# Research on Co-Channel Interference Cancellation for Underwater Acoustic MIMO Communications
## 2 MIMO System Design with CoI Cancellation
### Projection Based Channel Estimation  
At present, the CoI cancellation is typically performed in channel equalization; however, the literature investigating the role of CoI cancellation in channel estimation is very limited. For CE-DFE, the filter taps are obtained from channel estimates; thus, the accuracy of the channel estimation determines the channel equalization performance. In this subsec  
Figure 1: The system overview for the proposed underwater acoustic MIMO communication. tion, we introduce a projection-based channel estimation. The signal received in baseband at \(n\)-th hydrophone can be expressed as:  
\[y_{n}(i)=\sum_{m=1}^{M}\big{(}\sum_{l=0}^{L-1}x_{m}(i-l)h_{m,n}(l)\big{)}+w_{n}( i). \tag{1}\]  
In (1), \(x_{m}(i)\) is transmitted symbols from \(m\)-th transmitter, where \(m=1,2,\cdots,M\), and \(M\) is the number of total transmitters. Notation \(y_{n}(i)\) is the received signal at \(n\)-th hydrophone; \(h_{m,n}(i)\) is the channel from \(m\)-th transmitter to \(n\)-th receiver; \(w_{n}(i)\) is the ambient noise. \(L\) is the channel length. It is assumed that the channel is time invariant over a number of \(S\) samples; then, (1) can be written in matrix form, such as  
\[\mathbf{y}_{n}=\sum_{m=1}^{M}\mathbf{A}_{m}\mathbf{h}_{m,n}+\mathbf{w}_{n}\, \tag{2}\]  
where  
\[\mathbf{A}_{m}=\left( \begin{array}{cccc}x_{m}(L-1)&x_{m}(L-2)&\cdots&x_{m}(0 )\\ x_{m}(L)&x_{m}(L-1)&\cdots&x_{m}(1)\\ \vdots&\vdots&\ddots&\vdots\\ x_{m}(L+S-2)&x_{m}(L+S-3)&\cdots&x_{m}(S-1)\end{array} \right)\, \tag{3}\]  
\[\mathbf{y}_{n}=\left( \begin{array}{cccc}y_{n}(L-1)&y_{n}(L)&\cdots&y_{n}(L+ S-2)\end{array} \right)^{T}, \tag{4}\]  
\[\mathbf{h}_{m,n}=\left( \begin{array}{cccc}h_{m,n}(L-1)&h_{m,n}(L-2)&\cdots&h _{m,n}(0)\end{array} \right)^{T}\, \tag{5}\]  
\[\mathbf{w}_{n}=\left( \begin{array}{cccc}w_{n}(L-1)&w_{n}(L)&\cdots&w_{n}(L+ S-2)\end{array} \right)^{T}. \tag{6}\]  
When estimating the channel from \(i\)-th transmitter \(\mathbf{h}_{i,n}\), the signals received from other transmitters are the co-channel interference. Traditional channel estimation methods do not consider the Col cancellation; the Col is treated as noise, and the SNR received from a specific transmitter is low. This results in a lower channel estimation and channel equalization accuracy. The orthogonal projection method is used to cancel such Cols. The main idea of the orthogonal projection is that a projection matrix is used to span the received signal to a new subspace, where the interference is free. Our previous work in [29] first applied the projection method to channel estimation, but only two transmitters in the near-far network were considered. In this paper, we extend more than two transmitters for underwater acoustic MIMO channel estimation using the orthogonal projection method. It is supposed that channel \(\mathbf{h}_{m,n}\) needs to be estimated, and the interference is canceled in order. That is to say, the interference from transmitter \(1\) is canceled first, and the interference from transmitter \(M\) is canceled last. The cancellation is iteratively performed until the signal from \(m\)-th transmitter remains, and interference from other transmitters is removed. When canceling the Col from transmitter \(1\), the projection is defined as \(\mathbf{P}_{1}=\mathbf{I}-\mathbf{A}_{1}(\mathbf{A}_{1}^{H}\mathbf{A}_{1})^{- 1})\mathbf{A}_{1}^{H}\). Left multiplying \(\mathbf{P}_{1}\) to (2), it can be obtained  
\[\mathbf{P}_{1}\mathbf{y}_{n}=\mathbf{P}_{1  
The process continues until all the interference is cancelled.  
In general, when sequentially canceling the interference from \(k\)-th transmitter, the projection matrix \(\mathbf{P}_{k}\) is defined as  
\[\mathbf{P}_{k}=\mathbf{I}-\mathbf{B}_{k}(\mathbf{B}_{k}^{H}\mathbf{B}_{k})^{-1} \mathbf{B}_{k}^{H}. \tag{9}\]  
The matrix \(\mathbf{B}_{k}\) is defined as  
\[\mathbf{B}_{k}=(\prod_{i=1}^{k}\mathbf{P}_{i-1})\mathbf{A}_{k}\, \tag{10}\]  
where \(\mathbf{P}_{0}=\mathbf{I}\), and \(\mathbf{P}_{m}=\mathbf{I}\).