## 4 Uncoded BER of APD OOK Receivers  
We consider the bit decision of an OOK data stream with an APD-RFE, including its noise processes. Figure 6 shows a typical pseudo-random bit sequence (PRBS) with 100 Mbps detected with an APD-RFE. Regard the stronger shot-noise caused by the OOK signal during the reception of the binary on bits. Further information on direct detection receivers can be found in [45; 46; 47; 48; 49; 50; 51; 52]. Noise is assumed to be Gaussian-distributed with a different noise sigma for 1 s and 0 s (Figure 7). A detection threshold current level, \(I_{Thr}\), between the curves around the 0 s and 1 s, decides for a binary 0 and 1. Decisions to the wrong side from \(I_{Thr}\), will lead to bit errors. Their probability is linear to the ratio of the stippled areas \(A+B\) related to the total curve-area. The areas on the wrong side of \(I_{Thr}\) are minimized with an optimum decision threshold level. For an analytically applicable approximation, \(I_{Thr}\) is defined by the intersection of both curves, which leads to  
\[I_{Thr}=\frac{\sigma_{0}\cdot\langle s_{1}\rangle+\sigma_{1}\cdot\langle s_{0} \rangle}{\sigma_{0}+\sigma_{1}} \tag{11}\]  
Figure 6: Typical binary OOK data bit stream at 100 Mbps, as monitored by an oscilloscope: upper plot transmitted data, lower plot is the received signal from an APD-RFE. Figure 7: Generic situation of the Gaussian-shaped noise distributions at binary 1 and 0. The quality factor \(Q\) can then be derived as  
\[Q=\frac{\langle s_{1}\rangle-\langle s_{0}\rangle}{\sigma_{1}+\sigma_{0}} \tag{12}\]  
Assuming \(<\)\(s_{0}\)\(>\) = 0 (no offset), by integrating over the tails of the two Gaussian distributions, we get the bit error probability, \(p_{BE}\). \[p_{BE}=\frac{1}{2}\cdot erfc\bigg{(}\frac{1}{\sqrt{2}}\cdot\frac{\langle s_{1 }\rangle}{\sigma_{0}+\sigma_{1}}\bigg{)}=\frac{1}{2}\cdot erfc\bigg{(}\frac{1 }{\sqrt{2}}\cdot Q\bigg{)} \tag{13}\]  
where \(\langle s_{1}\rangle=M\ R\ \hat{P}_{Rx}\) is given by the optical power during an on-symbol, which is, again, twice the average power \(\langle P_{Rx}\rangle\). The noise current during \"0\" and \"1\" consists of the same thermal noise current:  
\[\sigma_{t}=i_{t}\cdot\sqrt{B} \tag{14}\]  
and the shot noise from background light and dark currents is  
\[\sigma_{s,\rho}=\sqrt{B\cdot 2q[M^{2}F_{A}(I_{dm}+R\cdot P_{BGL})+I_{du}]} \tag{15}\]  
where during a binary \"1\", we see additional signal shot noise:  
\[\sigma_{s,1}=\sqrt{B\cdot 2q\{M^{2}F_{A}[R\ \cdot(2\langle P_{Rx}\rangle)+I_{ dm}+R\cdot P_{BGL}]+I_{du}\}} \tag{16}\]  
\(Q\) then becomes  
\[Q_{st,APD}=\frac{M\ R\ \cdot(2\langle P_{Rx}\rangle)}{\sqrt{\sigma_{s,0} ^{2}+\sigma_{t}^{2}}+\sqrt{\sigma_{s,1}^{2}+\sigma_{t}^{2}}} \tag{17}\]  
When neglecting the small unmultiplied dark current \(I_{du}\), we find  
\[ \begin{array}{ll}Q_{st,APD}&=\frac{M\ R\cdot(2\langle P_{Rx}\rangle)}{\sqrt {B}\cdot\sqrt{2 qM^{2}F_{A}(RF_{BGL}+I_{dm})+i_{t}^{2}+\cdots}}\\ &\frac{\sqrt{B}\cdot\sqrt{2 qM^{2}F_{A}(R(2\langle P_{Rx}\rangle+P_{BGL})+I_{ dm})+i_{t}^{2}}}\end{array}  \tag{18}\]  
_Sensitivity Estimation without Background Light_  
We find a general simplified (no \(P_{BGL}\), no \(I_{d}\)) receiver sensitivity formula in terms of the Q-factor, with signal power during the reception of a binary 1 being \(P_{1}=2\langle P_{Rx}\rangle\), and the signal level during a binary 0 being is \(P_{0}=0\). We regard the noise during the binary 0 and 1 as (\(\sigma_{0}=\sigma_{thermal}\) and \(\sigma_{1}=\sqrt{\sigma_{shot-1}^{2}+\sigma_{thermal}}\)) and photon energy \(E_{Ph}=hc/\lambda\), and assume equal distribution of 1s and 0s in the bit-stream:  
\[ \begin{array}{ll}Q_{APD}&=\frac{M\ R\cdot(2\langle P_{Rx}\rangle)}{\sigma_ {t}+\sqrt{\sigma_{shot-1}^{2}+\sigma_{t}^{2}}}=\\ &=\frac{\sqrt{B\cdot M}\ R\cdot(N\cdot)\cdot E_{Ph}}{i_{u}+\sqrt{4 qM^{2}F_{A} \cdot R\cdot(N)\cdot E_{Ph}+i_{u}^{2}}}\end{array}  \tag{19}\]  
From \(Q\), we can derive \(p_{BE}\) via (13).