## 3 RX Phased Array Design
### Array G/T Analysis  
The equivalent RX chain of the 2048-element array is illustrated in Figure 13. The 8-channel silicon RX beamformer with a compact dimension of 6 mm \(\times\) 5.5 mm was available in a wafer level chip scale package (WLCSP), which employed 0.25 mm diameter balls as pins so that it can provide the smallest chip size and easy surface mounting to the PCB. The key parameters of the 8-channel beamformer are detailed in Table 2.  
The electronic gain (\(G_{\text{R}}\)) of the chain was is 27 dB at 11.7 GHz, as calculated using the equation:  
\[G_{\text{R}}=G_{\text{BF}}-L_{\text{WI}}+G_{\text{LNA}}-L_{\text{W2}} \tag{15}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Typical Value** & **Unit** \\ \hline Frequency & 10.5–13 & GHz \\ Gain & 19 & dB \\ Noise Figure & 2.3 & dB \\ IP\({}_{\text{1 dB}}\) & \(-25\) & dBm \\ Phase Shifter & 6 & bit \\ Phase Step & 5.625 & \(\circ\) \\ RMS Phase Error & 2 & \(\circ\) \\ Attenuator Range & 0–31.5 & dB \\ Attenuation Step & 0.5 & dB \\ RMS Attenuation Error & 0.3 & dB \\ Package & WLCSP & \\ Size & \(6\times 5.5\times 0.5\) & mm \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Key Parameters of the 8-channel Beamformer. Figure 12: Simulated co– and cross–polarization level of the antenna for the V– and H–polarization. (**a**) Azimuth plane. (**b**) Elevation plane. Figure 13: Equivalent RX chain of the 2048-element array. where \(G_{\rm BF}\) is the electronic gain of the beamformer chip, \(L_{\rm WI}\) is the loss of the 32:1 Wilkinson combiner microstrip on the multilayer PCB, \(G_{\rm LNA}\) is the electronic gain of the linear amplifier, and \(L_{\rm W2}\) includes both the loss at the external 16:1 Wilkinson power combiner module and that in the cables connecting the sixteen 128-element subarrays. The calculated NF is 1.76 (NF\({}_{\rm R}=2.45\) dB) of the RX chain at 11.7 GHz, according to the equation:  
\[F_{\rm R}=F_{\rm BF}+\frac{L_{\rm WI}-1}{G_{\rm BF}}+\frac{F_{\rm LNA}-1}{G_{ \rm BF}/L_{\rm WI}}+\frac{L_{\rm W2}-1}{G_{\rm BF}G_{\rm LNA}/L_{\rm WI}} \tag{16}\]  
where \(F_{\rm BF}\) is the NF of the beamformer chip, and \(F_{\rm LNA}\) is the NF of the linear amplifier. Since \(D_{\rm element}\) is 4.4 dBi at 11.7 GHz from Equation (14), the array gain (\(G_{\rm array}\)) is 37.1 dBi and can be calculated as  
\[G_{\rm array}=10\log_{10}(2048)+(D_{\rm element}-L_{\rm ant}) \tag{17}\]  
where \(L_{\rm ant}\) is the total loss between the antenna and the beamformer, and simulated to be 0.4 dB at 11.7 GHz (See Figure 12). The system noise temperature of the proposed RX phased array was calculated using the equation:  
\[T_{\rm array}=T_{\rm A}+T_{\rm R} \tag{18}\]  
where \(T_{\rm A}\) is the antenna noise temperature, and \(T_{\rm R}\) is the equivalent noise temperature of the RX chain. \(T_{\rm A}\) and \(T_{\rm R}\) were defined as:  
\[T_{\rm A} =\eta_{\rm ant}T_{\rm ant}+T_{0}(1-\eta_{\rm ant}) \tag{19}\] \[T_{\rm R} =T_{0}(F_{\rm R}-1) \tag{20}\]  
where \(T_{0}\) = 290 K, \(\eta_{\rm ant}=10^{-\frac{L_{\rm ant}}{10}}\) is the efficiency of the antenna, and \(T_{\rm ant}\) is the equivalent brightness temperature of the background seen by the main beam of the array. \(T_{\rm ant}\) = 20 K while the array was scanning a cold region of empty sky during satellite reception, and \(T_{\rm ant}\) = 290 K while the array was measured at room temperature in an anechoic chamber. By substituting Equations (19) and (20) into Equation (18), \(T_{\rm array}\) was calculated as:  
\[T_{\rm array} =\eta_{\rm ant}T_{\rm ant}+T_{0}(1-\eta_{\rm ant})+T_{0}(F_{\rm R}-1) \tag{21}\] \[=(10^{\frac{-0.4}{10}})T_{\rm ant}+290(1-(10^{\frac{-0.4}{10}}))+2 90(1.76-1)\]  
It was thus determined that \(T_{\rm array}\) = 263.9 K when \(T_{\rm ant}\) = 20K, and \(T_{\rm array}\) = 510.1 K when \(T_{\rm ant}\) = 290 K.  
The G/T value of the proposed array at 11.7 GHz was calculated using the equation:  
Figure 12:0  
which revealed that G/T = 12.9 dB/K when \(T_{\rm ant}\) = 20K, and G/T = 10.0 dB/K when \(T_{\rm ant}\) = 290 K.