## 3 Performance Analysis
### Path Diversity Analysis
Eigenvalue decomposition of the cross-correlation matrix should be firstly performed, ahead of solving the beamforming vectors \(\mathbf{w}\) and \(\mathbf{v}\) to maximize the SNR of receiver. It can be represented as:  
\[\mathbf{H}^{H}\mathbf{H}=\left[\mathbf{u}_{1},\mathbf{u}_{2},\cdots,\mathbf{u}_{ N_{H}}\right]diag\left(\lambda_{1},\lambda_{2},\cdots,\lambda_{N_{H}}\right) \left[\mathbf{u}_{1},\mathbf{u}_{2},\cdots,\mathbf{u}_{N_{H}}\right]^{H} \tag{14}\]  
where \(\lambda_{1}>\lambda_{2}>\cdots\lambda_{N}>0\) are the nonzero distinct eigenvalues, and \(\lambda_{1}\) and \(\mathbf{u}_{1}\) denote the maximal eigenvalue and its corresponding characteristic vector respectively. The optimal beamforming vector can be given by:  
\[\mathbf{w}=\mathbf{u}_{1} \tag{15}\]  
\[\mathbf{v}=\mathbf{H}\mathbf{u}_{1}/\left\|\mathbf{H}\mathbf{u}_{1}\right\|_{F} \tag{16}\]  
where \(\left\|\ \cdot\ \right\|_{F}\) represents the Frobenius norm operator. Further, the maximum SNR of the receiver is:  
\[\gamma=\lambda_{1}\gamma_{s} \tag{17}\]  
The beamforming schemes in Equations (15) and (16) are the maximal-ratio combining (MRC) and maximal-ratio transmitting (MRT) respectively. Letting \(N_{rH}=\min\left\{N_{H},N_{r}\right\}\), \(N_{Hr}=\max\left\{N_{H},N_{r}\right\}\), the PDF of \(\lambda_{1}\) can be then represented as:  
\[f_{\lambda_{1}}(x)=\frac{\exp(-tr(\uplambda))|\mathbf{\Psi}(x)|tr(\mathbf{ \Psi}^{-1}(x)\mathbf{\Psi}_{N_{rH}}(x))}{|V|\left(\Gamma(N_{Hr}-N_{rH}+1) \right)^{N_{rH}}}U(x) \tag{18}\]  
where \(\uplambda=diag\left(\lambda_{1},\lambda_{2},\cdots,\lambda_{N_{H}}\right)\), \(|\cdot|\) denotes the determinant operator, \(tr(\cdot)\) denotes the trace operator, \(\mathbf{\Psi}(x)\) is the \(N_{rH}\times N_{rH}\) matrix function, with entries \(\{\mathbf{\Psi}(x)\}_{i,j}=\int\limits_{0}^{x}z^{t-i}\exp\left(-z_{0}\right)_ {0}F_{1}(N_{Hr}-N_{rH}+1;\lambda_{j}z)dz,i,j=1,2,\cdots,N_{rH}\). \({}_{p}F_{q}(\cdot;\cdot)\) is the generalized hypergeometric function with parameters \(p\) and \(q\). \(\Gamma\left(\cdot\right)\) is the gamma function. \(|V|=\prod\limits_{i<j}^{N_{rH}}\left(\lambda_{i}-\lambda_{j}\right)\). \(U(x)\) is the unit step function. \(\mathbf{\Psi}_{N_{rH}}(x)\) is the \(N_{rH}\times N_{rH}\) matrix, with entries \(\{\mathbf{\Psi}_{N_{rH}}(x)\}_{i,j}=x_{N_{rH}}^{N_{Hr}-i}\exp\left(-x\right)_ {0}F_{1}(N_{Hr}-N_{rH}+1;\lambda_{j}x),i,j=1,2,\cdots,N_{rH}\). Using Equation (18), the PDF of the receiving SNR \(\gamma\) in Equation (13) can be obtained to be:  
\[f_{\gamma}\left(x\right)=\frac{1}{\gamma_{s}}f_{\lambda_{1}}\left(x/\gamma_{s }\right)=\frac{\exp(-tr(\uplambda))|\mathbf{\Psi}(x\sigma^{2}/\delta^{2} \gamma_{s})|tr(\mathbf{\Psi}^{-1}(x\sigma^{2}/\delta^{2}\gamma_{s})\mathbf{ \Psi}_{rH}(x\sigma^{2}/\delta^{2}\gamma_{s}))}{P_{s}\delta^{2}/\sigma^{2}| \mathbf{V}|\left(\Gamma(N_{Hr}-N_{rH}+1)\right)^{N_{rH}}}U(x) \tag{19}\]  
Therefore, the CDF of the receiving SNR \(\gamma\) can be written as:  
\[F_{\gamma}\left(x\right)=\int\limits_{0}^{x}f_{\gamma}\left(t\right)dt=\frac{ \exp(-tr(\uplambda))|\mathbf{\Psi}(x\sigma^{2}/\delta^{2}\gamma_{s})}{| \mathbf{V}|\left(\Gamma(N_{Hr}-N_{rH}+1)\right)^{N_{rH}}} \tag{20}\]  
There are two special cases that can simplify the process above. In the first case, if the Rician factor is large enough, _i.e._, the LOS component of \(\mathbf{H}\) is stronger and the signal is only slightly shadowed, then the NLOS component can be ignored. Then the matrix \(\mathbf{H}\) has only a single nonzero eigenvalue and the PDF of \(\lambda_{1}\) can be reduced to:  
\[f_{\lambda_{1}}(x)=\frac{\exp(-\lambda_{1})|\mathbf{\Psi}(x)|tr(\mathbf{\Psi}^ {-1}(x)\mathbf{\Psi}(x))}{\Gamma(N_{Hr}-N_{rH}+1)\uplambda_{1}^{N_{rH}-1} \prod\
olimits_{k=1}^{N_{rH}-1}\Gamma(N_{Hr}-k)\Gamma(N_{rH}-k)}U(x) \tag{21}\]In the second case, if the Rician factor is small enough, _i.e._, the NLOS component of \(\mathbf{H}\) is stronger and the signal is deeply shadowed, then the LOS component can be ignored. Then the \(\mathbf{H}\) matrix has up to \(N_{rH}\) nonzero eigenvalues.