## 5 Results
#### Transmission Loss Threshold  
Now we compute the transmission loss threshold \(TL_{Th}\) or maximum propagation loss for properly receiving the transmitted signal given by (9). The link quality in underwater communication is severely affected by multipath phenomena, multipath fading and the refractive properties of the sound channel [3]. As a result, the bit error rates of the acoustic links are often high and efficient error control schemes are needed. The error control schemes automatic repeat request (ARQ), forward error correction (FEC) block codes and FEC convolutional codes have been selected and evaluated in terms of the packet error rate (PER).  
According to [3], OFDM transmission with QAM modulation has been used for our analysis. The scalable OFDM with 16-QAM modulation has been successfully tested in the settings with bandwidths 12 kHz, 25 kHz and 50 kHz, leading to data rates about 12 kbps, 25 kbps and 50 kbps respectively [23]. The BER for the modulation scheme 16-QAM is given by:  
\[p_{b}^{16 QAM}=\frac{3}{8}\textit{erfc}\Bigg{(}\sqrt{\frac{4}{10}\frac{E_{b}}{N _{0}}}\Bigg{)},\,\,E_{b}\,/\,N_{0}=\psi\,\frac{B_{N}}{R} \tag{24}\]  
where \(\psi\)=\(10^{\textit{wtB(d)/10}}\), \(B_{N}\) is the noise bandwidth and \(R\) is the data rate.  
Based on the bit error rate \(p_{b}\), the PER for the error control schemes can be calculated as follows:  
For ARQ, the cyclic redundancy check (CRC) block code detection mechanism is deployed. Assuming detection of all possible packet errors, the PER of a single transmission for a packet of \(l\) bits is computed as:  
\[PER^{CRC}\big{(}l\big{)}=1-\big{(}1-p_{b}\big{)} \tag{25}\]  
A BCH (Bose, Ray-Chaudhuri, Hocquenghem) and a RS (Reed-Solomon) FEC block code is represented by \((n,\,k,\,t)\), where \(n\) is the block length, \(k\) is the payload length, and \(t\) is the error correcting capability in bits. For the BCH and RS codes, the block error rate (BLER) is given by:  
\[BLER\big{(}n,k,t\big{)}=\sum_{i=t+1}^{n}\binom{n}{i}p_{b}^{i}\big{(}1-p_{b} \big{)}^{n-i} \tag{26}\]  
Since a packet can be larger than the block length \(n\), the PER for BCH and RS block codes is given by:  
Figure 12: Corresponding ray trace for a source depth of 350 m.  
\[PER^{BC}\left(l,n,k,t\right)=1-\left(1-BLER\left(n,k,t\right)\right)^{\left[\frac{t }{2}\right]} \tag{27}\]  
where \(\left\lceil l/k\right\rceil\) is the number of blocks required to send \(l\) bits and \(\left\lceil\.\.\right\rceil\) is the ceiling function.  
For FEC convolutional codes, the PER of a single transmission for a packet of \(l\) bits is given by:  
\[PER^{convolutional}\left(l,n,k\right)=1-\left(1-p_{b}\right)^{\left\lfloor\frac{t }{2}\right\rfloor} \tag{28}\]  
where \(p_{b}\) is the bit error probability of an encoded data packet of length of length \(\left\lceil l/R_{c}\right\rceil\) and code rate \(R_{c}=k/n\).  
The relationship between the PER and \(TL_{Th}\) for ARQ, three BCH, three RS and two convolutional codes is shown in Figure 13 for shallow water and in Figure 14 for deep water. For a target PER, FEC block codes can support \(TL_{Th}\) values higher than ARQ and convolutional codes. We notice that the maximum \(TL_{Th}\) value is increased with the error correcting capability of the FEC block codes and the code rate of the FEC convolutional codes. For a particular \(TL_{Th}\) value, the PER is equal or lower for error control codes in deep water in comparison with shallow water.  
The FEC convolutional code with \(R_{c}=1/2\) has the lowest \(TL_{Th}\) value. This is the worst case, because the maximum transmission loss is restricted to lower values. Therefore, with this error control  
Figure 14: PER as a function of transmission loss for deep water.  
Figure 13: PER as a function of transmission loss for shallow water.  
scheme for a target PER of \(10^{-2}\) the maximum \(TL_{Th}\) of 112.5 dB for shallow water and 129 dB for deep water have been selected. More details about the performance of error control schemes in UWSNs can be found in [20].