## 4 GLRT Based on Kronecker Product
### MPKP-GLRT  
In Section 4.1, we have approximated \(\mathbf{\Sigma}_{1}\) in the form of two smaller covariance matrices through the SPKP model (9). The aforementioned SPKP model can also be written as:  
\[\mathbf{\Sigma}_{1}=\mathbf{\Sigma}_{\text{A}}\otimes\sum_{k=1}^{K}\lambda_{k}\mathbf{S}_{k}, \tag{14}\]  
where \(\mathbf{S}_{k}\triangleq\mathbf{v}_{k}\mathbf{v}_{k}^{H}\) is an orthonormal basis component represented as a singular matrix [19]. Furthermore, by using the identity \(\mathbf{B}\otimes\left(\mathbf{O}+\mathbf{D}\right)=\mathbf{B}\otimes \mathbf{O}+\mathbf{B}\otimes\mathbf{D}\), we can write  
\[\mathbf{\Sigma}_{1}=\sum_{k=1}^{K}\mathbf{\Sigma}_{\text{A}}\otimes\lambda_{k}\mathbf{S}_{ k}=\sum_{k=1}^{K}\lambda_{k}\mathbf{\Sigma}_{\text{A}}\otimes\mathbf{S}_{k}. \tag{15}\]  
The expression makes it obvious that each spatial component \(\mathbf{S}_{k}\) has the same inter-antenna covariance matrix. Hence, the SPKP covariance model is based on a very rigid assumption since it assumes similar inter-antenna spatial covariance structures at different SUs. However, in a more realistic case, different SUs are likely to have different inter-antenna spatial covariance. Therefore, to relax this rigid assumption and better explain the separation of inter-receiver and inter-antenna correlation, we consider a more general _multi-pair_ Kronecker product (MPKP) model as:  
\[\mathbf{\Sigma}_{1}=\sum_{k=1}^{K}\mathbf{A}_{k}\otimes\mathbf{S}_{k}. \tag{16}\]  
Note that by setting \(\mathbf{A}_{k}=\lambda_{k}\mathbf{\Sigma}_{\text{A}}\) we can see how Equation (16) subsumes the SPKP model. Moreover, it is to be noted that the inter-receiver correlation components \(\left\{\mathbf{S}_{k}\right\}_{k=1}^{K}\) are (\(K\times K\)) rank-1 matrices and their corresponding inter-antenna correlation matrices \(\mathbf{A}_{k}\) are full rank (\(L\times L\)) matrices. The estimate of \(\mathbf{S}_{k}\) or the spatial orthogonal components the \(\left\{\mathbf{v}_{k}\right\}_{k=1}^{K}\) can be found by singular value decomposition (SVD) of \(\mathbf{X}_{V}=\left[\mathbf{X}^{T}\left(1\right),\mathbf{X}^{T}\left(2 \right),\cdots,\mathbf{X}^{T}\left(N\right)\right]^{T}\in\mathbb{C}^{LN\times K}\) as: \(\mathbf{X}_{V}=\mathbf{U}\mathbf{\Psi}\mathbf{V}^{H}\)[19]. Where \(\mathbf{U}\) is a (\(LN\times K\)) orthogonal matrix, \(\mathbf{\Psi}\) is a (\(K\times K\)) diagonal matrix with singular values \(\left\{\eta_{k}\right\}_{k=1}^{K}\) of \(\mathbf{X}_{V}\) as diagonal elements and \(\mathbf{V}\) is a (\(K\times K\)) orthogonal matrix of the spatial components. Each row of \(\mathbf{V}^{H}\) is \(\mathbf{v}_{k}\) that forms \(\mathbf{S}_{k}\) as: \(\mathbf{S}_{k}=\mathbf{v}_{k}\mathbf{v}_{k}^{H}\) for the model (16). Similarly, the estimate of \(\mathbf{A}_{k}\) can be found based on the MLE paradigm as [19]:  
\[\hat{\mathbf{A}}_{k}=\frac{1}{N}\sum_{n=1}^{N}\mathbf{X}\left(n\right)\hat\mathbf{{S}}_{k}\mathbf{X}^{H}\left(n\right), \tag{17}\]  
and \(\hat{\mathbf{S}}_{k}=\mathbf{v}_{k}\mathbf{v}_{k}^{H}\). Hence, we can write \(\hat{\mathbf{\Sigma}}_{1}=\sum_{k=1}^{K}\hat{\mathbf{A}}_{k}\otimes\hat{\mathbf{S}}_{k}\) and the MPKP-GLRT becomes:  
\[\Lambda_{\text{MPKP}}\left(\mathbf{X}_{N}\right)=\frac{\left|\sum_{k=1}^{K} \hat{\mathbf{A}}_{k}\otimes\hat{\mathbf{S}}_{k}\right|}{\left|\hat{\mathbf{\Sigma}}_{ 0}\right|}\gtrless_{\mathcal{H}_{1}}^{\mathcal{H}_{0}}\gamma. \tag{18}\]Compared to the SPKP-GLRT in Equation (13), the MPKP-GLRT in Equation (18) can better account for the different inter-antenna structures, since it is not constrained to the assumption of identical inter-antenna correlation at different SUs. However, the computational cost of the detection scheme Equation (18) is slightly higher than that of Equation (13). This is because the number of free parameters in the covariance matrix to be estimated under hypothesis \(\mathcal{H}_{1}\) has increased to \(L^{2}+LK\left(K-1\right)/2\)[19].