# Research on Co-Channel Interference Cancellation for Underwater Acoustic MIMO Communications
## 2 MIMO System Design with CoI Cancellation
### Performance Analysis  
For convenience, and based on the fact that most underwater acoustic MIMO communication systems utilize two transmitters, in this subsection, we analyze the performance using two transmitters (\(M=2\)). Consider the proposed channel model in (12); we used the LS method to analyze the performance. This assumed that the estimated channel is \(\hat{\mathbf{h}}_{1,n}\), which can be achieved by least square,  
\[\hat{\mathbf{h}}_{1,n}=(\mathbf{\Phi}_{1}^{H}\mathbf{\Phi}_{1})^{-1}\mathbf{ \Phi}_{1}\mathbf{p}_{\mathbf{y}\mathbf{a}1}. \tag{22}\]  
Notation \(\mathbf{\Phi}_{1}\) is defined as \(\mathbf{P}_{2}\mathbf{A}_{1}\) and \(\mathbf{p}_{\mathbf{y}\mathbf{a}1}\) is defined as \(\mathbf{P}_{2}\mathbf{y}_{n}\), where \(\mathbf{P}_{2}=(\mathbf{I}-\mathbf{A}_{2}(\mathbf{A}_{2}^{H}\mathbf{A}_{2})^{ -1}\ \mathbf{A}_{2}^{H})\). Defining the channel mean square error (MSE) is  
\[MSE=\mathbb{E}\|\hat{\mathbf{h}}_{1,n}-\mathbf{h}_{1,n}\|_{2}^{2}\, \tag{23}\]  
where \(\mathbf{h}_{1,n}\) is the true channel estimate. Then, the channel MSE of \(\hat{\mathbf{h}}_{1,n}\) is \[MSE_{\mathbf{h}_{1,n}} =\mathbb{E}\{\|(\mathbf{\Phi}_{1}^{H}\mathbf{\Phi}_{1})^{-1}\mathbf{ \Phi}_{1}(\mathbf{\Phi}_{1}\mathbf{h}_{1,n}+\mathbf{P}_{\mathbf{w}_{1}})- \mathbf{h}_{1,n}\|_{2}^{2}\} \tag{24}\] \[=\mathbb{E}\{\|(\mathbf{\[PERSON]}_{1}^{H}\mathbf{\[PERSON]}_{1})^{-1} \mathbf{\Phi}_{1}\mathbf{P}_{\mathbf{w}_{1}}\|_{2}^{2}\}\] \[=\mathbb{E}\{tr(\mathbf{P}_{\mathbf{w}_{1}}^{H}\mathbf{\Phi}_{1}( \mathbf{\Phi}_{1}^{H}\mathbf{\[PERSON]}_{1})^{-2}\mathbf{\[PERSON]}_{1}^{H}\mathbf{P}_{ \mathbf{w}_{1}})\}\] \[=tr(\mathbf{\[PERSON]}_{1}^{H}\mathbb{E}\{\mathbf{P}_{\mathbf{w}_{1}} \mathbf{P}_{\mathbf{w}_{1}}^{H}\}\mathbf{\[PERSON]}_{1}(\mathbf{\[PERSON]}_{1}^{H} \mathbf{\[PERSON]}_{1})^{-2})\] \[=tr(\mathbb{E}\{\mathbf{P}_{\mathbf{w}_{1}}\mathbf{P}_{\mathbf{w }_{1}}^{H}\}(\mathbf{\[PERSON]}_{1}^{H}\mathbf{\[PERSON]}_{1})^{-1})=\frac{\mathbb{E}\{ \mathbf{P}_{2}\mathbf{w}_{n}\mathbf{w}_{n}^{H}\mathbf{P}_{2}^{H}\}}{tr( \mathbf{A}_{1}^{H}\mathbf{P}_{2}^{H}\mathbf{P}_{2}\mathbf{A}_{1})}\] \[=\frac{\sigma_{\mathbf{w}_{n}}^{2}tr(\mathbf{P}_{2}\mathbf{P}_{2} ^{H})}{tr(\mathbf{A}_{1}^{H}\mathbf{P}_{2}^{H}\mathbf{P}_{2}\mathbf{A}_{1})}\;.\]  
In (24), notation \(\sigma_{\mathbf{w}_{n}}^{2}\) is the variance of ambient noise \(\mathbf{w}_{n}\). As \(\mathbf{P}_{2}^{H}\) is a projection matrix, it satisfies  
\[\mathbf{P}_{2}^{H}\mathbf{P}_{2}=\mathbf{P}_{2}\mathbf{P}_{2}^{H}=\mathbf{P}_ {2}^{H}\approx\frac{P-L}{P}\mathbf{I}\;. \tag{25}\]  
Appendix A I proves (25). Then, we can easily obtain  
\[MSE_{\mathbf{h}_{1}}=\frac{\sigma^{2}}{tr(\mathbf{A}_{1}^{H}\mathbf{A}_{1})}\;. \tag{26}\]  
Analogously, consider the traditional way that does not consider CoI; the MSE for \(\mathbf{h}_{1,n}\) is  
\[MSE=\mathbb{E}\|\hat{\mathbf{h}}_{1,n}-\mathbf{h}_{1,n}\|_{2}^{2}\, \tag{23}\]0 \[MSE=\mathbb{E}\|\hat{\mathbf{h}}_{1,n}-\mathbf{h}_{1,n}\|_{2}^{2}\, \tag{23}\]1  
It is assumed that the transmitted symbols are white sequences and independent of ambient noise \(\mathbf{w}_{n}\) and channel estimate; therefore, we can obtain  
\[MSE=\mathbb{E}\|\hat{\mathbf{h}}_{1,n}-\mathbf{h}_{1,n}\|_{2}^{2}\, \tag{23}\]2  
As a result, the channel MSE, obtained in the traditional way, is  
\[MSE=\mathbb{E}\|\hat{\mathbf{h}}_{1,n}-\mathbf{h}_{1,n}\|_{2}^{2}\, \tag{23}\]3  
Comparing (26) to (29), one may see that the MSE obtained by the proposed method is far less than that obtained by traditional method, because the proposed method adopts the orthogonal projection method to remove the CoI. The more accurate channel estimate will provide more precise channel information to CE-DFE. It is assumed that the estimated channel consists of the true channel estimate \(\hat{\mathbf{G}}_{m,n}\) and the erroneous channel estimate matrix \(\mathbf{E}_{\mathbf{G}_{m,n}}\).