# K/Ka-Band Very High Data-Rate Receivers: A Viable Solution for Future Moon Exploration Missions
## 2 Basic Link Budget  
Before presenting a detailed analysis, some basic considerations are introduced here in order to estimate the receiver signal levels. Figure 1 shows a simplified link geometry where the receiver is on the Moon (M) surface, exactly at the zenith of a ground station on the Earth (E). In this figure \(G_{a}^{TX}\) and \(G_{a}^{RX}\) are the gains of the ground station and of the satellite antennas respectively, \(P_{TX}\) is the transmitted power and \(R\) is the average Earth-to-Moon distance. The received power \(P_{RX}\) can be determined applying the Friis' formula to such a link geometry:  
\[P_{RX}=G_{a}^{TX}\,G_{a}^{RX}\,\left(\frac{\lambda}{4\,\pi\,R}\right)^{2}\,P_{ TX} \tag{1}\]  
where \(\lambda\) is the carrier wavelength. In this relationship, the atmospheric attenuation is not considered because the scope of the basic analysis is the determination of the maximum received signal level. Table 1 reports an evaluation of the path losses in both the K and Ka frequency bands for \(R=384.4\times 10^{3}\) km. Note that going from the K to the Ka band leads to a 2 dB increase of the path loss  
Figure 1: Link geometry. To estimate the maximum signal level the receiver is assumed on the Moon (M) surface, perfectly at the ground station Zenith on Earth (E). Link parameters: \(R=384.4\times 10^{3}\) km, transmitted power \(P_{TX}=100\) W, transmitting antenna gain \(G_{a}^{TX}=65\) dBi (10 m dish antenna with a 60% efficiency), receiving antenna gain \(G_{a}^{RX}\approx 40\) dBi. (i.e., from about 231 to about 233 dB), but that such an increase is exactly counteracted by a greater gain of the fixed-diameter receiving antenna. Although reflectarrays [10], reconfigurable phased arrays [11] or flat-plate antenna solutions with efficiency as high as 85% [12] could be employed, a 50 cm dish antenna is considered here for simplicity. Such a size is compatible with the deployable parabolic reflectors recently developed by several authors [7, 8] for cubesat applications. The antenna gain \(G_{a}\) and the half power beam width \(\Theta_{h}\) can be estimated, for a given diameter \(D\) and efficiency \(\eta_{a}\), with the following relationships:  
\[G_{a}=\eta_{a}\,\left(\frac{\pi\,D}{\lambda}\right)^{2} \tag{2}\]  
\[\Theta_{h}=\frac{2\,\lambda}{\sqrt{\pi\,D}} \tag{3}\]  
A receiving antenna with \(D_{RX}=50\,\)cm has gain of about 39 dB in K-band and reaches 41 dB in the Ka-band, as reported in Table 1.  
As a first assumption, the antenna diameter at the ground station is \(D_{TX}=10\,\)m (transmitting antenna, uplink). Considering an antenna efficiency \(\eta_{a}^{TX}=60\,\%\), a 65 dBi gain and a 5.1 arcmin half-power beam width are obtained, the latter value being about 1/6 of the the angular diameter of the Moon seen from the Earth. A reasonable value for the transmitted power is \(P_{TX}=100\,\)W, that corresponds to \(+50\,\)dBm. As a consequence the effective isotropic radiated power (EIRP) at the ground station is:  
\[EIRP_{TX}=G_{a}^{TX}\,P_{TX}\approx 65.4+50=115.4\,\text{dBm} \tag{4}\]  
where this quantity is evaluated at 22.85 GHz, i.e., in the K-band. Other EIRP values are quoted in Table 2 for different antenna diameters and different frequencies. The same table also quotes the power levels received on the Moon. These levels are the absolute maximum and should be significantly above the receiver noise floor in order to guarantee for non zenithal links and atmospheric attenuation. As an example, a rain rate of 10 mm/h at 23 GHz results in a specific atmospheric loss of 2 dB/km. Integrating over a typical rain path of 5 km, an atmospheric attenuation of 10 dB is estimated, according to International Telecommunication Union, ITU-R P.838-2 [13].