# Estimation of BER from Error Vector Magnitude for Optical Coherent Systems
## 4 BER Estimation with Correction Factor  
The simulation results shown in Figure 2 reveal that the relationship Equation (2) underestimates the actual BER at the target BER of \(10^{-3}\). The BER estimated from EVM analysis, \(BER_{EVM}\), can be calibrated using  
\[BER_{cal}=k\cdot BER_{EVM} \tag{5}\]  
where \(k\) is a correction factor given as  
\[k=\frac{BER_{ideal}}{BER_{EVM}} \tag{6}\]  
Figure 1: Sensitivity penalty at the target bit error ratio (BER) of \(10^{-3}\) for QSPK, 16-QAM and 64-QAM.  
Figure 2: Estimated BER from error vector magnitude (EVM) analysis for different modulation formats at the target BER of \(10^{-3}\).  
at the calibration point and \(BER_{ideal}\) is the target BER. The residual sensitivity penalty shown in Figure 1 is due to the differential coding penalty and the implementation penalty of the carrier recovery process. For the square-QAM constellations, the differential coding penalty is given as [13]  
\[F=1+\frac{\log_{2}(M)}{2(\sqrt{M}-1)} \tag{7}\]  
Thus, the values of \(F\) are 2, 1.67, and 1.43 for QPSK, 16-QAM, and 64-QAM, respectively. For data-aided reception with an ideal carrier phase recovery process, \(F\) can be used as the correction factor to take into account the above differential coding penalty. In this work, the correction factor, \(k\), to calibrate the BER estimated from EVM analysis using Equation (5) includes the contributions from the differential coding penalty and also the additional implementation penalty of the carrier phase recovery process.  
Figure 3 shows the values of \(k\) for the different modulation formats at the target BER of \(10^{-3}\) following the compensation of laser phase noise with differential decoding as discussed in Section 3. The values of \(k\) were found to be steady for the respective modulation formats with \(\Delta v\cdot T_{s}<10^{-5}\) and higher than the differential coding penalty given by Equation (7) with the carrier phase recovery.  
Finally, the accuracy of \(BER_{cal}\) to track BER fluctuations up to a BER of \(\sim 10^{-2}\) was investigated for reliable performance monitoring. The values of \(\Delta v\cdot T_{s}\) were set to \(10^{-4}\) for QPSK/16-QAM and \(10^{-5}\) for 64-QAM. Figure 4a-c show the BER estimated from EVM analysis for QPSK, 16-QAM, and 64-QAM, respectively, with and without the correction factor. The insets show the recovered constellations at the BER of \(10^{-3}\). For QPSK as well as 16-QAM, the values of \(k\) derived at the target BER of \(10^{-2}\) and \(10^{-3}\) were found to be similar with \(\Delta v\cdot T_{s}=10^{-4}\) for the respective modulation formats. For 64-QAM with \(\Delta v\cdot T_{s}=10^{-5}\), the mean value of \(k\) derived at the target BER of \(10^{-2}\) and \(10^{-3}\) was chosen to calibrate the estimated BER for the results shown in Figure 4c. Without the correction factor, the relationship (2) underestimates the BER at the different SNR. However, the calibrated BER using Equation (5) can be seen to be in close agreement with the actual BER obtained by error counting. The calibrated BER from EVM analysis can thus be used to reliably monitor the performance near the target BER for the different modulation formats. Moreover, although the estimated BER from Equation (2) assumes data-aided reception, the simulation results in Figure 4 demonstrate that the relationship can also be applied to nondata-aided reception with the correction factor even when the symbol sequences are unknown and SNR is relatively low near the target BER of \(10^{-3}\).  
Figure 3: Correction factor, \(k\), for different modulation formats at the target BER of \(10^{-3}\).