## 2 Problem Formulation  
We consider a cognitive network that has \(K\) secondary users, in which the \(k\)-th SU is equipped with \(L\) antennas. In the presence of the primary signal (macrocell user signal), the received signal at the output of the \(L\) antennas of the \(k\)-th SU can be expressed as: \(\mathbf{x}_{k}\left(n\right)=\mathbf{s}_{k}(n)+\mathbf{w}_{k}\left(n\right),\)\(n=1,\cdots,N\), where the (\(L\times 1\)) vector \(\mathbf{s}_{k}(n)\) contains the samples of the primary signal and vector \(\mathbf{w}_{k}\left(n\right)\) consists of the complex additive white Gaussian noise samples at time \(n\). We assume that measurements from SUs \(\left\{\mathbf{x}_{k}\left(n\right)\right\}_{k=1}^{K}\) are sent to the fusion center (_i.e.,_ the base station) via dedicated links. Which is in general a reasonable assumption, as in most of the cases SUs and the fusion center would have mutually agreed upon digital communication links. Moreover, the SUs will have better error correction schemes to overcome the impairments of their corresponding channels with the fusion center. One particular example where such assumption is completely applicable is the cognitive-enabled femtocell networks (network of base stations of femtocells). Femtocells are small cellular telecommunications base stations that can be installed in residential or business environments either as single stand-alone items or in clusters to provide improved cellular coverage within a building [20]. In most cognitive-enabled femtocell networks, there exists a wired back-haul connection between HeNBs (HeNB is the 3 GPP's term for a LTE femtocell or Small Cell) that can be considered as an ideal link between receivers and the fusion center [20], as shown in the Figure 1. The fusion center stacks the received signal vectors \(\left\{\mathbf{x}_{k}\left(n\right)\right\}_{k=1}^{K}\) in the (\(L\times K\)) matrix \(\mathbf{X}\left(n\right)\). By using the \(vec\) operator as \(\mathbf{x}\left(n\right)=vec\left\{\mathbf{X}\left(n\right)\right\}\), the two hypotheses can be represented as:  
\[\mathcal{H}_{0}: \mathbf{x}\left(n\right)=\mathbf{w}\left(n\right),\text{ \ \ \ \ \ \ \ }n=1,\cdots,N \tag{1}\] \[\mathcal{H}_{1}: \mathbf{x}\left(n\right)=\mathbf{s}(n)+\mathbf{w}\left(n\right), \text{ }n=1,\cdots,N\]  
where  
\[\mathbf{x}\left(n\right)=\left[\mathbf{x}_{1}^{T}\left(n\right),\mathbf{x}_{2 }^{T}\left(n\right),\cdots,\mathbf{x}_{K}^{T}\left(n\right)\right]^{T} \tag{2}\]  
\[\mathbf{s}\left(n\right)=\left[\mathbf{s}_{1}^{T}\left(n\right),\mathbf{s}_{2 }^{T}\left(n\right),\cdots,\mathbf{s}_{K}^{T}\left(n\right)\right]^{T} \tag{3}\]  
\[\mathbf{w}\left(n\right)=\left[\mathbf{w}_{1}^{T}\left(n\right),\mathbf{w}_{2 }^{T}\left(n\right),\cdots,\mathbf{w}_{K}^{T}\left(n\right)\right]^{T} \tag{4}\]  
Therefore, we have the covariance matrix \(\mathbf{\Sigma}_{1}=\mathbf{E}\left[\mathbf{x}\mathbf{x}^{H}\right]\) as:  
\[\mathbf{\Sigma}_{1}=\left[ \begin{array}{cccc}\mathbf{\Sigma}_{11}&\mathbf{ \Sigma}_{12}&\cdots&\mathbf{\Sigma}_{1K}\\ \mathbf{\Sigma}_{21}&\mathbf{\Sigma}_{22}&\cdots&\mathbf{\Sigma}_{1K}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{\Sigma}_{K1}&\mathbf{\Sigma}_{K2}&\cdots&\mathbf{\Sigma}_{KK}\end{array}  \right]\in\mathbb{C}^{KL\times KL} \tag{5}\]  
where the covariance matrices \(\mathbf{\Sigma}_{k,k}=E\left[\mathbf{x}_{k}\mathbf{x}_{k}^{H}\right]\), \(k=1,\ldots,K\) capture the inter-antennas correlation present in the signal received at \(L\) antennas of the \(k\)-th SU. Similarly, non-zero off-diagonal blocks \(\mathbf{\Sigma}_{k,j}=E\left[\mathbf{x}_{k}\mathbf{x}_{j}^{H}\right]\) with \(k\
eq j\) indicate the presence of inter-receiver spatial correlation. Furthermore, in most practical working conditions, it seems reasonable to consider the ensemble vectors \(\left\{\mathbf{x}\left(n\right)\right\}_{n=1}^{N}\) to be zero mean complex Gaussian distributed, a case that is particularly well-suited for orthogonal frequency division multiplexing (OFDM) [21].