## 3 Mathematical Model  
The sensitivity in the presented DD SCM/SSB scheme is the result of the following trade-off. OMI needs to be high to obtain a low CSPR, which reduces the relative power wasted in the optical carrier, thereby achieving a better sensitivity. However, higher values of OMI translate into higher IMD due to the nonlinear response of the optical modulator, which can translate into a penalty in the sensitivity if the IMD is higher than the noise floor originating from the ASE and obtained in the photocurrent. Therefore, a mathematical model that considers only the main source of optical noise, namely the ASE, obtains the best sensitivity that is potentially achievable as a function of OMI. Such a model was presented in [10] for the case of optical IQ modulators, and is employed in this work. As IMD at a given subchannel is the result of summing the interfering intermodulation products (IMP), a perfect model would require considering all the individual electrical phases in the system. However, for simplicity, IMD is modelled as random noise with Gaussian distribution, and, accordingly, discrepancies with the model are expected for higher OMIs. In the conditions stated in Section 2, the best achievable optical sensitivity is [10]:  
\[P_{IN}=\frac{2Q_{F}^{2}Fh\
u B}{\Big{(}J_{0}^{N-1}(2m)J_{1}(2m)\Big{)}^{2} \big{(}1-Q_{F}^{2}D_{IMD}\big{)}} \tag{1}\]  
where \(F\) represents the noise figure of the EDFA, \(h\) is [PERSON]'s constant, \(\
u\) is the optical frequency, \(B\) is the electrical baseband bandwidth of the subchannels, \(Q_{F}\) represents the quality factor of the digital communication, \(N\) states the number of electrical subchannels, \(J_{n}(x)\) stands for the \(n^{\text{th}}\) order Bessel function of the first kind, \(m\) represents the rms OMI per subchannel (\(m=\pi V_{AC}/2V_{\pi}\) where \(V_{\pi}\) is the half-wave voltage of the MZMs and \(V_{AC}\) is the peak voltage of a tone in theory but represents  
Figure 1: **(a)** Direct Detection (DD) electro-optical scheme compatible with subcarrier multiplexing (SCM)/single side band (SSB) and orthogonal subcarrier multiplexing (OSCM)/SSB transmission based on optical IQ modulator and pre-amplified optical receiver; **(b)** example of transmitted optical spectrum for the case of SCM/SSB consisting of three 2.7 Gbaud quadrature phase shift keying (QPSK) subchannels; **(c)** example of transmitted optical spectrum for the case of OSCM/SSB consisting of three orthogonally overlapping 2.7 Gbaud QPSK subchannels. the rms voltage per subchannel at the electrical input of the MZMs in practice [10]), and, finally, \(D_{IMD}\) represents the distortion associated with IMD. Typically, the main sources of IMD are second order intermodulation (\(\Omega=\Omega_{i}-\Omega_{j}\)), third order intermodulation (\(\Omega=\Omega_{i}\pm 2\Omega_{j}\)), and a different case of third order intermodulation that is called triple beat (\(\Omega=\Omega_{i}\pm\Omega_{j}\pm\Omega_{k}\)), where \(\Omega_{i}\), \(\Omega_{j}\), and \(\Omega_{k}\) are any three arbitrary frequencies [15]. For systems consisting of many subchannels, triple beat is the dominant third order intermodulation [15], but for the presented case which consists of a low number of subchannels, the described subdivision of the third order IMD is necessary. For each subchannel in a given frequency plan, there can be a different number of interfering intermodulation products of each kind. These values are denoted as \(N_{CSO}\) for the second order intermodulation, \(N_{IM3}\) for the third order intermodulation, and \(N_{CTB}\) for the triple beat. In the case under analysis, with the MZMs biased at quadrature, the received photocurrent is free of second order intermodulation [10]. For that reason, only \(N_{IM3}\) and \(N_{CTB}\) are analysed, and are reported in Table 1 for the two frequency plans under analysis.