# Energy-Efficient Channel Coding Strategy for Underwater Acoustic Networks
## 3 Channel and System Model  
In this section, we model the underwater acoustic channel in terms of path loss, additive noise and fading. Moreover, we also discuss the bit error probability (BEP) of binary non-coherent frequency shift keying modulation and the relationship between acoustic and electric transmit power. The path loss experienced in an underwater acoustic channel can be written as [26]  
\[A(l,f)=(1000\cdot l)^{k}a(f)^{l}, \tag{1}\]  
where \(l\) is the transmit distance in km, \(f\) is the frequency in kHz, \(k\) is the spreading factor and \(a(f)\) is the absorption coefficient. The spreading factor describes the geometry of the propagation. We assume a practical spreading factor of \(k=1.5\). The absorption coefficient is calculated according to the [PERSON]'s expression as [27]  
\[10\log_{10}\big{[}a(f)\big{]}=\frac{0.11f^{2}}{1+f^{2}}+\frac{44f^{2}}{4100+f^ {2}}+\frac{2.75f^{2}}{10^{4}}+0.003\ \ \mathrm{dB/km}. \tag{2}\]  
The additive noise in the acoustic underwater channel is modeled considering four basic sources: thermal noise \(N_{th}(f)\), waves \(N_{w}(f)\), shipping \(N_{s}(f)\) and turbulence \(N_{t}(f)\). The following empirical formulas define these noise components, in dB re \(\mu\)Pa per Hz [6]:  
\[10\log_{10}N_{th}(f)=-15+20\log_{10}(f), \tag{3}\]  
\[10\log_{10}N_{w}(f)=50+7.5\sqrt{w}+20\log_{10}(f)-40\log_{10}(f+0.4), \tag{4}\]  
\[10\log_{10}N_{s}(f)=40+20(s-0.5)+26\log(f)-60\log_{10}(f+0.03), \tag{5}\]  
\[10\log_{10}N_{t}(f)=17-30\log_{10}(f). \tag{6}\]  
In \(N_{w}(f)\), the wind speed \(w\) is given in m/s. The shipping activity parameter \(s\) in \(N_{s}(f)\) has a range from 0 to 1, from low to high activity, respectively. The overall noise power spectral density is thus  
\[N(f)=N_{th}(f)+N_{w}(f)+N_{s}(f)+N_{t}(f)\ \ \ \mathrm{Pa/Hz}. \tag{7}\]  
Given that the acoustic transmit and received power are, respectively, \(P_{r}\) and \(P_{t}\), the average received signal-to-noise ratio (SNR) is  
\[\gamma(l,f)=\frac{P_{r}}{N(f)B(f)}=\frac{P_{t}/A(l,f)}{N(f)B(f)}\propto\frac{ 1}{A(l,f)N(f)}, \tag{8}\]  
where \(B(f)\) is the usable signal bandwidth around the center frequency. We assume a narrow-band model, where the channel gain can be approximated as being constant in the signal bandwidth [28]. The \(A(l,f)N(f)\) product shows the frequency-dependent part of the SNR. Therefore, for a specific transmit distance \(l\), there exists an optimal carrier frequency that maximizes the SNR. According to experimental results, it was shown in [29] that the small-scale propagation effects, due to multipath fading, in the underwater acoustic channel can be well modeled by the \(K\)-distribution probability density function,  
\[f(x)=\frac{4}{\sqrt{\alpha}\ \Gamma(\
u)}\left(\frac{x}{\sqrt{\alpha}}\right)^{ \
u}K_{\
u-1}\left(\frac{2x}{\sqrt{\alpha}}\right), \tag{9}\]  
where \(\
u\) is a shape parameter, \(\alpha\) is a scale parameter, \(K_{\
u-1}(\cdot)\) is the modified Bessel function of second kind and order \(\
u-1\), and \(\Gamma(\cdot)\) is the Gamma function. We assume a scenario where the transmission system operates at a fixed frequency and under a fast fading channel [30]. Moreover, in our subsequent analysis, we consider a non-coherent binary FSK modulation scheme, whose BEP in the \(K\)-distributed fading is given by [29]  
\[P_{b}(\tilde{\gamma})=\frac{\
u}{\Gamma(\
u)}\ \int_{0}^{\infty}\frac{u^{(v-1)}e ^{-u}}{2\
u+u\ \tilde{\gamma}}\ du, \tag{10}\]  
where the average received SNR is expressed as \(\tilde{\gamma}=E_{b}/N(f)\). The parameter \(E_{b}=P_{r}/R_{b}\) represents the average energy per bit. The electric transmit power \(P_{PA}\) of the power amplifier (PA) can be obtained from the acoustic transmit power \(P_{t}\) (in \(\mu\)Pa) using the following relation [31]  
\[10\log_{10}\big{[}a(f)\big{]}=\frac{0.11f^{2}}{1+f^{2}}+\frac{44f^{2}}{4100+f^ {2}}+\frac{2.75f^{2}}{10^{4}}+0.003\ \ \mathrm{dB/km}. \tag{2}\]0  
where \(10^{-17.2}\) is a conversion factor and \(\varphi\) is the overall efficiency of the electric power amplifier plus transducer.