## 2 System Model and GD Test Statistics
### System Model
The jointly sufficient statistics of the mean and variance of the likelihood ratio is obtained in the case of GASP implementation, while the classical and modern signal processing theories can deliver only a sufficient statistics of the mean or variance of the likelihood ratio. Thus, the implementation of GASP allows us to obtain more information about the input process or received information signal (the PU signal). Owing to this fact, an implementation of receivers constructed based on the GASP basis allows us to improve the spectrum sensing performance of CR wireless networks in comparison with employment of other conventional receivers at the sensing node. The GD flowchart is presented in Figure 2. As we can see from Figure 2, the GD consists of three channels:  
* The GD correlation channel--the PF, multipliers 1 and 2, model signal generator MSG;
* The GD ED channel--the PF, AF, multipliers 3 and 4, summator 1;
* The GD compensation channel--the summators 2 and 3 and accumulator \(\Sigma\). As follows from Figure 2, under the hypothesis \(\
ot{\pi}_{1}\) (a \"yes\" PU signal), the GD correlation channel generates the signal component \(s_{i}^{\text{mod}}[k]s_{i}[k]\) caused by interaction between the model signal, the reference signal at the GD model signal generator (MSG) output, and the incoming information signal (the PU signal) and the noise component \(2s_{i}^{\text{mod}}[k]\xi_{i}[k]\) caused by interaction between the model signal \(s_{i}^{\text{mod}}[k]\) and the noise \(\xi_{i}[k]\) (the PF output). Under the hypothesis \(\
ot{\pi}_{1}\), the GD ED generates the information signal energy \(s_{i}^{\text{}}[k]\) and the random component \(2s_{i}[k]\xi_{i}[k]\) caused by interaction  
Figure 2: GD flowchart. between the information signal \(s_{i}[k]\) and the noise \(\xi_{i}[k]\). The main purpose of the GD compensation channel is to cancel in the statistical sense the GD correlation channel noise component \(2s_{i}^{\text{mod}}[k]\xi_{i}[k]\) and the GD ED random component \(2s_{i}[k]\xi_{i}[k]\) between each other based on the same nature of the noise \(\xi_{i}[k]\). To describe the GD flowchart we consider the discrete-time processes without loss of any generality. Evidently, the cancelation in the statistical sense between the GD correlation channel noise component \(2s_{i}^{\text{mod}}[k]\xi_{i}[k]\) and the GD ED random component \(2s_{i}[k]\xi_{i}[k]\) is possible only based on the same nature of the noise \(\xi_{i}[k]\) satisfying the condition of equality between the signal model \(s_{i}^{\text{mod}}[k]\) and incoming PU signal \(s_{i}[k]\) over the whole range of parameters. The condition  
\[s_{i}^{\text{mod}}[k]=s_{i}[k] \tag{8}\]  
is the main GD functioning condition. Naturally, in practice, the signal parameters are random. The complete matching between the model signal \(s_{i}^{\text{mod}}[k]\) and the incoming signal \(s_{i}[k]\) (the PU signal), especially by amplitude, is a very hard problem in practice and only in the ideal case the complete matching is possible. How the GR sensing performance can be deteriorated under mismatching between the model signal \(s_{i}^{\text{mod}}[k]\) and the incoming (PU) signal \(s_{i}[k]\) is discussed in this paper. Under the hypothesis \(\mathcal{H}_{0}\), _i.e._, a \"no\" information signal (the PU signal), satisfying the GD main functioning condition given by (8), we obtain only the background noise \(\eta_{i}^{\ast}[k]-\xi_{i}^{\ast}[k]\) at the GD output. The GD PF bandwidth is matched with the bandwidth of the information signal (the PU signal) \(s_{i}[k]\). The threshold apparatus (THRA) device defines the GD threshold. The GD PF and AF can be considered as the linear systems, for example, as the bandpass filters, with the impulse responses \(h_{{}_{PF}}[m]\) and \(h_{{}_{AF}}[m]\), respectively. For simplicity of analysis, we assume that these filters have the same amplitude-frequency characteristics or impulse responses by shape.