# K/Ka-Band Very High Data-Rate Receivers: A Viable Solution for Future Moon Exploration Missions
## 4 Receiver Architecture
### Phase Noise Impairments
The numerator of such an expression can be seen as an effective input noise power \(N_{i}^{eff}\) accounting for the phase-noise:  
\[N_{i}^{eff}\simeq N_{i}+S_{i}\,\Phi_{rms}^{2} \tag{13}\]  
Now, expressing \(N_{i}^{eff}\) in terms of an effective noise figure \(F_{RX}^{eff}\) and \(N_{i}\) in terms of the conventional noise figure \(F_{RX}\) one gets:  
\[k_{B}\,F_{RX}^{eff}\,T_{0}\,B_{IF}=k_{B}\,F_{RX}\,T_{0}\,B_{IF}+S_{i}\,\Phi_{rms }^{2} \tag{14}\]  
Finally, assuming \(S_{i}=P_{RX}\) (true for constant envelope signals like QPSK), the above relationship can be used to derive the effective noise figure \(F_{RX}^{eff}\) in terms of the other system parameters:  
\[F_{RX}^{eff}=F_{RX}+\frac{P_{RX}}{k_{B}\,T_{0}\,B_{IF}}\,\Phi_{rms}^{2} \tag{15}\]  
This equation states a very interesting fact: In order to account for the local oscillator phase-noise, the receiver noise figure is increased by a quantity proportional to the squared phase error multiplied by the input power referred to the noise power in the receiver bandwidth. As a result, there are two asymptotic cases: Low quality signals and high quality signals. In the first case the effective signal to noise ratio is limited by the receiver noise figure whereas, in the second case, it is limited by the local oscillator phase-noise. From (15) follows that the effective SNR in (9) can be evaluated, considering the antenna noise temperature \(T_{a}\), as for the conventional noise figure:  
\[\text{SNR}_{eff}\simeq\frac{P_{RX}}{k_{B}\,\left[T_{a}+\left(F_{RX}^{eff}-1 \right)\,\,T_{0}\right]\,\,B_{IF}} \tag{16}\]To apply the above theory we consider, as first local oscillator, the ADF5356 microwave frequency synthesizer (Note that only the phase noise of the first local oscillator is considered. This because the second conversion happens inside the SDR, whose local oscillator has a much better phase noise than the ADF5356). Its phase-noise is shown in Figure 5 for a 13.6 GHz carrier (output B of the device) with a loop filter having a 2 kHz bandwidth. The reference frequency to the phase-frequency detector (PFD) is 61.44 MHz. The black line with diamonds represents the measurements reported in the device data-sheet ([21], p. 13). According to ([20], p. 41), the closed-loop synthesizer phase-noise can be modeled as:  
\[S_{\phi}(f)=\left\{ \begin{array}{ll}\beta&\text{if $f<f_{L}$}\\ \frac{\beta}{(ff_{L})^{3}}&\text{if $f_{L}\leq f<f_{C}$}\\ \frac{\beta}{(f_{C}/f_{L})^{3}\,(f/f_{C})^{2}}&\text{if $f\geq f_{C}$}\end{array} \right. \tag{17}\]  
with the following model parameters: \(\beta=1.25\times 10^{-7}\,\)s, \(f_{L}=6\,\)kHz and \(f_{C}=100\,\)kHz. The fitting between measurements and model is also illustrated in Figure 5 (purple line). The phase-noise model reported in (17) can now be exploited to compute the integral in (10) and, thus, the phase error. Assuming \(f_{min}\geq f_{L}\) and \(f_{max}\geq f_{c}\) one gets:  
\[\Phi_{rms}^{2}=\frac{n^{2}\,\beta\,f_{L}}{2}\,\left[\left(\frac{f_{L}}{f_{min}} \right)^{2}+\left(\frac{f_{L}}{f_{C}}\right)^{2}-2\,\frac{f_{L}^{2}}{f_{C}\,f _{max}}\right] \tag{18}\]  
On the other hand, for \(f_{min}<f_{L}\) and \(f_{max}\geq f_{c}\) one obtains:  
\[\Phi_{rms}^{2}=n^{2}\,\beta\,\left(f_{L}-f_{min}\right)+\frac{n^{2}\,\beta\,f _{L}}{2}\,\left[1+\left(\frac{f_{L}}{f_{C}}\right)^{2}-2\,\frac{f_{L}^{2}}{f_{ C}\,f_{max}}\right] \tag{19}\]  
To deal with a numerical case, a QPSK signal is considered with 16200-bits DVB-S2 short frames, i.e., with \(M_{s}=8100\) symbols per frame (the QPSK implies 2 bits per symbol). The minimum frequency is assumed equal to the frame frequency: \(f_{min}=f_{s}\,/\,M_{s}\), where \(f_{s}\) is the symbol rate. For the computations it is assumed \(n=2\) since, in the developed receiver, a sub-harmonic mixer (with an internal frequency doubler) is adopted.