## 4 Numerical Results
Regarding performance comparison, we considered the competitor of the proposed adaptive transmission scheme to be one employing OFDM where pilot symbols are transmitted each frame to perform estimation, without considering prediction. Moreover, we assumed that, due to the channel non-reciprocity, estimation operated only at the receiver side, with channel coefficients being sent back to the transmitter via a bipolar modulation. Specifically, the information about the channel coefficients in the frequency domain was quantized with 32 bits before being fed back to the transmitter. During this signaling phase, potential errors might occur, impacting on bit-loading process as well. As a performance metric, we introduced \(\sigma\) as the difference between the number of bits allocated via the ideal case (channel perfectly known at both the transmitter and the receiver) and the methods under investigation (that is, the proposed one and the frame-by-frame estimation with quantized feedback, respectively). The results are shown for two different channel realizations, referred as case (a) and (b) in Figure 5, respectively. The numerical evaluation is reported on a per OFDM sub-carrier basis, that is, \(\sigma_{k}=b_{ideal}(k)-b_{proposed}(k)\) are represented with red circled markers, while \(\sigma_{k}=b_{ideal}(k)-b_{estimation-only}(k)\) are highlighted with blue markers (\(k\) = 1, 2, \(\ldots\), \(N_{sc}\)). From Figure 5, it is possible to appreciate that the mismatch between ideal and proposed bit-loading was always equal to  
Figure 4: Values of \(L_{adapt}\) in Equation (18) when different channel realizations are considered. zero in channel case (a), while only a few errors were made in case (b). Considering the scheme with frame-by-frame channel estimation, we can observe that there were several mis-allocated bits; in fact, the number of non-zero differences were 20 and 17 for cases (a) and (b), respectively. Mis-allocation in terms of bits leads to a situation that can be problematic. In fact, if the transmitter and receiver share mismatched channel information or, even worse, different information about the modulation format, the performance, in terms of rise in error rate, rapidly lowers. Such a result can be inferred from Figure 6, where we report the BER values by considering the evolution in terms of channel realization. In this regard, we want to emphasize that, in the bit-loading procedure we set an SNR-margin [54] granting a BER value of \(10^{-6}\). By investigating Figure 6, for most of the channel realizations, target BER performances were granted by the proposed method. When this was not possible (due to time-varying propagation conditions not being perfectly compensated) the reliability lowered and BER increased to values around less than \(10^{-4}\), with the exception of very few realizations where the channel was really bad. On the other hand, OFDM transmission with frame-by-frame estimation and quantized feedback was not able to achieve those values. In fact, the average BER was a bit higher than \(2\times 10^{-4}\), as is possible to appreciate by the distribution of blue markers in Figure 6.  
Figure 5: Difference between ideal bit-loading and the proposed scheme compared with the difference between ideal bit-loading and quantized information feedback link in channel realizations (**a**,**b**). Another interesting aspect to evaluate is the impact of the number of FFT points \(N_{SC}\), that is, the number of considered OFDM sub-channels, on the communication performance. In this regard, by referring to different values of \(N_{SC}\), we report in Table 3 the average BER evaluated among all the channel realizations, as well as the maximum BER and minimum one. From the results, it can be emphasized that, using an increasing number of \(N_{SC}\) led to a reduction in the average BER. This was true for \(N_{SC}\leq 128\). However, when \(N_{SC}=256\), the average BER increased. We can explain this non-monotonic behavior as follows.