## 3 Simulation Results
### SNR Penalty in the Presence of Phase Error  
We then evaluate the influence of phase error which is measured in units of radians. Figure 10a,b shows photos of signal constellations for 1024- and 4096-QAM, respectively, at a phase error of 0.08. In general, the BER worsens as the phase error increases, because higher-order QAM has a weakness for errors in amplitude and phase as a result of the shortening Euclidean distance. In some cases, the QAM can no longer meet a BER of \(1\times 10^{-2}\) even if SNR is increased.  
Figure 11 shows the BER of 1024-QAM as a function of phase error at a coding rate of 1/3. At phase errors of 0.10 and 0.15, the SNRs needed to achieve a BER of \(1\times 10^{-2}\) are increased by approximately 1.5 dB and 5 dB, respectively, relative to the no-phase-error case. When the phase error is increased to 0.20, 1024-QAM can no longer achieve a BER of \(1\times 10^{-2}\) even if the SNR is increased.  
Figure 12 shows the SNR penalty needed to achieve a BER of \(1\times 10^{-2}\) against phase error for 1024-QAM at various coding rates. It is clearly observed that the SNR penalty increases as the phase error increases. The SNR penalty is approximately 6 dB at a phase error of 0.05 and a coding rate of 3/4.  
Figure 8: Fundamental BER performance of 4096-QAM as a function of coding rate [26].  
Figure 9: BER performance for QPSK to 4096-QAM at coding rates of 1/3 and 3/4. QPSK = quadrature phase shift keying.  
To reduce the SNR penalty to below 3 dB, the phase error cannot exceed 0.04 at coding rates of 3/4 or below.  
Figure 11: BER performance of 1024-QAM as a function of phase error at a coding rate of 1/3 [26].  
Figure 12: SNR penalty against phase error for 1024-QAM [26].  
Figure 10: Signal constellation at a phase error of 0.08 radian for (a) 1024-QAM and (b) 4096-QAM.  
Figure 13 shows the BER of 4096-QAM as a function of phase error at a coding rate of 1/3. At phase errors of 0.08 and 0.12, the SNRs needed to achieve a BER of \(1\times 10^{-2}\) are increased by approximately 2 dB and 4 dB, respectively, relative to the no-phase-error case. When the phase error is increased 0.16, 4096-QAM can no longer achieve a BER of \(1\times 10^{-2}\), even if the SNR is increased. Figure 14 shows the BER of 4096-QAM as a function of phase error at a coding rate of 3/4. Relative to the case for which the coding rate of 1/3, the BER is worse, and when the phase error is increased to 0.04, 4096-QAM can no longer achieve a BER of \(1\times 10^{-2}\) even if the SNR is increased.  
Figure 15 plots the SNR penalty needed to achieve a BER of \(1\times 10^{-2}\) against phase error for 4096-QAM at various coding rates. At a phase error of 0.03 and a coding rate of 2/3, the SNR penalty of 4096-QAM is approximately 6 dB. To reduce the SNR penalty to less than 3 dB, the phase error must be no greater than 0.024 at a coding rate of 3/4 or below.  
Figure 16 plots the SNR penalty required to achieve a BER of \(1\times 10^{-2}\) against phase error for QPSK, 16-, 64-, 256-, 1024-, and 4096-QAM at a coding rate of 1/3. Phase error severely impacts 1024- and 4096-QAM compared with existing QAM and QPSK schemes. To limit the SNR penalty to below 3 dB, the phase errors for 16-, 64-, 256-, 1024-, and 4096-QAM must be no greater than 0.39, 0.31, 0.18, 0.12, and 0.11, respectively.  
Figure 14: BER performance of 4096-QAM as a function of phase error at a coding rate of 3/4.  
Figure 13: BER performance of 4096-QAM as a function of phase error at a coding rate of 1/3 [26].