## 3 Performance Analysis
### BER Source Analysis  
Noise and multipath interference in the maritime communication channel will affect the system performance. Taking the additive Gaussian white noise (AWGN) channel as an example, the received signal can be expressed as  
\[r(t)=s(t)+n(t), \tag{12}\]  
where \(r(t)\), \(s(t)\) and \(n(t)\) represent the received signal, transmitted signal and Gaussian white noise vectors, respectively. It can be seen that the presence of noise will cause the coherent computation to induce a deviation, which can be denoted as \(\Delta G\), and therefore the solution of the system of linear equations must have a deviation, \(\Delta A\). Then, Equation (5) will become  
\[A+\Delta A= R^{-1}(G+\Delta G). \tag{13}\]  
Since \(RA\equiv G\), we have  
\[\Delta A= R^{-1}\Delta G. \tag{14}\]  
It can be seen that noise directly causes the deviation in the system of linear equations. According to the nature of the matrix norm, we have  
\[\|\Delta A\|\leq\|R^{-1}\|\cdot\|\Delta G\|. \tag{15}\]  
It can be seen that the deviation \(\Delta A\) can be decreased by reducing the norm of the matrix \(R^{-1}\). From Equations (2) and (3), it can be seen that the elements of matrix \(R\) are related to carrier parameters, such as the frequency \(f\), delay \(\tau\), and the total number of carriers \(N\). Therefore, to ensure good system performance, the carrier parameters must be designed appropriately. In the maritime multipath channel, the performance of the TDOM will be severely deteriorated because delays have been artificially introduced into the communication symbol. These delays, superimposed on the delays in the multipath channel, will cause serious interference to the system of linear equations constructed at the receiver. Assuming that the total number of paths is \(L\), the delay of each path is \(\tau_{l}\) (\(l=1,\ldots,L\)), and the path loss is \(\alpha_{l}\), the received signal \(r(t)\) will become  
\[ \begin{array}{rl}r(t)&=\sum\limits_{l=1}^{L}\alpha_{l}s(t-\tau_{l})+n(t)\\ &=a_{0}\sum\limits_{l=1}^{L}\alpha_{l}sin[2\pi f_{0}(t-\tau_{l})]+a_{1}\sum \limits_{l=1}^{L}\alpha_{l}sin[2\pi f_{1}(t-\tau-\tau_{l})]+\ldots\\ &+a_{N}\sum\limits_{l=1}^{L}\alpha_{l}sin[2\pi f_{N}(t-N\tau-\tau_{l})]+n(t), \end{array}  \tag{16}\]  
At this point, the coherence computation will become  
\[ \begin{array}{rl}\mathbf{G}_{n}&=\int_{0}^{T}sin[2\pi f_{n}(t-n\tau)]\times r (t)\\ &=\int_{0}^{T}sin[2\pi f_{n}(t-n\tau)]\times\left\{a_{0}\sum\limits_{l=1}^{L} \alpha_{l}sin[2\pi f_{0}(t-\tau_{l})]\right.\left.+a_{1}\sum\limits_{l=1}^{L} \alpha_{l}sin[2\pi f_{1}(t-\tau-\tau_{l})]+\ldots\\ &+a_{N}\sum\limits_{l=1}^{L}\alpha_{l}sin[2\pi f_{N}(t-N\tau-\tau_{l})]+n(t) \right\}dt\\ &=a_{0}r_{n0}^{\prime}+a_{1}r_{n1}^{\prime}+\ldots+a_{N}r_{nN}^{\prime}+ \Delta\mathbf{G}_{n_{r}}(n=0,\ldots,N)\\ &\text{where}\\ &r_{ij}^{\prime}=\int_{0}^{T}sin[2\pi f_{i}(t-i\tau)]\times\sum \limits_{l=1}^{L}\alpha_{l}sin[2\pi f_{i}(t-j\tau-\tau_{l})]dt,(i=0,\ldots,N,j=0,\ldots,N)\\ &\Delta\mathbf{G}_{n}=\int_{0}^{T}sin[2\pi f_{n}(t-n\tau)]\times n(t)dt.\end{array}  \tag{19}\]  
From the above analysis, it can be seen that the right-side term of the equation \(RA=G\) contains not only the interference from noise, but also the inter-carrier interference (ICI) from multipath channels. Compared with the noise, the interference of the multipath channel will create great errors in the solution of the linear equation system. Meanwhile, the TDOM system will also suffer severe inter-symbol-interference (ISI) in the maritime multipath channel. Therefore, in multipath channels, channel estimation and channel equalization methods must be adopted to weaken the ISI and ICI to ensure good communication performance. It is worth noting that, for the TDOM, a simpler channel estimation and channel equalization method that does not require the accurate knowledge of the channel state information can be implemented.