c}\tau) \tag{6}\]  
\[S_{if\_c}(f,t_{m})=|S_{b}(f)|^{2}\exp[-j2\pi(f_{c}+f)\tau] \tag{7}\]  
In the formulas above, \(\tau\) is the function of slow time, and \(\sin\c(x)=\sin(\pi x)/(\pi x)\) is the sinc function. In Equation (6) the peak of the sinc function represents the radial position of scatter. The Doppler information which is used for cross-range resolution represents in the phase of \(\exp(-j2\pi f_{c}\tau)\). Assuming that the frequency transfer function of the channel is \(H_{TR}(f)\), the echo signals are modulated by \(H_{TR}(f)\), and the spectrum of the HRRP can be expressed as:  
\[S_{com\_chn}(f,t_{m})=|S_{b}(f)|^{2}\cdot H_{TR}(f)\cdot\exp[-j2\pi(f_{c}+f)\tau] \tag{8}\]  
Due to the influence of the non-ideal characteristics of the transmitting and receiving channels, the baseband echo signals are not ideal LFM signal. Taking the inverse Fourier transform (IFFT) to Equation (8), the same sinc function as in Equation (6) cannot be obtained. The main lobe is broadened and the side lobe level is increased. Figure 1 shows the HRRP results of the single point target in bistatic radar system. Figure 1a is the HRRP obtained with the ideal channel, and Figure 1b is the HRRP obtained with the non-ideal channel. By contrast, it was seen that the main lobe of the HRRP in Figure 1b is significantly broadened, and the peak amplitude was also reduced because of the channels non-ideal characteristics. The essential reason for this phenomenon is that the channel non-ideal characteristics distort the LFM signal of the radar. If the ideal reference signal without calibration was used in pulse compression process as usual, the mismatch would degrade the pulse compression performance. The two-dimensional ISAR image was also defocused in Figure 2. It is obvious that the influence of non-ideal channel characteristics on echo cannot be ignored.