# MZM Optimization of PAM-4 Transmission in Data Center Interconnect
## 2 Mathematical Presentation  
Optical communication systems are subject to many kinds of impairments, which can be categorized mainly as: signal-independent (additive) noise, signal-dependent (multiplicative) noise, bandwidth limited induced intersymbol interface (ISI), non-linear distortions, and limitation on the signal strength such as a finite extinction ratio (ER). If signal-dependent noise dominates, higher power level is received with more noise and vice versa lower power level is received with lower noise [6; 7]. Typical examples of signal-dependent noise include relative intensity noise (RIN) and amplified spontaneous emission (ASE) in optically amplified systems, and shot noise (quantum noise). The total variance associated with these noises can be written as:  
\[\sigma_{l}^{2}=\sigma_{th}^{2}+\sigma_{sp-sp}^{2}+\sigma_{si}^{2}+\sigma_{ sig-sp_{l}}^{2}+\sigma_{RIN_{i}}^{2},\ 0\leq l\leq 3 \tag{1}\]  
where \(l\) is an integer that represents one of the four levels of the PAM-4 signal, \(\sigma_{th}^{2}\), \(\sigma_{sp-sp}^{2}\) are the thermal noise variance, and spontaneous-spontaneous beat noise variance, respectively. \(\sigma_{si}^{2}\), \(\sigma_{ig-spl}^{2}\) and \(\sigma_{RINl}^{2}\), are the shot noise variance, signal-spontaneous variance and RIN variance, respectively, which are multiplicative noise terms and are the dominate contributors in Equation (1), while the contribution of thermal and shot noises in many cases is neglected [8]. Therefore, higher power signals suffer from higher noise. Assuming that the noise distribution is Gaussian, the conditional probability density functions (CPDF) associated with each level is expressed by [9]:  
\[p(r/s_{l})=\frac{1}{\sigma_{l}\sqrt{2\pi}}e^{-\frac{(r-m_{l})^{2}}{2\sigma_{l} ^{2}}} \tag{2}\]  
where \(m_{l}\) is the received signal mean assuming symbol \(s_{l}\) (level \(l\)) was transmitted. Figure 1 presents an example of the CPDF of a PAM-4 signal with the parameters of the lab experiment discussed in Section 4, at a bit error rate (BER) of 5e-3. In Figure 1, the mean power levels (associated with the PAM-4 symbols) are equally spaced, while the signal dependent (multiplicative) noise dominates. It is clearly observed that the CPDF of higher-level power contains higher variance as compared to lower level power. Consequently, the symbols associated with the higher levels powers are subject to more errors than the symbols associated with the lower level powers, which results in suboptimum bit error rate (BER) performance. Optimization of the BER performance in such case of signal-dependent noise may be achieved by power level spacing optimization. The optimization is done by introducing larger spacing between the higher power levels and vice versa lower spacing between the lower power levels, so that the symbols error rate (SER) at the higher levels will be similar to the SER at low levels.  
In this paper we combine the power level spacing optimization in an \"optical uneven PAM-4\" manner, followed by electrical digital signal processing (DSP) using MLSE [3]. The DSP compensates for the non-linear distortions and ISI introduced by the band limited MZM and other channel components.  
Figure 1: Conditional probability density functions (CPDF) of a PAM-4 signal, within signal-dependent noise.