## 2 System Model and Connection Setup
### Cyclic Prefix Length Adaptive Tuning  
The cyclic prefix is a fundamental element in OFDM, allowing the system to be (hopefully) ISI- and ICI-free [45]. Due to possible relative motion between nodes, as well as sea stream changes, the channel results are unavoidably time-varying over consecutive frames. Therefore, the use of a fixed CP length during the whole communication may be ineffective to counterbalance multipath. On the other hand, the capability of tuning the CP length allows the transmission to be adapted to the propagation conditions.  
In this regard, we have the number of sub-carriers \(N_{SC}\) employed for OFDM signaling defining the symbol length \(T_{x}=1/\Delta_{f}\), where \(\Delta f\) is the sub-channel width obtained by partitioning the whole system bandwidth \(B\) in \(N_{SC}\) portions, so that \(\Delta_{f}=B/N_{SC}\). However, when CP is considered, the whole OFDM symbol time becomes \(T_{s}=T_{x}+T_{CP}\), where \(T_{CP}\) is the time length of cyclic prefix that must be set according to the channel delay spread \(\tau_{ds}\). So, the OFDM symbol length corresponds to the slot time length \(T_{s}\) within a communication frame. Here, we recall that the delay spread is a measure of the length of the channel response, that is, the time difference between the earliest channel path and the last one. Hence, analytically speaking, \(T_{CP}\) value can be a multiple integer of delay spread. Of course, the use of CP unavoidably causes transmission rate reduction. In order to provide a theoretical example, if the system must be set up to grant a rate of \(R\) bits/s, the information bits must be allocated so that \(\sum_{c=1}^{N_{SC}}b_{c}=RT_{s}\), where \(b_{c}\) is the number of bits allocated on the \(c\)-th sub-channel. The presence of CP entails \(T_{CP}>0\), leading \(T_{s}\) to increase. From this fact, it appears evident that when the OFDM symbol length grows, due to delay spread (cyclic prefix), the use of richer modulation formats is required in the sub-channels to achieve the target data rate. As a consequence, higher power must be spent in order to guarantee robustness to ISI and ICI. So, communication reliability is paid for in terms of power efficiency.  
The channel knowledge is the key to estimate the delay spread, that can be obtained by measuring the _memory_ of the channel; that is, how long the effect of a signal emission is present at the receiver. Based on Equation (9), describing the pilot received during the estimation phase, it suffices to measure the cross-correlation between the emitted signal and the received one in order to test the signal time spread due to propagation. Analytically speaking, the cross-correlation is defined as:  
\[y_{r,s}[m]=\frac{1}{E_{s0}(2T_{s}/T_{c}-1)}\sum_{p=1}^{T_{s}/T_{c}}r_{e}^{(m)*} [p]s[p+m] \tag{13}\]  
where \(y_{r,s}[0]\) corresponds to the signal energy on the first path. Then, by means of a threshold \(\vartheta,0<\vartheta<1\), we can also define a reference energy level expected for the signal on the last path, expressed as a fraction, \(y_{r,s}[0]\). In other words, once \(y_{r,s}[0]\) is selected as the cross-correlation value referred to the first path, we define, as delay spread, the time to achieve a percentage in terms of cross energy equal to \(\phi y_{r,s}[0]\). Hence, formally we have:  
\[\tilde{\tau}_{ds}=T_{c}\arg\min_{m=1_{\sim},T_{s}/T_{c}}|y_{r,s}[m]-\vartheta y _{r,s}[0]| \tag{14}\]  
returns the estimate of delay spread.