## 2 System Model and GD Test Statistics
### System Model
where  
\[\mathbf{X}=[x_{i}[0],\ldots,x_{M}[0],\ \ldots\ x_{1}[N-1],\ldots,x_{M}[N-1]]^{T} \tag{12}\]  
is the stochastic process vector at the GD PF output and \(\mathit{THR}_{{}_{GD}}\) is the GD detection threshold. We can rewrite Equation (11) in the vector form:  
\[T_{{}_{GD}}(\mathbf{X})=2\mathbf{S}^{\mathrm{mod}}\mathbf{X}-\mathbf{X}^{2}+ \boldsymbol{\eta}^{2}\geq_{\mathbf{\Psi}_{0}}^{N}\mathit{THR}_{{}_{GD}} \tag{13}\]  
where  
\[\mathbf{X}=[\mathbf{x}(0),\ldots,\mathbf{x}(N-1)] \tag{14}\]  
is the \(\mathit{M}\times 1\) vector of the random process at the GD PF output with elements defined as  
\[\mathbf{x}[k]=[x_{i}[k],\ldots,x_{u}[k]]^{T} \tag{15}\]  
\[\mathbf{S}^{\mathrm{mod}}=[\mathbf{s}^{\mathrm{mod}}(0),\ldots,\mathbf{s}^{ \mathrm{mod}}(N-1)] \tag{16}\]  
is the \(\mathit{M}\times 1\) vector of the process at the MSG output with the elements defined as  
\[\mathbf{s}^{\mathrm{mod}}[k]=[s_{1}^{\mathrm{mod}}[k],\ldots,s_{M}^{\mathrm{ mod}}[k]]^{T} \tag{17}\]  
\[\boldsymbol{\eta}=[\boldsymbol{\eta}(0),\ldots,\boldsymbol{\eta}(N-1)] \tag{18}\]  
is the \(\mathit{M}\times 1\) vector of the random process at the AF output with the elements defined as  
\[T_{{}_{GD}}(\textbf{X})=\sum_{k=0}^{N-1}\sum_{i=1}^{M}2x_{i}[k]s_{i}^{\text{ mod}}[k]-\sum_{k=0}^{N-1}\sum_{i=1}^{M}x_{i}^{2}[k]+\sum_{k=0}^{N-1}\sum_{i=1}^{M} \eta_{i}^{2}[k]\underset{\gamma_{\Psi_{0}}}{\underbrace{\gamma_{\Psi_{0}}}} \text{THR}_{{}_{GD}} \tag{11}\]0  
and \(\mathit{THR}_{{}_{GD}}\) is the GD detection threshold. According to GASP and GD structure shown in Figure 2 and the main GD functioning condition (8), the GD test statistics takes the following form under the hypotheses \(\mathit{\Psi}_{1}\) and \(\mathit{\Psi}_{0}\), respectively:  
\[T_{{}_{GD}}(\mathbf{X})= \begin{cases}\sum_{k=0}^{N-1}\sum_{i=1}^{M}s_{i}^{2}[ k]+\sum_{k=0}^{N-1}\sum_{i=1}^{M}\boldsymbol{\eta}_{i}^{2}[k]-\sum_{k=0}^{N-1} \sum_{i=1}^{M}\xi_{i}^{2}[k]&\Rightarrow\mathit{\Psi}_{1}\\ \sum_{k=0}^{N-1}\sum_{i=1}^{M}\eta_{i}^{2}[k]-\sum_{k=0}^{N-1}\sum_{i=1}^{M} \xi_{i}^{2}[k]&\Rightarrow\mathit{\Psi}_{0}\end{cases}  \tag{20}\]  
The term \(\sum_{k=0}^{N-1}\sum_{i=1}^{M}s_{i}^{2}[k]\) is the average energy of received signal and the term \(\sum_{k=0}^{N-1}\sum_{i=1}^{M}\eta_{i}^{2}[k]-\sum_{k=0}^{N-1}\sum_{i=1}^{M} \xi_{i}^{2}[k]\) presents the background noise at the GD output that is a difference between the noise power at the GD PF and AF outputs. It is important to mention that the GD main functioning condition is the equality between parameters of the model signal \(s_{i}^{\mathrm{mod}}[k]\) and the PU signal \(s_{i}[k]\) (see Equation (8)) over all range of parameters and, in particular, by amplitude. How we can satisfy this condition in practice is discussed in detail in [17] and ([18], Chapter 6, pp. 611-621 and Chapter 7, pp. 631-695) when there is no _a priori_ information about the signal \(s_{i}[k]\). Additionally, a practical implementation of the GD decision statistics requires an estimation of the noise variance \(\sigma_{w}^{2}\) using the reference noise \(\eta_{i}[k]\) at the GD AF output. The mean \(m_{\eta_{\mathrm{\Psi}_{0}}}^{{}_{GD}}\) and variance \(Var_{\eta_{\mathrm{\Psi}_{0}}}^{{}_{GD}}\) of the test decision statistics \(T_{{}_{GD}}(\mathbf{X})\) under the hypothesis \(\mathit{\Psi}_{0}\) are given in the following form ([19], Chapter 3):\[\left\{ \begin{array}{l}m_{\Psi_{6}}^{ GD}=E[T_{ GD}({\bf X})\big{|}\mathcal{H}_{0}]=0\\ Var_{\Psi_{6}}^{ GD}=Var[T_{ GD}({\bf X})\big{|}\mathcal{H}_{0}]=4 NM \sigma^{4}\end{array} \right. \tag{21}\]  
The above-mentioned discussion is correct for the case when the noise variances at the GD AF and GD PF outputs are the same, _i.e._, \(\sigma_{\xi}^{2}=\sigma_{\eta}^{2}=\sigma^{2}\).