## 3 Building Blocks Designs  
### Low Noise Amplifier (LNA)  
Figure 2 shows the schematic of the two-stage LNA for the proposed receiver front-end. The first stage of the LNA is a common-source amplifier stage with inductive peaking load. In contrast to the conventional inductor degenerated LNA [13; 14], the common-source amplifier with parasitic coupling capacitance (C\({}_{gd}\)) is employed to achieve the input matching without sacrificing noise performance due to a degenerated inductor at the source.  
The schema of the LNA's first stage, calculating the input matching condition, is provided in Figure 3. C\({}_{D}\) denotes the combination of interconnected line parasitic, ac-coupling and second-stage gate-source capacitances. The real part of input impedance (Re(\(Z_{in}\))) seen at the gate of trans-conductance (g\({}_{m}\)) stage is derived here to show the input matching condition. The effect of C\({}_{gs}\) is neglected since they merely functions as a shunt components in parallel with the derived Re(\(Z_{in}\)) expressed as  
\[Re(Z_{in})=\frac{(C_{D}+C_{gd})(g_{m1}L_{D}\omega^{2}+g_{m1}R_{D}^{2}+R_{D})-(R _{D}C_{D}+g_{m1}L_{D})}{A}\omega\, \tag{1}\]  
where \(A=(R_{D}C_{D}+g_{m1}L_{D})^{2}\omega^{2}+(g_{m1}R_{D}-L_{D}C_{D}\omega^{2}+1)^ {2}\). As shown in Equation (1), with the help of the reactive load of the LNA (L\({}_{D}\),C\({}_{D}\)) and C\({}_{gd}\), \(Z_{in}\) exhibits the real impedance suitable for input matching. To achieve 50 \(\Omega\) input matching and avoid potential instability, Re(\(Z_{in}\)) should have a positive value. \(A\) always has a positive sign, so the numerator of Re(\(Z_{in}\)) determines the stability. The numerator of Re(\(Z_{in}\)) has a positive value, regardless of frequencies, when \(D=(C_{D}+C_{gd})(g_{m1}R_{D}^{2}+R_{D})-(R_{D}C_{D}+g_{m1}L_{D})\) is larger than 0. Through some calculations, it can be proven that the proposed LNA exhibits positive real impedance if \(R_{D}>\sqrt{L_{D}/C_{D}}\).  
Figure 4 shows the stable and unstable region in terms of the value of R\({}_{D}\). Table 1 provides the values of real design parameters. To achieve the stable condition, R\({}_{D}\) should be higher than around 33 \(\Omega\). Based on these derivations of Re(\(Z_{in}\)), R\({}_{D}\) was selected to 44 \(\Omega\) in our design. The input inductor, L\({}_{M}\), cancels the reactive part of \(Z_{in}\) and thus achieves broadband input matching over 10 GHz of RF bandwidth. The second-stage provides additional gain of LNA with a 2:1 turn ratio passive Balun as a stagger-tuned load [15]. Note that the receiver front-end needs to cover a 30-40 GHz input bandwidth and the first-stage LNA load works as the low-Q band-pass filter centered at 25 GHz. Then, the stagger-tuned load in the second stage helps to compensate the gain at high-frequency regimes  
Figure 2: Schema of the LNA.  
frequency. The combinational effect due to two-stage stagger-tuned implementation gives the center frequency located at 38 GHz.  
The simulation result shows that the entire LNA exhibits a 28-45 GHz wide bandwidth, as depicted in Figure 5.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Description** \\ \hline g\({}_{m1}\) & 76.2 mS & transconductance of M\({}_{1}\) \\ L\({}_{D}\) & 230 pH & load inductor \\ R\({}_{D}\) & 44 \(\Omega\) & load resistor \\ C\({}_{gd}\) & 20 fF & parasitic capacitance between gate and drain \\ C\({}_{D}\) & 200 fF & parasitic gate capacitance of M\({}_{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: LNA design parameters.  
Figure 4: The condition of R\({}_{D}\) for LNA input matching and stability.  
Figure 3: Schema of the LNA’s first stage.