# MZM Optimization of PAM-4 Transmission in Data Center Interconnect
## Appendix A Analytical Derivation of PAM-4 Signal with Optimum Transmission Levels  
In PAM-4 transmission, due to noise, the received signal \(r\) may be incorrectly classified, which in turn results in decision errors given by:  
\[P_{\epsilon}=\frac{1}{4}P(\epsilon/0)+\frac{1}{4}P(\epsilon/1)+\frac{1}{4}P( \epsilon/2)+\frac{1}{4}P(\epsilon/3) \tag{1}\]  
Figure 5: (**a**) Uneven PAM-4 gain vs. MZM bias point. (**b**) OMAouter improvement (only), by keeping power levels spacing of both curves identical.  
where \(P\)(e/l), \(0\leq l\leq 3\) is the conditional probability of error given that symbol \(l\) was transmitted.  
_Receiver optimization:_ Assuming that the conditional probability density functions (CPDFs) are Gaussian and that the decision threshold is optimized to achieve minimum BER, the probability of error can be expressed by:  
\[\begin{array}{l}P_{e}=\frac{1}{8}erfc\Big{(}\frac{I_{1}-I_{0}}{\sigma_{1}+ \sigma_{0}}\Big{)}+\frac{1}{8}\Big{[}erfc\Big{(}\frac{I_{1}-I_{0}}{\sigma_{1}+ \sigma_{0}}\Big{)}+erfc\Big{(}\frac{I_{2}-I_{1}}{\sigma_{2}+\sigma_{1}}\Big{)} \Big{]}\\ +\frac{1}{8}\Big{[}erfc\Big{(}\frac{I_{2}-I_{1}}{\sigma_{2}+\sigma_{1}}\Big{)} +erfc\Big{(}\frac{I_{3}-I_{2}}{\sigma_{3}+\sigma_{2}}\Big{)}\Big{]}+\frac{1}{ 8}erfc\Big{(}\frac{I_{3}-I_{2}}{\sigma_{3}+\sigma_{2}}\Big{)}\end{array} \tag{1}\]  
where _erfc_ is the complementary error function, \(I_{l}\) (\(0\leq l\leq 3\)) is the received signal level assuming symbol \(s_{l}\) (level \(l\)) was transmitted, \(\sigma_{l}\) (\(0\leq l\leq 3\)) is the noise variance associated with each received level \(I_{l}\).  
_Transmitter optimization:_ minimum BER is obtained by manipulating the transmitted levels of the PAM-4 signal. The optimization scheme is achieved by maintaining all the six _erfc_ terms of Equation (1) identical. The resulting optimization scheme is expressed mathematically by [6]:  
\[\frac{I_{1}-I_{0}}{\sigma_{1}+\sigma_{0}}=\frac{I_{2}-I_{1}}{\sigma_{2}+\sigma _{1}} \tag{2}\]  
and  
\[\frac{I_{2}-I_{1}}{\sigma_{2}+\sigma_{1}}=\frac{I_{3}-I_{2}}{\sigma_{3}+\sigma _{2}} \tag{3}\]  
where \(I_{0}\) and \(I_{3}\) are the outer PAM-4 levels and are obtained by the average optical input power at the receiver input, photodiode responsivity and the signal voltage swing at the modulator input, and \(I_{1}\) and \(I_{2}\) are independent variables that form the PAM-4 level spacing optimization.  
Analytical Comparison of PAM-4 Levels Optimization with Equal Field Spacing Assuming Signal-Dependent Noise  
Assuming that the amplitudes of the optical field are equally spaced, we can introduce the received signal levels such that:  
\[I_{0}=A^{2},I_{1}=(A+d)^{2},I_{2}=(A+2d)^{2},I_{3}=(A+3d)^{2} \tag{4}\]  
where \(A\) (\(A=\sqrt{I0}\)) is the amplitude of optical field associated with level 0 and \(d\) is the optical field spacing. Substituting Equation (4) into Equation (2) yields:  
\[\frac{(A+d)^{2}-A^{2}}{\sigma(A+d)+\sigma A}=\frac{(A+2d)^{2}-(A+d)^{2}}{ \sigma(A+2d)+\sigma(A+d)} \tag{5}\]  
where \(\sigma\) is the variance of the optical noise.  
The result of both sides of Equation (5) is \(\frac{d}{\sigma}\), implying that the equality of Equation (2) is met, i.e., the optimization requirement of Equation (2) is satisfied.  
Similarly, substituting Equation (3) into Equation (4) yields the same result \(\frac{d}{\sigma}\), i.e., implying that the equality of Equation (3) is also met.  
Thus, equal optical field spacing introduces minimum BER assuming that signal-dependent noise dominates.