## 2 Optical Receiver Mathematical Design Model  
However, the derivation in this paper involves the optical receiver elements and signal plus noise components which are shown in Figure 3.  
In a typical optical digital communication system, the transmitted signal \(s(t)\) can be depicted as:  
\[s(t)=\sum_{k=-\infty}^{\infty}a_{k}\;h_{p}(t-kT). \tag{2}\]  
where \(T\) is the signalling period. If \(\tau\) is the timespan of each bit within a symbol. For binary (OOK) signal format, \(\mathrm{T}=\tau\), so \(1/\mathrm{T}\) is the bit rate. The \(a_{k}\) is the energy received in the \(\mathrm{k}^{\text{th}}\) symbol, whereas for binary system \(a_{k}\in\)[0, 1], the \(h_{p}(t)\) represents a transmitted optical pulse. \(s(t)\) suffers temporal distortions while propagating through a medium channel (air, fiber optic, or water) depending on the characteristic impulse response of the channel \(h_{c}(t)\). Hence, the received optical signal \(r_{opt}(t)\), which describes the temporal impact of \(h_{c}(t)\) on \(s(t)\) signal, is given as (see Figure 2):  
\[r_{opt}(t) =s(t)*h_{c}(t)\;, \tag{3a}\] \[=\sum_{k=-\infty}^{\infty}a_{k}\;h_{p}(t-kT)*h_{c}(t),\] (3b) \[= a_{0}h_{p}(t)*h_{c}(t)+\sum_{k=-\infty}^{\infty}a_{k}\;h_{p}(t- kT)*h_{c}(t),\] (3c) \[k\
eq 0\]  
where \(\star\) denotes the convolution operation. The \(r_{opt}(t)\) is the received optical signal which provides the optical input to the photodetector. The first term of Equation (3c) is depicted in Figure 2b,d. We can notice that \(h_{c}(t)\) causes a temporal broadening in \(s(t)\) (which means narrowing the \(r_{opt}(t)\) compared with the bandwidth of \(s(t)\)). In this work, we are considering a binary direct detection optical receiver system (Rx) depicted in Figure 3. Without losing generality, we will assume that the Rx uses a PIN photodetector when the internal gain (\(g\)) equals one. The photodetector converts the input photons of \(r_{opt}(t)\) into photoelectrons that form an output electronic signal \(r_{sig}(t)\)[26]. \[r_{sig}(t)=\sum_{j=1}^{N(t)}g_{j}\;h_{d}\big{(}\;t-t_{j}\big{)}, \tag{4}\]  
\(\{t_{j}\}\) denotes the photoelectron emission times. \(g_{j}\) is photodetector avalanche gain. In this work, we are assuming the use of a PIN detector where \(g_{j}=1\;\forall\;j\). The technological advancement in photodetector (PIN and APD) manufacturing permits us to assume that \(h_{d}(t)\) is a delta function \(\delta(t)\) (i.e., vast bandwidth). In Figure 3, the amplifier's role elevates the signal strength. However, from the conceptual perspective, the equalizer is expected to compensate for any temporal changes caused by the Rx front end. Alternatively, the \"optimum\" filter should be able to do that explicitly. From Equation (4), we can conclude the filter output as follows:  
\[r_{f}(t) =r_{sig}(t)*\;h_{f}(t)+r_{th}(t), \tag{5a}\] \[=\sum_{j=1}^{N(t)}g_{j}\;h_{d}\big{(}\;t-t_{j}\big{)}*h_{f}(t)+r_{th}(t), \tag{5b}\]  
where we can assume safely that \(h_{d}(t)=\delta(t)\). The first term represents the signal component \(r_{sig}(t)\), and the second term is the AGTN \(r_{th}(t)\). In Equations (4) and (5), the \(\{t_{j}\}\) is a set of photoelectrons' arrival times that follows Poisson statistics governed by \(N(t)\). \(N(t)\) represents the stochastic counting process of the generated photoelectrons during \(\{t_{0}=0\), \(t\}\). This counting process is inhomogeneous [27; 28] with the time-varying rate intensity of \(\{a_{k}\}\) in Equation (2). This makes \(r_{sig}(t)\) a doubly stochastic marked and filtered Poisson process (DSPP) [27]. \(N(t)\) is directly proportional to the received optical intensity \(\lambda(t)\)[16; 27] which can be expressed as follows:  
\[s(t)=\sum_{k=-\infty}^{\infty}a_{k}\;h_{p}(t-kT). \tag{2}\]0where \(\varphi_{dc}\) is the photoelectron count due to the detector's dark current. It is a stationary background noise with a flat power spectrum \(N_{0b}\). Hence, the optical receiver of bandwidth \(BW_{Rx}\) collects a noise power \(=N_{0b}.BW_{Rx}\)[27].