## 4 Bionic Morse Coding Method
### Information Modulation
#### 4.2.2 Step 2: Channel Estimation and Equalization  
The acoustic channel in the underwater sea environment degrades the communication signal poorly due to multipath arrivals, frequency and time variance [39]. These variations are adjusted by estimating the channel and equalizing the errors through the estimated channel impulse response. The goal of channel estimation and equalization is to enhance the performance of system. In this research, the matching pursuit (MP) estimation technique is opted for due to its robustness in changing the SNR and higher computational efficiency [40]. It sequentially identifies and estimates channel tap coefficients. Let us consider that the multipath channel has Additive White Gaussian Noise (AWGN) with variance \(\sigma^{2}\). The received signal can be written as  
\[y(t)=\sum_{l=0}^{N-1}x(t-l)*h(l)+n(t) \tag{4}\]  
where \(t\) = 0, 1, 2..., \(N-1\), \(x(t)\) is the transmitted symbol, \(h(t)\) is the \(t\)th channel tap and \(n(t)\) is the AWGN. The equation can be expanded as  
\[y(t)=\sum_{l=0}^{N-1}x(t-l)*h(l)+n(t) \tag{4}\]  
where \(t\) = 0, 1, 2..., \(N-1\), \(x(t)\) is the transmitted symbol, \(h(t)\) is the \(t\)th channel tap and \(n(t)\) is the AWGN. The equation can be expanded as  
\[\underbrace{\left[ \begin{array}{c}y(0)\\ y(1)\\ \vdots\\ y(N-1)\end{array} \right]}_{Y}=\underbrace{\left[ \begin{array}{cccc}x(0)&0& \cdots&0\\ x(1)&x(0)&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ x(N-1)&\cdots&\cdots&x(N-l)\end{array} \right]}_{X}\underbrace{\left[ \begin{array} []{c}h(0)\\ h(1)\\ \vdots\\ h(N-1)\end{array} \right]}_{B}+v(n) \tag{5}\]  
Figure 8: Detection of the synchronization waveform at the receiver. It can be seen from Equation (4) that the majority of the components of channel tap are extremely small or zero. Therefore, the columns of \(\mathbf{X}\) are added to get the received data. The residual \(\mathbf{r}_{i-1}\) is acquired by the difference of \(\mathbf{Y}\) and \(\mathbf{X}\) for \(i-1\) iterations, considering \(\mathbf{r}_{0}=\mathbf{Y}\). At \(i\)th iteration, the column of \(\mathbf{X}\) onto which the residual \(\mathbf{r}_{i-1}\) has the maximal rank-one projection denoted as \(x_{i}\) is equated as  
\[x_{i}=\underset{j\
eq i_{l-1}}{\operatorname{argmax}}\frac{\left|\mathbf{X}_{ j}^{H}\mathbf{r}_{i-1}\right|^{2}}{\left\|\mathbf{X}_{j}\right\|^{2}} \tag{6}\]  
where, \(\mathbf{X}_{j}\) represents the column of \(\mathbf{X}\) and superscript \(H\) is defined for Hermitian Transpose. \(\mathbf{I}_{i}\triangleq\{x_{1},x_{2},x_{3},\ldots,x_{i-1}\}\) is the index set of all previously selected columns. The component of \(\mathbf{h}\) termed as \(\hat{h}_{i}\) associated by \(x_{i}\) is estimated as  
\[\hat{h}_{i}=\frac{\mathbf{X}_{x_{i}}^{H}\mathbf{r}_{i-1}}{\left\|\mathbf{X}_ {x_{i}}\right\|^{2}} \tag{7}\]  
Hence, the residual vector is calculated as  
\[\mathbf{r}_{i}=\mathbf{r}_{i-1}-\hat{h}_{i}\mathbf{X}_{x_{i}} \tag{8}\]  
The objective of the channel estimation and equalization is to enhance the performance of the system. It reduces the bit error rate by evaluating the effect of multipath, time and frequency variance in an underwater acoustic channel. The channel equalization adds the distributed energy of multipath arrivals projected by the channel estimation technique which reduce errors. We equalize the errors using a virtual time reversal method (VTRM). The framework of the VTRM is shown in Figure 9. The estimated channel \(h^{\prime}(t)\) is inverted in time domain \(h^{\prime}(-t)\) and convolved with \(x(t)\). The final virtual signal received is acquired by convolving the received signal by the inverted estimated channel. Therefore:  
\[x_{r}(t)=y(t)\otimes h^{\prime}(-t) \tag{9}\]  
\[x_{r}(t)=\left[x(t)\otimes h(t)+n(t)\right]\otimes h^{\prime}(-t) \tag{10}\]  
\[x_{r}(t)=x(t)\otimes\left[h(t)\otimes h^{\prime}(-t)\right]+\left[n(t)\otimes h ^{\prime}(-t)\right] \tag{11}\]  
where \(x_{r}(t)\) is the equalized signal and \(\otimes\) represents the convolution operator.