# A Temperature-Robust Envelope Detector Receiving OOK-Modulated Signals for Low-Power Applications
## 3 ED and PTAT Block Description
### Temperature Compensation through the PTAT Block  
Actually, \(P_{SEN}\) as in (8) depends on temperature through \(f_{N}\) as well, which in turn is inversely proportional to the baseband output resistance \(R_{out}=Nr_{DS}=N^{2}R_{in}\). Therefore, the integrated noise at the output of the ED becomes  
\[\int V_{N,ED}^{2}(f)df\propto\frac{4k_{B}TN^{2}R_{in}}{N^{2}R_{in}}=4k_{B}T, \tag{10}\]  
which is independent of \(R_{in}\). As for \(NF\) of the baseband amplifier following the ED, we shall assume the amplifier to be standard, for instance a common-source with an active load, with negligible residual flicker noise and transconductance \(g_{m}\), for the sake of simplicity, and to operate in the subthreshold region. Let us also assume the baseband amplifier to have a bandwidth close to bitrate \(f_{S}\), which is in turn close to the ED noise bandwidth \(f_{N}\). In this case, (7) results in  
\[NF_{SE}=1+\frac{2\times\frac{4k_{B}T}{g_{m}}\left(g_{m}R_{out}\right)^{2}f_{S} }{\left(g_{m}R_{out}\right)^{2}\times 4k_{B}TN^{2}R_{in}f_{N}}=1+\frac{2}{g_{m}N^{2}R _{in}}, \tag{11}\]  
If \(N\) or \(g_{m}\) are big enough, e.g., a \(3.3-\mu\)S\(g_{m}\) is enough for a \(2.2\)-k\(\Omega\)\(R_{in}\) and \(N=60\) (see Section 4.2.1), \(NF\) is close enough to unity at room temperature and its variations with temperature are negligible. This results in \(A_{v}\), as in (5), being the only element with a dependence on \(R_{in}\) in (8). In turn, \(R_{in}\) is a very rapidly varying function of temperature \(T\) through \(r_{DS}\) as in (9), which ultimately results in  
\[P_{SEN}\propto\sqrt{\frac{T^{3}}{\left(A_{v}\left(R_{in}(T)\right)\right)^{4}}}. \tag{12}\]  
The purpose of the proposed compensation mechanism is to make \(R_{in}\), and thus \(A_{v}\), constant with temperature leaving \(P_{SEN}\propto\sqrt{T^{3}}\). No further attempt has been made to cancel this residual temperature dependence. An additional effect is making \(f_{N}\) constant as well. The following analysis is carried out assuming a small input signal, i.e., close to sensitivity, resulting in source voltages roughly equal to \(V_{C}\) for all diodes. As illustrated in Figure 2, the PTAT block generates a \(V_{GS}\) for the diodes so as to make their \(r_{DS}\), and thus \(R_{in}\), roughly constant with \(T\). This also results in a roughly constant \(A_{v}\) so that no changes to the matching network are required in case of temperature changes. The compensation mechanism is again based on the standard subthreshold current model for the diodes (1), (2). The PTAT provides the diodes with the same \(V_{GS}\) as \(M4\), i.e., \(V_{GS}=V_{GS,M4}=V_{B}-V_{C}\). Assuming the diodes to have a small \(V_{DS}\) and assuming the difference in \(V_{DS}\) between the diodes and \(M4\) not to significantly affect \(V_{TH}\),  
\[r_{DS}=\frac{V_{T}}{I_{S}\left(\frac{N}{T}\right)_{MD}}e^{\frac{-V_{GS}}{V_{T }}}=\frac{V_{T}}{I_{S}\left(\frac{W}{T}\right)_{MD}}e^{\frac{-V_{GS,M4}}{V_{T }}}. \tag{13}\] \begin{table}
\begin{tabular}{c c} \hline ED input resistance & \(R_{in}=r_{DS}/N\) \\ ED input capacitance & \(C_{in}=NC_{gs}+C_{pad}\) \\ Matching network gain & \(A_{v}=\sqrt{\frac{R_{in}}{R_{s}}}/\sqrt{\left(1+\frac{\omega R_{n}C_{in}}{C_{ ind}}\right)}\) \\ Single-ended ED output voltage & \(V_{ED}=\frac{NV_{GS}^{2}}{4\mu^{2}}\) \\ ED delay & \(\tau\sim CR_{in}\frac{N^{2}(N+1)}{2}\) \\ Sensitivity & \(P_{SEN}=\sqrt{\frac{SNR_{in}Nr(4v_{T})^{2}k_{B}T_{in}f_{S}}{A_{v}^{2}R_{s}}}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Useful formulas for sensitivity optimization—Section 3.1.