## 2 Fundamentals of SAR for Antenna Sizing and Transmitting Power Consideration
### Selection of Operating Frequency  
The selection of the operating frequency band for a spaceborne SAR satellite is influenced by various factors, including system performance, mission requirements, and technical considerations. The selection of the frequency band, such as the X-band, for a SAR satellite affects the antenna sizing process. There are some details on how the band selection influences antenna sizing and how the X-band can help in reducing the antenna size. The operating frequency of the SAR system is inversely proportional to the wavelength. The X-band, with a frequency range of approximately 8-12 GHz, has a relatively shorter wavelength compared to lower frequency bands like the L-band or S-band [33; 36; 37; 38; 39; 40; 41; 42; 43]. Radar in space typically employs frequencies between the L- and X-band. The X-band is often chosen, especially for small SAR satellites since it offers a good balance between spatial resolution and penetration capabilities. It provides sufficient resolution for imaging fine ground details while maintaining some surface penetration through vegetation and dry soil. X-band signals exhibit moderate surface reflectivity, making them suitable for various land surface types, including urban areas, forests, and agricultural regions. The signals are reflected well by these surfaces, allowing for the detection of different features and structures [33; 36; 37; 38]. X-band frequencies generally allow for smaller antenna dimensions compared to lower frequency bands, such as L-band frequencies. This is advantageous for small SAR satellites, as it helps in reducing the overall size and weight of the antenna subsystem, which is often a critical factor in small satellite design. Moreover, higher frequency bands, such as the X-band, offer the advantage of simpler antenna designs and shorter pulse durations, resulting in less complex SAR system architectures. This simplicity can be beneficial for small satellite missions with limited resources and short development times. Current power amplifiers have also been improved to support high frequencies. Furthermore, the X-band is well-suited for various antenna designs, including microstrip antennas, patch antennas, or aperture-based designs. These antenna technologies provide flexibility in terms of miniaturization and integration with small satellite platforms. The selection of the X-band can help in meeting certain system constraints. Small SAR satellites often have limitations in terms of the available space, mass, and electric power. It has an advantage of antenna gain for small and lightweight antennas due to wavelength. Choosing the X-band allows for smaller and more lightweight antennas, which can better fit within the overall constraints of the satellite system. However, the antenna sizing process is also influenced by various other factors, including the desired resolution, swath coverage, and signal-to-noise ratio requirements. The specific mission objectives, imaging performance goals, and other technical considerations also play a role in determining the optimal antenna size [42; 43; 44; 45; 46; 47]. The X-band is a widely allocated frequency band for Earth observation and remote-sensing applications, with well-defined regulatory frameworks. This allocation enables the acquisition of necessary frequency authorizations and ensures compatibility with other satellite systems and users. The X-band is well-suited for downlinking SAR data from the satellite to the ground stations. It provides higher data transfer rates compared to lower frequency bands, allowing for efficient and timely data transmission. As a type of remote sensing mission, spaceborne SAR systems use microwave regions of the electromagnetic spectrum. Higher frequency like Ku-band has the disadvantage of rain attention. Moreover, we can easily see the usage of the X-band (around 9.6 GHz or \(\lambda\) = 0.03 m) in recent small SAR satellites.