## 2 System Model and Connection Setup
### Channel Model, Estimation and Statistics Evaluation  
After interference acquisition, both the nodes must proceed with channel estimation as the second phase of CS, depicted in Figure 1. So, node \(u_{1}\) starts sending some known pilot symbols to \(u_{2}\), that replies with an identical transmission back to \(u_{1}\). The pilot is modeled as:  
\[x(t)=\sqrt{E_{s_{0}}}s(t) \tag{6}\]  
where \(E_{s_{0}}\) is the associated energy and \(s(t)\) is the equivalent baseband signal given by:  
\[s(t)=\frac{1}{\sqrt{2\pi\alpha}}e^{\frac{-(t-\frac{T_{c}}{2})^{2}}{2\gamma}}, \ \ \ 0\leq t\leq T_{s} \tag{7}\]  
where \(\gamma\) takes care of signal shape (and, consequently, its bandwidth), while \(\alpha\) is set so to have a unit energy signal \(s(t)\), since the signal \(x(t)\) in Equation (6) has \(E_{s_{0}}\) energy. The pilot length is supposed to be equal to \(T_{s}\), with \(s(t)\) being a single carrier signal, so as to allow the estimation of the whole channel. The choice of a Gaussian-like shape, described in Equation (7), is not mandatory and depends on the transmitting device capability to generate different shaping signals. Changing the signal shape (from Gaussian-like to Nyquist-like) would reflect on performance variations that are negligible if the bandwidthand transmit power levels do not change. If the term \(\gamma\) is sufficiently large, the signal is wide in bandwidth, since in the time domain it presents quick amplitude variations. At this point, such single-carrier pulse sent for estimation risks appearing a bit counter-intuitive since we base the communication on OFDM. However, by using such a pilot, we focus on a _whole_ channel impulse response in order to optimize the cyclic prefix. Additionally, through the use of filter-bank (equating the number of OFDM sub-carriers) we are able to estimate the time response of each sub-channel. This scheme does not avoid the frequency domain description of the channel through the Discrete Fourier Transform (DFT). With such bidirectional pilots transmission, each node can perform its own channel estimation. Especially regarding multipath, signal propagation along the two communication directions is different, that is, the channel is, in general, not reciprocal [42]. As a consequence, channel estimates available at the link sides may be different as well. However, since the largest part of channel energy is generally carried by the first path [43], reciprocity can be reasonably assumed. Moreover, what is important to underline is that, when reciprocity cannot be assumed, the channels are different. In other words, forward and backward channels can be different. However, it is highly probable that the differences do not impact on performance in a severe way. Hence, the conclusion is that reciprocity is worth it, especially when very particular propagation scenarios are not present [42], like, for example, obstacles that reflect a signal in a specific direction and block the signal in the opposite direction. In order to describe the received signal during pilot transmission, let us now introduce the channel model. In detail, the underwater acoustic channel is typically affected by a frequency-selective fading phenomenon that scatters the energy of the transmitted signal over a (usually) not so small number of paths generated by reflections from ground and sea surfaces, or due to propagation effects that can be described as curved rays. By also taking into account its time-variability, the channel can be modeled according to the following expression [40]:  
\[h(t;\tau)\triangleq\sum_{d=1}^{\rho(t)}\beta_{d}(t)a_{d}\delta(t-\tau_{d}(t))e ^{|2\pi\
u_{d}t} \tag{8}\]  
where \(a_{d}\) is the complex coefficient accounting for losses over the \(d\)-th path [40]. \(\rho(t)\) is the time-varying number of paths, each one characterized by a propagation delay \(\tau_{d}(t)\).