## 3 Simulation Results  
### Fundamental Bit Error Rate (BER) Performance under Additive White Gaussian Noise (AWGN)  
Figure 4(a),b shows the signal constellation for 1024- and 4096-QAM, respectively, at the transmitter side. As the multilevel (order) of QAM increases, the Euclidean distance among signal constellations naturally shortens, as shown in Figure 5. Figure 6 shows the observed instantaneous OFDM spectrum for 1200 subcarriers containing 1024- or 4096-QAM symbol modulation signals.  
Figure 7 plots the fundamental BER against received SNR of 1024-QAM at various coding rates, R, under an AWGN channel condition. The BER is measured by comparing the demodulated serial bits with the original binary serial bits. In this work, we assume a benchmark BER of \(1\times 10^{-2}\). When turbo coding and soft-decision Viterbi decoding are used, the SNRs needed to achieve a BER of \(1\times 10^{-2}\) at coding rates R of 1/3, 1/2, 2/3, and 3/4, which are approximately 14 dB, 19 dB, 23 dB, and 25 dB, respectively. The BER at a coding rate of one, i.e., without coding (w/o), is also plotted as a reference. Figure 8 plots the fundamental BER against received SNR of 4096-QAM at various coding rates under the condition of AWGN channels. The SNRs needed to achieve a BER of \(1\times 10^{-2}\) at coding rates of 1/3, 1/2, 2/3, and 3/4 are approximately 16 dB, 22 dB, 28 dB, and 30 dB, respectively. Relative to 1024-QAM, the SNRs needed to achieve the benchmark BER of \(1\times 10^{-2}\) for 4096-QAM are increased by  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Symbol modulation & QPSK to 4096-QAM \\ \hline \multirow{2}{*}{Encoder} & Turbo encoding \\  & Coding rate R = 1/3, 1/2, 2/3, 3/4, 1 \\  & Constraint length = 4 \\ \hline Signal mapping & Gray code \\ \hline Transmission bandwidth & 20 MHz \\ \hline IFFT/FFT size & 2048 \\ \hline Cyclic prefix length & 144 \\ \hline The number of transmitter/receiver antennas & 1/1 \\ \hline Decoder & Soft-decision Viterbi decoding \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation parameters. QPSK = quadrature phase shift keying.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Type** & **Parameter** & **Value** \\ \hline \multirow{2}{*}{EPA} & Path delay (ns) & 0, 30, 70, 90, 110, 190, 410 \\ \cline{2-3}  & Path gain (dB) & 0, \(-\)1, \(-\)2, \(-\)3, \(-\)8, \(-\)17.2, \(-\)20.8 \\ \hline \multirow{2}{*}{EVA} & Path delay (ns) & 0, 30, 150, 310, 370, 710, 1090, 1730, 2510 \\ \cline{2-3}  & Path gain (dB) & 0, \(-\)1.5, \(-\)1.4, \(-\)3.6, \(-\)0.6, \(-\)9.1, \(-\)7.0, \(-\)12.0, \(-\)16.9 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Multipath fading channels used in the simulation. EPA = extended pedestrian A; EVA = extended vehicular A.  
approximately 2 dB, 3 dB, 5 dB, and 5 dB at coding rates of 1/3, 1/2, 2/3, and 3/4, respectively. Figure 9 shows an overall comparison of BERs for QPSK, 16-, 64-, 256-, 1024-, and 4096-QAM at coding rates of 1/3 (blue lines) and 3/4 (red lines). The SNRs required for 4096-QAM at these coding rates are, respectively, 17 dB and 26 dB higher than those required for QPSK. Current LTE mobile systems use a link adaptation technology that can choose a modulation scheme and coding rates depending on the channel quality, i.e., received SNR or SINR. The results in Figure 9 suggest that if 4096-QAM is used for the \"modulation and coding sets (MCS)\", 4096-QAM at coding rates of 1/3 or 3/4 can be assigned to a UE at received SINRs of more than 16 dB or 30 dB, respectively. Similarly, 1024-QAM at a coding rate of 1/3 can be assigned to a UE at received SINR of more than 14 dB.  
Figure 5: Signal constellation at transmitter side for (**a**) 1024-QAM and (**b**) 4096-QAM [26].  
Figure 6: OFDM spectrum at transmitter side.  
Figure 7: Fundamental bit error rate (BER) performance of 1024-QAM as a function of coding rate. SNR = signal-to-noise ratio.