## 4 Experiments with Ensemble-Average SNR  
We now describe the procedure for performing simulations with a given ensemble-average SNR value. Essentially, an experiment for a specified SNR value consists of at least 50 trials. In each trial, a new realization of the specified channel model is generated. The noise power is calculated based on the average signal power in (7), the given SNR value, and the assumption of unity channel gain. The reported BER is computed by averaging the BER over all trials. Following this procedure, we conducted an experiment with 10,000 trials, each corresponding to the duration of one DL frame. In the experiment, each value of \(|H[n,k]|\), for all \(n\) and \(k\) in all trials, is used as a realization of the magnitude of channel frequency response to obtain the empirical probability density function (pdf) of \(|H[\cdot,\ \cdot]|\). Figure 4 shows the pdfs of the magnitude of the channel frequency response (i.e., \(|H[n,k]|\)), derived from the 10,000 trials mentioned earlier, for the three studied channel models. Each pdf plot is obtained from the contour of a histogram with a bin width of 0.01. Recall that perfect side information is assumed to be available; therefore, no channel estimation is performed in the experiments. For a more realistic evaluation, advanced channel estimation methods could be utilized [21; 22]. From Figure 4, we observe that the three empirical pdfs are almost identical. Since \(\mathbf{H}[n,\ k]\) is the DFT of the sampled version of channel impulse response, \(\mathbf{H}[n,\ k]\) is a linear combination of path gains, \(\mathbf{h}_{l}\), \(l=0,1,\ldots,L-1\). Thus, it is straightforward to prove that \(|\mathbf{H}[n,\ k]|\) has a normalized Rayleigh distribution (with unity ensemble-average power gain) for all three channel models, which is confirmed by the figure. Figure 4: Theoretical and empirical pdfs of the magnitude of channel frequency response. Figure 3: A block diagram of the receiver used in simulations. Using the same experiment setup, we also obtained the BER performance after demodulation (referred to as the demodulated BER), and the results are plotted in Figure 5. It is observed that all three channel models yield the same demodulated BER performance. As the three models have the same statistical distribution for the magnitude of channel frequency response, this result is predictable. It is known that the analytical BER of QPSK modulation via a Rayleigh fading channel [23] is  
\[\text{BER}_{\text{QPSK, Rayleigh}}=\frac{1}{2}(1-\sqrt{\frac{E_{b}/N_{0}}{1+E_{b }/N_{0}}}) \tag{10}\]  
where \(E_{b}\) is the average energy of each coded bit. For example, if SNR = 0 dB, then \(E_{b}/N_{0}=\frac{2048}{600}\cdot\frac{1}{2}=1.71\), where \(\frac{2048}{600}\) is the ratio of FFT size to active subcarriers. Therefore, \(\text{BER}_{\text{QPSK, Rayleigh}}=\frac{1}{2}\Big{(}1-\sqrt{\frac{1.71}{1+1.71}} \Big{)}=0.10\). To be complete, the analytic results are also plotted in Figure 5. The high degree of agreement between theoretic and experimental results in Figures 4 and 5 confirms the correctness of the simulation program. Since \(|\mathbf{H}[n,\ k]|\) has been proven to have a normalized Rayleigh distribution for all three channel models, the simulation BERs agree with the analytical BERs. Figures 4 and 5 illustrate that all three simulated channels exhibit the same probabilistic characteristics and demodulated BER, regardless of the channels' delay profiles. Different results will be observed when applying the short-term SNR to repeat the same experiment in Section 5. Using the MATLAB code that implements the receiver shown in Figure 3, we obtain the BER performance after decoding (referred to as the decoded BER throughout this paper), as presented in Figure 6. It is observed that the BERs for all three channels are different. This outcome is counter-intuitive. Given that the three channel models produce identical demodulated BER performance, one would expect similar decoded BER performances as well (after six decoding iterations).