## 2 Theory of Video SAR
### Frame Rate  
The frame rate is one of the most important parameters in the ViSAR system. In ViSAR, the frame rate is proportional to the inverse of the SAR video frame time, which is a fraction of the synthetic aperture time (SAT) and therefore can be expressed as:  
\[f_{v}=\frac{1}{T_{f}} \tag{1}\]  
Figure 2: The collection geometry of video SAR on a circular path: top view. CDU: cardinal direction up. where \(T_{f}\) is the SAR video frame time, which is defined as:  
\[T_{f}=F_{o}T_{a} \tag{2}\]  
where \(T_{a}\) is the SAT and \(F_{o}\) is the overlap ratio between SAR video frames. Note that \(F_{o}=1\) describes the non-overlap processing condition, which will be applied throughout this paper. For spotlight SAR mode (or partial measurement of circular SAR) on a circular path, the synthetic aperture time for spotlight SAR can be expressed as [2] Equation (21)):  
\[T_{a}=\frac{L}{v}=\frac{\lambda R_{a}K_{a}}{v\cdot 2\rho_{a}\sin\left(\alpha_{ dc}\right)} \tag{3}\]  
where \(L\) is the synthetic aperture length, \(\lambda\) is the wavelength at the transmitted frequency, \(R_{a}\) is the distance from the antenna phase center (APC) to the scene center, \(K_{a}\) is the beam broadening factor, \(v\) is the sensor velocity, \(\rho_{a}\) is the cross-range resolution, and \(\alpha_{dc}\) is the antenna cone angle, which is the antenna squint angle and is assumed to be 90 degrees throughout this paper. For the rest of this paper, it is convenient to assume that, in general, the radar is in a broadside condition. This assumption simplifies the subsequent mathematics. Therefore, the frame rate in the spotlight SAR mode can be represented as:  
\[f_{v}=\frac{1}{T_{a}}=\frac{v\cdot 2\rho_{a}\sin\left(\alpha_{dc}\right)}{ \lambda R_{a}K_{a}}=\frac{v\cdot 2\rho_{a}\sin\left(\alpha_{dc}\right)}{cR_{a}K_{a }}f_{c} \tag{4}\]  
where \(f_{c}\) is the transmitted frequency, and \(c\) is the speed of light. From the above equation, when the resolution and the distance to the observation point are fixed, the radar velocity or transmitted frequency should be increased in order to increase the frame rate of the SAR video. However, the radar velocity is limited by the Doppler bandwidth, which is constrained to avoid azimuth ambiguity. Figure 3a,b shows an example of the frame rate of the ViSAR system, which is calculated using Equation (4) at \(R_{a}=1000\) m and \(K_{a}=1\) with various resolutions and platform velocities. As shown in Figure 3a, the frame rate is 1.003 Hz at 94 GHz in the W band, which is 9.4 times as high as it is in the X band--which includes systems such as Global Hawk or NanoSAR--where the frame rate is 0.107 Hz at 10 GHz, according to Equation (4). Therefore, the frame rate can be increased significantly, by increasing the transmitted frequency. Additionally, a higher radar velocity leads to higher frame rate. For example, two times higher frame rates can be achieved by increasing the platform velocity from 20 m/s to 40 m/s; i.e., frame rates are increased from 1.003 Hz to 2.005 Hz, at 94 GHz. However, in conventional SARs, the radar velocity is limited, to prevent Doppler ambiguities from occurring. At this time, if the MCRA is applied and the number of receiving channels is increased, Doppler ambiguities can be eliminated, even if the Doppler bandwidth increases due to the increase in the radar velocity. Therefore, the radar platform velocity can be increased without increasing Doppler ambiguities, by increasing the number of receiving channels using the MCRA. In summary, according to Equation (4), the frame rate in the circular path can be increased by increasing the transmitted frequency or the platform velocity while maintaining the azimuth resolution. Pulse repetition frequency (PRF) is proportional to the frame rate, since this is proportional to the platform velocity. As the platform velocity increases, the increased Doppler bandwidth requires an increased PRF.