# Research on Co-Channel Interference Cancellation for Underwater Acoustic MIMO Communications
## 2 MIMO System Design with CoI Cancellation
### Channel Estimation Based Decision Feedback Equalization with IC  
Considering the channel equalization model, (1) can be written as  
\[\mathbf{y}_{n}=\sum_{m=1}^{M}\mathbf{G}_{m,n}\mathbf{x}_{m}+\mathbf{w}_{n}. \tag{13}\]  
where \(\mathbf{G}_{m,n}\) is the channel matrix from the \(m\)-th transmitter to the \(n\)-th hydrophone, and \(\mathbf{w}_{n}\) is the ambient noise. Notations \(\mathbf{y}_{n}\), \(\mathbf{x}_{m}\), \(\mathbf{w}_{n}\), and \(\mathbf{G}_{m,n}\) are defined as follows  
\[\mathbf{y}_{n}\triangleq(y_{n}(i+L_{c})\cdots y_{n}(i)\cdots y_{n}(i-L_{a}+1) )^{T}\, \tag{14}\]  
\[\mathbf{x}_{m}\triangleq(x_{m}(i+L_{c})\cdots x_{m}(i)\cdots x_{m}(i-L_{a}+1 ))^{T}\, \tag{15}\]  
\[\mathbf{w}_{n}\triangleq(w_{n}(i+L_{c})\cdots w_{n}(i)\cdots w_{n}(i-L_{a}+1 ))^{T}\, \tag{16}\]  
\[\mathbf{G}_{m,n}\triangleq\left(\begin{array}{c|c|c|c}h_{m,n}(0)&\cdots&0& \cdots&0\\ \vdots&\ddots&\vdots&\ddots&0\\ 0&\cdots&h_{m,n}(L-1)&\cdots&0\\ \vdots&\ddots&\vdots&\ddots&\vdots\\ 0&\cdots&h_{m,n}(0)&\cdots&h_{m,n}(L-1)\\ \end{array}\right), \tag{17}\]  
\[\mathbf{G}_{m,n}=\left(\mathbf{G}_{0_{1}}|\mathbf{g}_{1}|\mathbf{F}_{1}\right). \tag{18}\]  
In (14)-(16), \(L_{c}\) and \(L_{a}\) denote the number of causal and a-causal taps, respectively. The lengths of \(\mathbf{g}_{ff}\), \(\mathbf{g}_{fb}\), and \(\mathbf{g}_{ic}\) are \(L_{ff}\), \(L_{fb}\), and \(L_{ic}\), respectively. We partition the transmitted symbols into three groups \(\mathbf{x}_{m}=(\mathbf{x}_{fb_{m}}^{T},x_{m}(i),\mathbf{x}_{0_{m}}^{T})^{T}\). Notations \(\mathbf{x}_{fb_{m}}\) and \(\mathbf{x}_{0_{m}}\) are defined as \(\mathbf{x}_{fb_{m}}^{T}\triangleq(x_{m}(i+L_{fb}),x_{m}(i+L_{fb}-1),\cdots,x_ {m}(i+1))^{T}\) and \(\mathbf{x}_{0_{m}}\triangleq(x_{m}(i-1),\cdots,x_{m}(i-1),\cdots,x_{m}(i-1))^{T}\).  
\(L_{a}-L_{c}-L+1)^{T}\). Similarly, the channel matrices \(\mathbf{G}_{m,n}\) are partitioned into three groups, as shown in (18). Then, (13) can be written as  
\[\mathbf{y}_{n}=\sum_{m=1}^{M}(\mathbf{g}_{m}x_{m}(i)+\mathbf{F}_{m}\mathbf{x}_{ fb_{m}}+\mathbf{G}_{0_{m}}\mathbf{x}_{0_{m}})+\mathbf{w}_{n}. \tag{19}\]  
From (19), we can observe that the symbols from all transmitters interfere with each other; if no cancellation is applied, the signal-to-noise ratio for each branch will be significantly degraded. In (19), the terms \(\mathbf{g}_{m}x_{m}(i)\) are the transmitted symbols that need to be recovered. As an example to recover \(x_{1}(i)\), the second term \(\mathbf{F}_{1}\mathbf{x}_{fb_{1}}\) should be cancelled by feedback filter, and the terms \(\sum_{m=2}^{M}\mathbf{F}_{m}\mathbf{x}_{fb_{m}}\) should be cancelled by IC filters. The remaining terms are the effective observation noise that the feedforward filters must try to eliminate.  
According to Figure 1, \(x_{s_{m}}\) can be represented by  
\[x_{s_{m}}=\mathbf{g}_{ff_{m}}^{H}\mathbf{y}_{n}+\mathbf{g}_{fb_{m}}^{H} \mathbf{x}_{fb_{m}}+\sum_{i=1,i\
eq m}^{M}(\mathbf{g}_{ic_{m,i}}^{H}\mathbf{x} _{fb_{i}}). \tag{20}\]  
Finally, the coefficients of the filters are measured as (21) (see [16] for complete derivations).  
\[\mathbf{g}_{ff_{m}}=(\sum_{k=1}^{M}(\mathbf{g}_{k}\mathbf{g}_{k}^{H}+\mathbf{ D}_{k}))^{-1}\mathbf{g}_{m}. \tag{21a}\] \[\mathbf{g}_{fb_{m}}=-\mathbf{F}_{m}\mathbf{g}_{ff_{m}}\,\] (21b) \[\mathbf{g}_{ic_{m,k}}=-\mathbf{F}_{k}\mathbf{g}_{ff_{m}},\ k\
eq m. \tag{21c}\]  
From (21), as we expected, the filters of channel equalization are obtained via channel estimates. The traditional channel estimation method does not perform interference cancellation, and the filters \(\mathbf{g}_{ff}\), \(\mathbf{g}_{fb}\), and \(\mathbf{g}_{ic}\) will have plenty of noise; as a result, the channel equalization performance will degrade. In addition, traditional CE-DFE does not have IC filters to eliminate the CoI. In this paper, the proposed method has the functionality to remove the interference in both channel estimation and channel equalization; therefore, the communication performance will be significantly improved.