## 3 Circuit Design
### Transimpedance Amplifier
With the introduction of emitter degeneration, shown in Figure 2, the input-referred noise voltage spectrum of CE stage Q\({}_{1}\) with emitter degeneration \(\overline{v_{n,CE}^{2}}\) is expressed as  
\[\overline{v_{n,CE}^{2}}=4 kT\bigg{(}r_{b}+\frac{1}{2g_{m}}+R_{E}\bigg{)}, \tag{4}\]  
where \(r_{b}\) is the base resistor of Q\({}_{1}\). Then the input-referred noise current spectrum \(\overline{i_{n,in}^{2}}\) of the input TIA is given by  
Figure 3: The simplified small-signal equivalent circuit of the input TIA. \[ \begin{split}\vec{i_{n,in}^{2}}&=\Bigg{(}\frac{i_{in}}{v_{in }}\Bigg{)}^{2}\overline{v_{n,CE}^{2}}+\frac{4 kT}{R_{F}}\\ &=4 kT\Bigg{(}r_{b}+\frac{1}{2g_{m}}+R_{E}\Bigg{)}\frac{1+s^{2}(R_ {F}C_{T})^{2}}{R_{F}^{2}}+\frac{4 kT}{R_{F}}\end{split} . \tag{5}\]  
There are two terms in Equation (5); the first term is the shaped noise (\(f^{2}\) noise) of CE stage Q\({}_{1}\) with emitter degeneration, while the second term is the noise introduced by \(R_{F}\). There are inevitable trade-offs between the five parameters in Equation (5) for noise optimization, namely, \(r_{b}\), \(g_{m}\), \(R_{E}\), \(R_{F}\), and \(C_{T}\). However, conventional noise optimization methods [9] have several shortcomings: the noise introduced by \(R_{E}\) is not considered, nor are the relationships among \(I_{C}\) (the collector current of Q\({}_{1}\)), \(r_{b}\), \(g_{m}\), and \(C_{T}\). With multiple parameters to trade off, the conventional noise optimization method is complicated. Considering that the performance of the bipolar transistor varies a lot with the DC operating point in SiGe BiCMOS technology, Q\({}_{1}\) with given size has to be biased at the optimum operating point for high F\({}_{t}\) (characteristic frequency) and \(\beta\) (common emitter current gain). Therefore, the size of Q\({}_{1}\) can be determined by a given \(I_{C}\). Then, the sizes of Q\({}_{1}\), \(C_{b}\), and \(g_{m}\) are proportional to \(I_{C}\)[25], and \(R_{b}\) is inversely proportional to the sizes of Q\({}_{1}\) and \(I_{C}\). Hence, \(I_{C}\) determines the values of \(r_{b}\), \(g_{m}\), and \(C_{T}\) with given \(C_{EX}\). The improved noise optimization method modifies the noise model (Equation (5)) and gets the relationships among \(I_{C}\), \(r_{b}\), \(g_{m}\), and \(C_{T}\); then the optimum \(I_{C}\) can be obtained from given \(R_{F}\), \(R_{E}\), and \(C_{EX}\). As a rule of thumb, the value of \(C_{EX}\) is 100 fF. As given by Equation (5), the noise is also suppressed by large \(R_{F}\). However, large \(R_{F}\) would degrade the bandwidth of the TIA from Equation (3) and S11. As a result of a trade-off, \(R_{F}\) was selected to be 230 \(\Omega\). Although the noise and crosstalk coupled from the ground wire are suppressed by emitter degeneration, the noise introduced by emitter degeneration resistor \(R_{E}\) increases the input-referred noise current of the input TIA stage. Therefore, the value of \(R_{E}\) has to be small enough that it will not deteriorate the input-referred noise. The results of the noise optimization method by Matlab with \(R_{E}\) varying from 3 \(\Omega\) to 11 \(\Omega\) are shown in Figure 4. Considering the requirement of \(\xi\), \(R_{E}\) was selected to be 7 \(\Omega\) from Figure 5. Then, the optimum collector current is 4 mA, and the corresponding emitter length of Q\({}_{1}\) is 6 \(\upmu\)m. The value of \(A_{V}\) was derived from the given \(R_{F}\) and \(C_{T}\) from Equation (3) for the bandwidth requirement. As shown in Figure 6, the simulated \(BW\) of Figure 2 is 29 GHz. Meanwhile, the simulated \(i_{n,rms}\) of Figure 2 is 2.6 \(\upmu\)A, which confirms the accuracy of the noise model in Equation (5) and the improved noise optimization method. With the stability of the TIA enhanced by emitter degeneration, the simulated damping factor \(\xi\) (Equation (2)) is 0.74, which is stable enough for PVT variation. Figure 4: The noise optimization results of the input TIA.