## 2 System Model and GD Test Statistics
### System Model
Moreover, the GD AF central frequency is detuned with respect to the GD PF central frequency on such a value that the information signal (the PU signal) cannot pass through the GD AF. Thus, the PU signal and noise can appear at the GD PF output and the only noise is appeared at the GD AF output (see Figure 3). If a value of detuning between the GD AF and PF central frequencies is more than 4 or 5 \(\Delta\mathcal{J}_{s}\), where \(\Delta\mathcal{J}_{s}\) is the PU signal bandwidth, the processes at the GD AF and PF outputs can be considered as the uncorrelated and independent processes and, in practice, under this condition, the coefficient of correlation between GD PF and AF output processes is not more than 0.05 that was confirmed experimentally [31,32]. In the present paper, we consider the spectrum sensing problem of a single radio channel where the GD AF bandwidth is always idle and cannot be used by the SU because it is out of the useful spectrum of the PU network. There is a need to note that in a general case, the GD AF portion of the spectrum may be occupied by the PU signals from other networks and can be not absolutely unoccupied. In this case, the PU signals from other networks can be considered as interferences or interfering signals. Investigation and study of GD under this case is discussed in [20]. The processes at the GD AF and PF outputs present the input stochastic samples from two independent frequency-time regions. If the noise \(\mathcal{W}[k]\) at the GD PF and AF inputs is Gaussian, the noise at the GD PF and AF outputs is Gaussian, too, because the GD PF and AF are the linear systems, and we believe that these linear systems do not change the statistical parameters of the input process. Weuse this assumption for simplicity of theoretical analysis. Thus, the GD AF can be considered as a reference noise source with _a priori_ knowledge a \"no\" signal (the reference noise sample). Detailed discussion of the GD AF and PF can be found in [18, 19]. The noise at the GD PF and AF outputs can be presented in the following form:  
\[\left\{ \begin{array}{l}w_{{}_{PF}}[k]=\sum_{i=1}^{M}\xi_{i}[k]=\sum_{m=- \infty}^{\infty}h_{{}_{PF}}[m]w_{i}[k-m]\\ w_{{}_{AF}}[k]=\sum_{i=1}^{M}\eta_{i}[k]=\sum_{m=-\infty}^{\infty}h_{{}_{AF}}[m ]w_{i}[k-m]\end{array} \right. \tag{9}\]  
Under the hypothesis \(\
ot\!\!\gamma_{1}\), the signal at the GD PF output can be defined as \(x_{i}[k]=s_{i}[k]+\xi_{i}[k]\) (see Figure 2), where \(\xi_{i}[k]\) is the observed noise at the GD PF output and  
\[s_{i}[k]=h_{i}[k]\times s[k] \tag{10}\]  
Under the hypothesis \(\
ot\!\gamma_{0}\), and for all \(i\) and \(k\), the process \(x_{i}[k]=\xi_{i}[k]\) at the GD PF output is subjected to the complex Gaussian distribution and can be considered as the independent and identically distributed (i.i.d.) process. The process at the GD AF output is the reference noise \(\eta_{i}[k]\) with the same statistical parameters as the noise \(\xi_{i}[k]\) in the ideal case. We make this assumption for simplicity. In practice, the statistical parameters of the noise \(\xi_{i}[k]\) and \(\eta_{i}[k]\) are different, as a rule. We consider this case below. The decision statistics at the GD output presented in [17, 18] is extended to the case of antenna array employment when an adoption of multiple antennas and antenna arrays is effective to mitigate the negative attenuation and fading effects [20, 24]. The GD decision statistics can be presented in the following form:  
\[T_{{}_{GD}}(\textbf{X})=\sum_{k=0}^{N-1}\sum_{i=1}^{M}2x_{i}[k]s_{i}^{\text{ mod}}[k]-\sum_{k=0}^{N-1}\sum_{i=1}^{M}x_{i}^{2}[k]+\sum_{k=0}^{N-1}\sum_{i=1}^{M} \eta_{i}^{2}[k]\underset{\gamma_{\Psi_{0}}}{\underbrace{\gamma_{\Psi_{0}}}} \text{THR}_{{}_{GD}} \tag{11}\]  
Figure 3: Signals at the GD PF and AF outputs: (**a**) GD AF response and noise; (**b**) GD PF response and noise; (**c**) GD PF response and PU signal.