## 2 Problem Formulation  
### Model of MIMO Transmission  
Consider MIMO system with \(\mathbf{N}_{\mathrm{T}}\) transmitting antennas and \(\mathbf{N}_{\mathrm{R}}\) receiving antennas. Let \(\widetilde{\widetilde{\mathbf{x}}}\in\mathbb{C}^{\mathbf{N}_{\mathrm{T}}\times 1}\) (\(\mathbb{C}^{\mathbf{N}\times m}\) denotes the \(\mathbf{N}\times m\) complex spaces) be a transmitting signal vector, satisfying \(\mathbb{E}_{\widetilde{\widetilde{\mathbf{x}}}}\{\widetilde{\widetilde{x}}\}=0\) and \(\mathbb{E}_{\widetilde{\widetilde{\mathbf{x}}}}\{\widetilde{\widetilde{ \mathbf{x}}}\widetilde{\widetilde{\mathbf{x}}}^{\mathrm{H}}\}=I\), where \(\mathbb{E}_{(\bullet)}\{*\}\) stands for the statistical expectation of random \(*\) with respect to its variable \(\cdot;I\) and \(0\) denote an unit and zero matrix of appropriate dimensions, respectively. \((\bullet)^{\mathrm{H}}\) is the conjugate transpose of matrix \(\cdot\) MIMO transmission is generally modeled by  
\[\widetilde{\widetilde{\mathbf{y}}}=\widetilde{\widetilde{H}}\widetilde{ \widetilde{\mathbf{x}}}+\widetilde{\widetilde{w}} \tag{1}\]  
where preliminaries are made as below.  
1. \(\widetilde{\widetilde{\mathbf{H}}}\in\mathbb{C}^{\mathbf{N}_{\mathrm{R}}\times \mathbf{N}_{\mathrm{T}}}\) is a complex fading channel matrix between transmitting antenna and receiving antenna arrays. The doubly correlated MIMO Rayleigh fading channel is modeled by \(\mathbf{\Psi}_{\mathrm{R}}^{1/2}\widetilde{\widetilde{H}}_{\mathrm{WGC}} \mathbf{\Psi}_{\mathrm{T}}^{1/2}\in\mathbb{C}^{\mathbf{N}_{\mathrm{R}}\times \mathbf{N}_{\mathrm{T}}}\)[1], where \(\widetilde{\widetilde{H}}_{\mathrm{WGC}}\in\mathbb{C}^{\mathbf{N}_{\mathrm{R}} \times\mathbf{N}_{\mathrm{T}}}\) is a matrix consisted of independent and identically distributed \(\mathcal{CN}(0,1)\) complex Gaussian entries; \(\boldsymbol{\
abla}_{\text{T}}\) and \(\boldsymbol{\
abla}_{\text{R}}\) are transmitting and receiving correlation matrices, respectively. \(\boldsymbol{\
abla}_{\text{T}}\) and \(\boldsymbol{\
abla}_{\text{R}}\) can be expressed as  
\[\boldsymbol{\
abla}_{\text{T}}=\boldsymbol{U}_{\text{T}}\boldsymbol{\Sigma}_{ \text{T}}\boldsymbol{U}_{\text{T}}^{\text{H}}\text{ and }\boldsymbol{\
abla}_{\text{R}}= \boldsymbol{U}_{\text{R}}\boldsymbol{\Sigma}_{\text{R}}\boldsymbol{U}_{\text{ R}}^{\text{H}} \tag{2}\]  
where \(\boldsymbol{U}_{\text{T}}\) and \(\boldsymbol{U}_{\text{R}}\) are unitary matrices whose columns are eigenvectors of \(\boldsymbol{\
abla}_{\text{T}}\) and \(\boldsymbol{\
abla}_{\text{R}}\); \(\boldsymbol{\Sigma}_{\text{T}}\) and \(\boldsymbol{\Sigma}_{\text{R}}\) represent diagonal matrices whose diagonal entries are the eigenvalues of \(\boldsymbol{\
abla}_{\text{T}}\) and \(\boldsymbol{\
abla}_{\text{R}}\), respectively.
2. \(\widetilde{\widetilde{\boldsymbol{w}}}\in\mathbb{C}^{N_{\text{R}}\times 1}\) stands for AWGN corresponding to \(\text{N}_{\text{R}}\) receiving antennas, where each element is independent and identically complex Gaussian distributed, satisfying \(\mathbb{E}_{\widetilde{\widetilde{\boldsymbol{w}}}}\left\{\widetilde{ \widetilde{\boldsymbol{w}}}\right\}=0\) and \(\mathbb{E}_{\widetilde{\widetilde{\boldsymbol{w}}}}\left\{\widetilde{ \widetilde{\boldsymbol{w}}}\widetilde{\widetilde{\boldsymbol{w}}}^{\text{H}} \right\}=\sigma^{2}\boldsymbol{I}\).