# A Method for Underwater Wireless Data Transmission in a Hydroacoustic Channel under NLOS Conditions
## 3 Model of the Multipath Channel  
Let the discrete hydroacoustic signal \(x(n)\) and the channel impulse response (CIR) \(h(n)\) of the multipath channel in the SISO (single input single output) system be given. Assuming that the channel is stationary in the considered time interval, the received signal can be expressed as a sum of \(M\) multipath components of a certain amplitude \(a_{i}\) reaching the receiver with a certain delay \(n_{i}\) in the presence of noise \(w(n)\), i.e., in the form:  
\[y(n)=\sum_{i=1}^{M}a_{i}x(n-n_{i})+w(n), \tag{6}\]  
where: \(n\)--a discrete time (sample).  
The impulse response characterizes the propagation conditions of the acoustic wave along the path between the transmitter and receiver, including multipath conditions. The form of this response will depend on the mutual spatial position of the transmitter and receiver in the water area. Accordingly, the received signal may be represented as a convolution of the transmitted signal, with the impulse response of the channel in the presence of noise \(w(n)\)[22]:  
\[y(n)=h(n)*x(n)+w(n). \tag{7}\]  
In the case of a SIMO (single input multiply output) system, i.e., with one transmitter and \(K\) receivers, the relationship between the output and input signals can be expressed as a matrix equation [20]:  
\[\mathbf{y}(n)=(\mathbf{H}*x)(n)+\mathbf{w}(n), \tag{8}\]  
where: vector \(\mathbf{y}(n)\) having a dimension of \((K\times 1)\) contains instantaneous values of signals received at time \(n\) by individual hydrophones, vector \(\mathbf{w}(n)\) having a dimension of \((K\times 1)\) represents instantaneous noise values at time \(n\) in each receiving channel, vector \(\mathbf{H}(n)\) having a dimension of \((K\times 1)\) describes the propagation conditions of the hydroacoustic channel, where the elements \(h_{1}(n),\dots,h_{K}(n)\) of this vector represent the impulse response of the hydroacoustic channel between the transmitting hydrophone and the \(k\)-th receiving hydrophone.  
In wireless communication systems distributions such as Rayleigh, Weibull, Nakagami are commonly used to model propagation conditions.  
In the matrix model given by (8), a vector \(\mathbf{w}(n)\) was introduced, representing the noise recorded in each receiver. Most often it is assumed that the noise has a Gaussian distribution, where the real part and the imaginary part of the noise \(w(n)\) are independent random variables with zero mean value and variance \(\frac{\sigma_{w}^{2}}{2}\)[20]:  
\[\left\{\begin{array}{l}Re[w(n)]\sim N\Big{(}0,\frac{\sigma_{w}^{2}}{2}\Big{)} \\ Im[w(n)]\sim N\Big{(}0,\frac{\sigma_{w}^{2}}{2}\Big{)}\end{array}\right., \tag{9}\]  
The power of such defined noise in each receiver will be the same, i.e., \(\sigma_{w}^{2}\).  
The above analysis concerned the time domain. In the proposed method of diversity combining, the signal spectra from individual hydrophones are summed up. Then the variance \(\sigma_{f}^{2}\) of the noise observed in the spectrum of the diversity combined signal decreases according to the equation [21]:  
\[\sigma_{f}^{2}=\frac{2^{1/(2L)}\left(\sigma_{w}^{2}\right)^{2}}{M}, \tag{10}\]  
where: \(M\)--the number of spectrum components, \(L\)--the number of spectra used to derive the mean value of the spectrum.