## 2 Principle of Linearity Improvement  
### Analysis of the Principle of \"Reverse Uplift\"  
The Gilbert-cell-based structure is one of the most popular implementations of active mixers, consisting of a V/I conversion stage (converting voltage changes to current changes) and a switching core. The half Gilbert-cell is analyzed for simplification, as shown in Figure 1; we treat the transconductance stage as a memoryless non-linear system. Considering the harmonic distortion caused by non-linearity, the output signal can be approximately expressed as  
\[y(t)=\alpha_{1}x(t)+\alpha_{2}x^{2}(t)+\alpha_{3}x^{3}(t)+\ldots+\alpha_{n}x^{ n}(t) \tag{1}\]  
where \(\alpha_{1}\), \(\alpha_{2}\), \(\alpha_{3}\), \(\ldots\)\(\alpha_{n}\) represent the coefficients of each order component, respectively. In general, when a cosine signal with an amplitude of \(M\) and an initial phase of zero is fed into such a system, a fundamental component \(\left(\alpha_{1}M+\frac{3}{2}\alpha_{3}M^{3}\right)\cos(2\pi f_{in}t)\) and a second harmonic component \(\frac{\alpha_{2}\cdot M^{2}}{2}\cos(4\pi f_{in}t)\) are generated due to nonlinear effects, where \(f_{in}\) represents the signal frequency. The magnitude of the second harmonic component is proportional to \(M^{2}\), the energy of higher-order harmonics is weak enough to have a significant effect. If the system is stable, the gain of the fundamental component will be compressed as \(M\) increases, so the condition of \(\alpha_{1}\alpha_{3}<0\) needs to be satisfied. However, the actual circuit system is not conducting unidirectionally; for example, the junction capacitors of the transistors will cause reverse feedthrough, which will bring a part of the signal back to the input with some attenuation and phase shift. Then, the extra components will be generated through the effect of frequency mixing. Parameters \(k_{1}\) and \(k_{2}\) are defined as the feedback factors of the fundamental component and the second harmonic component from the output to the input through some passive devices, respectively. The possible reversal caused by the phase change is also considered, so its range is  
\[-1<k_{1},\ k_{2}<1 \tag{2}\]  
Therefore, the total fundamental and second harmonic components at the input can be expressed as \(Y_{1}(t)\) and \(Y_{2}(t)\). \[Y_{1}(t)=k_{1}\bigg{(}\alpha_{1}M+\frac{3}{4}\alpha_{3}M^{3}\bigg{)}\cos(2\pi f _{in}t)+M\cos(2\pi f_{in}t) \tag{3}\]  
\[Y_{2}(t)=k_{2}\frac{\alpha_{2}M^{2}}{2}\cos(4\pi f_{in}t) \tag{4}\]  
\[Y_{mix}(t)=K(Y_{1}(t)\times Y_{2}(t)) \tag{5}\]  
\(Y_{1}(t)\) and \(Y_{2}(t)\) will mix at the input, resulting in extra components as shown in Equation (5), where K represents the coefficient determined by specific circuits. The fundamental component in \(Y_{mix}(t)\) can be expressed as  
\[K\bigg{[}\frac{k_{2}\alpha_{2}M^{2}}{2}(1+k_{1}\alpha_{1})M^{3}+\frac{3}{8}k_ {1}k_{2}\alpha_{2}\alpha_{3}M^{5}]\cos(2\pi f_{in}t)\ \bigg{]} \tag{6}\]  
Additionally, the gain of the fundamental component can be updated as  
\[ \begin{split} Gain=\alpha_{1}+[\frac{3}{4}\alpha_{3}+\frac{2}{ 2}(1+k_{1}\alpha_{1})]M^{2}\\ +\frac{3}{8}k_{1}k_{2}\alpha_{2}\alpha_{3}M^{4},\ (-1<k_{1},\ k_{2}< 1,\ M>0)\end{split}  \tag{7}\]  
Compared with \(\alpha_{1}+\frac{3}{4}\alpha_{3}M^{2}\), the extra part in Equation (7) may increase the compression point to a certain extent. Where 1 dB compression should have occurred, the new gain component compensates for these losses, thereby increasing the 1 dB compression point. This possibility is analyzed below. Let  
\[T_{1}=\alpha_{1} \tag{8}\]  
\[T_{2}=\frac{3}{4}\alpha_{3}+\frac{k_{2}\alpha_{2}}{2}(1+k_{1}\alpha_{1}),\ (-1<k_{1}<1) \tag{9}\]  
\[T_{3}=\frac{3}{8}k_{1}k_{2}\alpha_{2}\alpha_{3},\ (-1<k_{1},\ k_{2}<1) \tag{10}\]  
Figure 1: Block of nonlinear system. As shown in Figure 2a,b, in order to compress the gain in the opposite direction as the input signal increases and make the system tend to be stable, it is necessary to satisfy \(T_{1}T_{3}<0\).