## 4 GLRT Based on Kronecker Product
### Spkp-Glrt  
In Section 2, we have discussed that the (\(KL\times 1\)) vector \(\mathbf{x}\)(n) contains \(K\) sub-vectors of dimension (\(L\times 1\)) corresponding to the arrays at the \(K\) different SUs. Consequently, we see that the covariance matrix \(\mathbf{\Sigma}_{1}\) has mainly two types of spatial correlation structures. Inter-receiver correlation structure appears between the SUs due to their proximity, and inter-antenna correlation appears between samples from the antenna elements of the same SU. Moreover, by assuming that the spatial correlation structure of arrays at different SUs is similar, the observation matrix \(\mathbf{X}\) can be assumed as a matrix normal with separable structure [23,24]. Keeping the above facts and discussion in [22,24] into considerations, the overall (inter-antenna plus inter-receiver) spatial covariance \(\mathbf{\Sigma}_{1}\) can be represented with the help of the SPKP model as:  
\[\mathbf{\Sigma}_{1}=\mathbf{\Sigma}_{\mathrm{A}}\otimes\mathbf{\Sigma}_{ \mathrm{S}}, \tag{9}\]  
where \(\mathbf{\Sigma}_{\mathrm{S}}\in\mathbb{C}^{K\times K}\) and \(\mathbf{\Sigma}_{\mathrm{A}}\in\mathbb{C}^{L\times L}\) capture the inter-receiver and inter-antenna spatial correlation, respectively. By using Equation (9) the SPKP-GLRT is [17]:  
\[\Lambda_{\mathrm{SPKP}}\left(\mathbf{X}_{N}\right)=\frac{\underset{\mathbf{ \Sigma}_{0}}{\text{max}}f_{\mathbf{x}}\left(\mathbf{X}_{N};\mathbf{\Sigma}_{0 }\right)}{\underset{\mathbf{\Sigma}_{\mathrm{S}},\mathbf{\Sigma}_{\mathrm{A}} }{\text{max}}f_{\mathbf{x}}\left(\mathbf{X}_{N};\mathbf{\Sigma}_{\mathrm{A}} \otimes\mathbf{\Sigma}_{\mathrm{S}}\right)}\gtrless_{\mathcal{H}_{1}}^{ \mathcal{H}_{0}}\gamma. \tag{10}\]Solving Equation (10), under the hypothesis \(\mathcal{H}_{1}\), we need to estimate covariance matrices \(\mathbf{\Sigma}_{\text{S}}\) and \(\mathbf{\Sigma}_{\text{A}}\) by using the MLE paradigm. The MLEs under the hypothesis \(\mathcal{H}_{1}\), can be written as [17,23,25]:  
\[\hat{\mathbf{\Sigma}}_{\text{A}} =\frac{1}{KN}\sum_{n=1}^{N}\mathbf{X}\left(n\right)\hat{\mathbf{ \Sigma}}_{\text{S}}^{-1}\mathbf{X}^{H}\left(n\right), \tag{11}\] \[\hat{\mathbf{\Sigma}}_{\text{S}} =\frac{1}{LN}\sum_{n=1}^{N}\mathbf{X}^{H}\left(n\right)\hat{ \mathbf{\Sigma}}_{\text{A}}^{-1}\mathbf{X}\left(n\right). \tag{12}\]  
Expressions Equations (11) and (12) suggest that \(\hat{\mathbf{\Sigma}}_{\text{A}}\) and \(\hat{\mathbf{\Sigma}}_{\text{S}}\) can be achieved using an iterative method such as the Flip-Flop algorithm [23,25]. The Flip-Flop algorithm is obtained by alternately maximizing \(\text{log}f_{\mathbf{x}}\left(\mathbf{X}_{N};\mathbf{\Sigma}_{\text{S}} \otimes\mathbf{\Sigma}_{\text{A}}\right)\) w.r.t. \(\mathbf{\Sigma}_{\text{S}}\) keeping the last available estimate of \(\mathbf{\Sigma}_{\text{A}}\) fixed and vice versa. In [23], it has been proved that for the case of large enough \(N\), asymptotic efficiency of the ML approach can be achieved by only performing steps 1 to 3, without iterating. Taking into account this fact, in order to get positive definite \(\hat{\mathbf{\Sigma}}_{1}\) we adopt a non-iterative Flip-Flop approach and only perform the steps given in Algorithm 1 with an initial value of \(\hat{\mathbf{\Sigma}}_{\text{S}}^{0}=\mathbf{I}_{K\times K}\). Finally, solving Equation (10), we get the expression:  
1. Choose a starting value for \(\hat{\mathbf{\Sigma}}_{\text{S}}^{0}\) as \(\mathbf{I}_{K\times K}\). 2. Estimate \(\hat{\mathbf{\Sigma}}_{\text{A}}^{1}\) from Equation (11) with \(\hat{\mathbf{\Sigma}}_{\text{S}}^{0}\). 3. Find the following * Estimate \(\hat{\mathbf{\Sigma}}_{\text{S}}\) from Equation (12) with \(\hat{\mathbf{\Sigma}}_{\text{A}}^{1}\). * Estimate \(\hat{\mathbf{\Sigma}}_{\text{A}}\) from Equation (11) with \(\hat{\mathbf{\Sigma}}_{\text{S}}\).