## 4 Experiments with Ensemble-Average SNR
However, Figure 6 shows significant variation among the channels. Notably, the decoded BER for the EPA channel model is substantially higher than those for the EVA and ETU models. Furthermore, if a particular channel model were to result in a significantly higher decoded BER, one would expect it to be the ETU model, due to its longer delay spread compared to the CP, which leads to inter-symbol interference (ISI). It is worth mentioning that our previous research on Worldwide Interoperability for Microwave Access (WiMAX) [15] also demonstrated that different channel models yield similar decoded BER performances, within the bounds of experimental uncertainty. Figure 5: Demodulated BERs using the ensemble-average SNR as a basis of comparison. Upon conducting in-depth studies, we discovered that the disparity between the demodulated and decoded BERs is caused by the collective distribution of \(|H[n,k]|\) across channel realizations. To simplify the discussion, let \(H^{(i)}[n,k]\) denote the realization of the channel frequency response random process \(\mathbf{H}[n,\ k]\) at the \(i\)th trial. By averaging the power gains of all 2048 subcarriers in an OFDM symbol, we obtain  
\[\overline{\left|\mathbf{H}[n,\cdot]\right|^{2}}=\frac{1}{2048}\sum_{k=0}^{204}\left| \mathbf{H}[n,k]\right|^{2}. \tag{11}\]  
According to [PERSON]'s theorem [24]:  
\[\overline{\left|\mathbf{H}[n,\cdot]\right|^{2}}=\sum_{l=0}^{L-1}\left|\mathbf{h}_{l} \right|^{2} \tag{12}\]  
Since the magnitude of path gain \(\left|\mathbf{h}_{l}\right|\) is a Rayleigh random variable with parameter \(\sigma^{2}=\widetilde{p}_{l}/2\), the random variable \(\mathbf{Z}_{l}=\left|\mathbf{h}_{l}\right|^{2}\) is an exponential random variable with mean \(\widetilde{p}_{l}\). Thus, the moment generation function of \(\overline{\left|\mathbf{H}[n,\cdot]\right|^{2}}\) is given by  
\[M(t)=\prod_{l=0}^{L-1}\frac{1}{1-\widetilde{p}_{l}t} \tag{13}\]  
From this, it follows that the mean and variance of \(\overline{\left|\mathbf{H}[n,\cdot]\right|^{2}}\) are  
Figure 3:0  
and  
Figure 3:1  
respectively. Recall that, following (4), we have \(\widetilde{p}_{0}=0.314\) and \(\widetilde{p}_{1}=0.249\) for the EPA channel. Using (15), we have \(\mathrm{Var}[\overline{\left|\mathbf{H}[n,\cdot]\right|^{2}}]=0.237\). With further calculations, we obtain variances approximately equal to \(0.177\) and \(0.129\) for the EVA and ETU channel models,  
Figure 3:2 Decoded BERs using ensemble-average SNR as a basis of comparison. respectively. According to (4) and (15), a channel with more evenly distributed power on each tap has smaller \(\text{Var}[\overline{\left|\mathbf{H}[n,\cdot]\right|^{2}}]\). To validate our analysis, we plot the theoretical and empirical pdfs of \(\overline{\left|\mathbf{H}[n,\cdot]\right|^{2}}\) in Figure 7 for the three channel models. The theoretical pdf of \(\overline{\left|\mathbf{H}[n,\cdot]\right|^{2}}\) is computed by convolving pdfs of \(L\) exponential random variables \(\mathbf{Z}_{t}\), \(l=0,1,\ldots,L-1\). The empirical pdf of \(\overline{\left|\mathbf{H}[n,\cdot]\right|^{2}}\) is obtained using  
Figure 3:3  
in each trial \(i\) as a realization. In the figure, both \(K=2048\) and \(K=600\) are considered for empirical pdf plots. In the \(K=2048\) case, \(\overline{\left|\mathbf{H}^{(i)}[n,\cdot]\right|^{2}}\) in (16) is an average over all subcarriers; in the \(K=600\) case, it is an average over the 600 subcarriers assigned to a transport block. Note that the theoretical and empirical pdfs for 2048 subcarriers are very close to each other, as predicted by (11) and (12). Figure 7 reveals that the EPA model exhibits the widest pdf shape around the mean, while the ETU model has the narrowest shape, as corroborated by their variances. Although we previously demonstrated that the pdfs of the magnitude of the channel frequency response (and thus, the pdfs of subcarrier power gain) for all three channel models are nearly identical (cf.