## 3 Simulation Results
### BER Performance in Multipath Fading Channels  
Next, we evaluate the BER performance of 1024- and 4096-QAM in the multipath fading channels listed in Table 2. Figure 17a,b show the signal constellations for 1024- and 4096-QAM, respectively, observed at a receiver side affected by EPA fading. Figure 18 shows the instantaneous OFDM spectrum observed at the receiver side. Channel estimation and channel equalization, as shown in Figure 2, play an important role to compensate these distorted signals at the receiver side. If the channel estimation and channel equalization do not work sufficiently, the BER performance worsens.  
Figure 19 plots the BER of 1024-QAM against received SNR under EPA fading at coding rates of 1/3, 1/2, 2/3, and 3/4. The broken line indicates the BER obtained without fading (w/o fading), i.e., under static conditions. At a coding rate of 1/3, the SNR needed to achieve a BER of \(1\times 10^{-2}\) is approximately 20 dB, which is approximately 6 dB higher than what is required under the static condition. This behavior is similar to that observed at the other coding rates. The performance degradation is assumed to be affected by Doppler frequency shift and by imperfections in the channel estimation process.  
Figure 16: SNR penalty against phase error for QPSK to 4096-QAM at a coding rate of 1/3.  
Figure 15: SNR penalty against phase error for 4096-QAM [26].  
Figure 20 plots the BER of 1024-QAM against received SNR under the EVA fading model. Compared to the EPA fading case, the BER performance is significantly degraded. At a coding rate of 1/3, the SNR required to achieve a BER of \(1\times 10^{-2}\) is approximately 22 dB, which is approximately  
Figure 19: BER performance of 1024-QAM at various coding rates under EPA fading.  
Figure 17: Signal constellation at receiver side under EPA fading for **(a)** 1024-QAM and **(b)** 4096-QAM.  
Figure 18: OFDM spectrum observed at receiver side under EPA fading.  
8 dB higher than that is required under the static condition. At coding rates above 2/3, 1024-QAM can no longer achieve a BER of \(1\times 10^{-2}\).  
Figure 21 plots the BER of 4096-QAM against received SNR under EPA fading at coding rates of 1/3, 1/2, 2/3, and 3/4. At a coding rate of 1/3, the SNR required to achieve a BER of \(1\times 10^{-2}\) is approximately 22 dB, which is approximately 7 dB higher than that is required under the static condition. This behavior is similar to that observed at the other coding rates. Relative to 1024-QAM, the SNRs required to achieve a BER of \(1\times 10^{-2}\) at coding rates of 1/3, 1/2, 2/3, and 3/4 under EPA fading are increased by approximately 2-5 dB.  
Figure 22 plots the BER of 4096-QAM against received SNR under EVA fading. At a coding rate of 1/3, the SNR required to achieve a BER of \(1\times 10^{-2}\) is approximately 24 dB, which is approximately 9 dB higher than that is required under the static condition. At coding rates greater than 1/2, 4096-QAM can no longer achieve a BER of \(1\times 10^{-2}\).  
Figure 21: BER performance of 4096-QAM at various coding rates under EPA fading.  
Figure 20: BER performance of 1024-QAM at various coding rates under EVA fading.