## 4 Numerical Results
When the number of sub-channels increases, the frequency domain description of the channel is more accurate, and the same is true for the prediction, since variations of the frequency response are measured with a dense sampling. On the other hand, having few FFT points (low values of \(N_{SC}\)) leads to a poor description of the channel in the frequency domain. So, channel changes in time are badly tracked and likewise for the equalization. The reason why for \(N_{SC}=256\) the average BER increased, is that the Doppler effect became more important, since the sub-channel bandwidth became narrower, and this induced mis  
Figure 6: BER for different channel realizations for the proposed scheme and the case with frame-by-frame estimation and quantized feedback. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**FFT Points** & **Average BER** & **Maximum BER** & **Minimum BER** \\ \hline \(N_{SC}=16\) & \(9.7\times 10^{-6}\) & \(5.6\times 10^{-2}\) & \(10^{-6}\) \\ \hline \(N_{SC}=32\) & \(6.1\times 10^{-6}\) & \(2.3\times 10^{-2}\) & \(10^{-6}\) \\ \hline \(N_{SC}=64\) & \(4.7\times 10^{-6}\) & \(1.4\times 10^{-2}\) & \(10^{-6}\) \\ \hline \(N_{SC}=128\) & \(3.6\times 10^{-6}\) & \(8.7\times 10^{-3}\) & \(10^{-6}\) \\ \hline \(N_{SC}=256\) & \(4.3\times 10^{-6}\) & \(1.1\times 10^{-2}\) & \(10^{-6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: BER performance as a function of \(N_{SC}\). detection pf events. By looking at the third column of Table 3, we note that the maximum BER followed the same behavior as the average BER, and such a high value was justified by the fact that sometimes the channel can change radically and very quickly. Finally, the fourth column of the table reports the same values for the minimum BER, since that was the target considered as the constraint for bit-loading. Furthermore, another important result must be highlighted. We recall that the proposed transmission framework considers transmitting and receiving nodes as operating separately on interference acquisition and channel estimation, without any mutual feedback. So, it may happen that the bit-loading performed at the transmit side does not match with that expected at the receiver. This fact may lead signal detection to completely fail, being based on a symbol constellation that may be different from that actually employed for transmission. However, the results in Figure 6 demonstrate the high reliability of the proposed hybrid channel estimation-prediction method, outperforming, also, the case where frequent channel estimation operated. Such an advantage can be finally appreciated in terms of communication efficiency as well. By referring to \(\eta\), as defined in Equations (23)-(25), for the transmission schemes under investigation, we show in Figure 7 that, when the message length increased, the OFDM system, where frame-by-frame estimation was performed, was unable to become more efficient, since the amount of pilot symbols to be sent increased with the number of transmitted frames. On the other hand, the proposed system paid off, in terms of efficiency, when the file was very short, since the time spent acquiring the statistics (CS stage) was long with respect to the transmission of a few bytes (10-30). However, the time spent in CS lost importance when the amount of information increased, as the largest part of the time was spent sending data (with an additional small percentage involved in re-estimating the channel).