## 2 System Model and GD Test Statistics  
### System Model  
The spectrum sensor has an antenna array with the number of elements equal to \(M\) and each antenna array element receives \(N\) samples during the sensing time. The spectrum sensing problem can be modeled as the conventional binary hypothesis test:  
\[ \begin{cases}\
ot{\sigma}_{0}\to z_{s}[k]=w_{s}[k],&i=1,...,M;\ k=0,...,N-1,\\ \
ot{\sigma}_{1}\to z_{s}[k]=h_{s}[k]s[k]+w_{s}[k],&i=1,...,M;\ k=0,...,N-1, \end{cases}  \tag{1}\]  
where \(z_{s}[k]\) is the discrete-time received signal at the spectrum sensor input; \(w_{s}[k]\) is the discrete-time circularly symmetric complex Gaussian noise with zero mean and variance \(\sigma_{w}^{2}\), _i.e._, \(w_{s}[k]\sim\zeta_{w}\gamma^{\prime}(0,\sigma_{w}^{2})\); \(h_{s}[k]\) is the discrete-time channel coefficients obeying the circularly symmetric complex Gaussian distribution with zero mean and variance equal to \(\sigma_{h}^{2}\), _i.e._, \(h_{s}[k]\sim\zeta_{w}\gamma^{\prime}(0,\sigma_{h}^{2})\); and \(s[k]\) is the discrete-time PU signal, _i.e._, the signal to be detected. We consider the same initial conditions with respect to \(s[k]\) as in [3]. The channel parameters are not varied during the sensing time and the channel coefficients \(h_{s}[k]\) are spatially correlated between each other. Throughout this paper, the PU signal \(s[k]\), the channel coefficients \(h_{s}[k]\), and the noise \(w_{s}[k]\) are independent and uncorrelated between each other. The same channel model is widely used in [25, 26, 27]. In general, the ED does not require channel state information (CSI) for spectrum sensing [28] and the GD shares this property with ED because the ED is a constituent of the GD. It is well known that information about the CSI allows us to obtain better spectrum sensing performance in comparison with unknown CSI case. The knowledge about CSI can be more useful and effective in the cooperative spectrum sensing case. Under the low SNR and noise power uncertainty conditions, we can claim that we have imperfect CSI [29]. When the noise power estimation is applied, we have partial knowledge about the CSI. In this paper, we assume that the coarse spectrum sensing is performed without knowledge about the CSI. Owing to its simplicity, the exponential matrix model is widely used to describe the spatial correlation between the adjacent antenna array elements [30]. The components of the \(M\times M\) antenna array element correlation matrix \(\mathbf{C}\) can be presented in the following form:  
\[\mathbf{C}_{ij}=\{\rho^{i-j}\}\,\quad i\leq j,\quad i,j=1,...,M \tag{2}\]  
where \(\rho\) is the coefficient of spatial correlation between the adjacent antenna array elements (\(0\leq\rho\leq 1\), the real values). Applying the results presented in [30], the coefficient of spatial correlation \(\rho\) can be given as  
\[\rho=\exp\{-23\Lambda^{2}(d\ /\lambda)^{2}\} \tag{3}\]  
where \(\Lambda\) is the angular spread, an important propagation parameter defining a distribution of multipath power of radio waves coming in at the receiver input from a number of azimuthal directions with respect to the horizon; \(\lambda\) is the wavelength; and \(d\) is the distance between two adjacent antenna array elements (the antenna array element spacing). The correlation matrix of antenna array elements \(\mathbf{C}\) given by Equation (2) is the symmetric Toeplitz matrix [25]. signal (the PU signal) and input noise are cancelled in the statistical sense. This GD feature allows us to obtain the better detection performance in comparison with other classical receivers or detectors. The specific feature of GASP is introduction of the additional noise source that does not carry any information about the incoming signal with the purpose to improve a qualitative signal detection performance. This additional noise can be considered as the reference noise without any information about the PU signal [17].