# Low-Complexity Progressive MIMO-OFDM Receiver for Underwater Acoustic Communication
## 4 ICI Aware Progressive MIMO-OFDM Iterative Receiver
#### ICI Equalization of MIMO System  
Based on Equation (5), the FFT outputs of the \(N_{r}\) receivers associated with the transmitted symbols on the \(m\)th subcarrier of all data streams can be written as:  
\[\left[ \begin{array}{c}\mathbf{z}_{1,m}\\ \mathbf{z}_{2,m}\\ \vdots\\ \mathbf{z}_{N_{r},m}\end{array} \right]=\left[ \begin{array}{ccc}\mathbf{H}_{1,1,m}^{D}&\cdots&\mathbf{H}_{1,N_{r},m}^{D}\\ \mathbf{H}_{2,1,m}^{D}&\cdots&\mathbf{H}_{N_{r},N_{r},m}^{D}\\ \vdots&\ddots&\vdots\\ \mathbf{H}_{N_{r},1,m}^{D}&\cdots&\mathbf{H}_{N_{r},N_{r},m}^{D}\\ \end{array} \right]\left[ \begin{array}{c}\mathbf{s}_{1,m}\\ \mathbf{s}_{2,m}\\ \vdots\\ \mathbf{s}_{N_{r},m}\end{array} \right]+\left[ \begin{array}{c}\mathbf{\eta}_ {1,m}\\ \mathbf{\eta}_{2,m}\\ \vdots\\ \mathbf{\eta}_{N_{r},m}\end{array} \right], \tag{19}\]  
where  
\[\mathbf{z}_{v,m}=[z_{v}[m-D],\ldots,z_{v}[m+D]]^{T}, \tag{20}\]  
\[\mathbf{s}_{v,m}=[s_{v}[m-2D],\ldots,s_{v}[m+2D]]^{T}, \tag{21}\]  
\[\mathbf{\eta}_{v,m}=[\eta_{v}[m-D],\ldots,\eta_{v}[m+D]]^{T} \tag{22}\]  
The channel matrix \(\mathbf{H}_{v,\mu}^{D}\) of dimensions (2D+1) \(\times\) (4D+1) can be expressed as:  
\[\mathbf{H}_{v,\mu}^{D}=\left[ \begin{array}{ccc}H_{v,\mu}^{D}[m-D,m-2D]& \cdots&H_{v,\mu}^{D}[m-D,m]\\ &\ddots&\cdots&\ddots\\ &&H_{v,\mu}^{D}[m+D,m]&H_{v,\mu}^{D}[m+D,m+2D]\end{array} \right] \tag{23}\]  
If the value of \(D\) is zero, it means there is no ICI in the system and the channel matrix will be a strip matrix having non-zero values only at the diagonals. With the increase in value of \(D\), the channel matrix gets complex and the number of non-zero entries increase. The purpose of the MIMO system ICI equalization is to estimate the transmitted information symbol \(\mathbf{s}\) while mitigating the effects of ICI in the system. This paper uses the ICI equalization method based on a priori information (mean and variance) and MMSE criteria [24]. The specific steps are described as follows:  
Let the expectation of the \(u\)th transmitter subcarriers be \(\vec{s}_{\mu}[m]:=E(s_{\mu}[m])\), the variance be \(\vec{\sigma}_{\mu}^{2}:=Cov(s_{\mu}[m],s_{\mu}[m])\), and \(\mathbf{h}_{v,\mu,m}\) be the (D+1)th column of the matrix \(\mathbf{H}_{v,\mu,m}^{D}\), then  
\[\mathbf{h}_{v,\mu,m}:=\left[H_{v,\mu}^{D}[m-D,m],\ldots,H_{v,  
To estimate \(s_{\mu}[m]\), we need to use the information of the adjacent subcarriers \(\mathbf{s}_{\mu,m}^{-}\) of the current data stream and the data symbols that will interfere with the estimated symbols from other data streams \((\mathbf{s}_{1,m},\ldots,\mathbf{s}_{\mu-1,m},\mathbf{s}_{\mu+1,m},\ldots, \mathbf{s}_{N_{\mu},m})^{T}\). The mean and variance of \(\mathbf{s}_{\mu,m}^{-}\) are:  
\[\eta_{\mu,m}^{-}=E(\mathbf{S}_{\mu,m}^{-}), \tag{27}\]  
\[\phi_{\mu,m}^{-}=Cov(\mathbf{S}_{\mu,m}^{-},\mathbf{S}_{\mu,m}^{-}) \tag{28}\]  
During the initial iteration of equalization, the mean and variance of \(s_{\mu}[m]\) are unknown and can be set to 0 and \(E_{s}\), respectively, i.e., \(E(s_{\mu}[m])=0\) and \(\sigma_{\mu}^{2}[m]=E_{s}\). In addition, the equivalent noise \(\mathbf{n}_{\gamma,m}\) is assumed to be Gaussian white noise with a covariance matrix of \(\hat{N}_{0,\sigma}\mathbf{I}_{2D+1}\), where \(v=1,2,\ldots,N_{r}\). Based on Equation (24), the MMSE estimate for \(s_{\mu}[m]\) is:  
\[ \begin{array}{ll}\hat{s}_{\mu}[m]&=Cov(s_{\mu}[m],\mathbf{z}_{m})Cov(\mathbf{ z}_{m},\mathbf{z}_{m})^{-1}(\mathbf{z}_{m}-\mathbf{H}_{\mu,m}^{D-}\eta_{\mu,m}^{ -}),\\ &=\mathbf{f}_{\mathbf{m}}\Big{(}\mathbf{z}_{m}-\mathbf{H}_{\mu,m}^{D-}\eta_{ \mu,m}^{-}\Big{)}\end{array}