# Co-Efficient Vector Based Differential Distributed Quasi-Orthogonal Space Time Frequency Coding
## 4 Pairwise Error Probability Analysis
Similarly, statistically independent samples of the relay-destination channel were defined as \(\mathbf{g}=[g_{1,t}(0),\ldots,g_{1,t}(L_{RD}-1),\ldots,g_{P,t}(0),\ldots,g_{P,t}(L_{ RD}-1)]\). Under the assumption that all \(f_{p,t}(l)\) and \(g_{p,t}(l)\) were independent identically distributed complex Gaussian variables, it could be implied that \(\mathbf{h}=[h_{1,t}(0),\ldots,h_{1,t}(L-1),\ldots,h_{P,t}(0),\ldots,h_{P,t}(L-1)]\), \(h_{p,t}(l)=f_{p,t}(l)\cdot g_{p,t}(l)\). For any \(k_{th}\) block, the STF codeword could be viewed as a collection of symbols transmitted across \(\Gamma\) time/frequency slots by \(P\) relay nodes. Based on this, the consecutively received signals at the destination in the \(k_{th}\) block and \((k+1)_{th}\) block could be rewritten as (24) under the constraint that the sub-channel gain of adjacent transmission blocks was almost constant. \[\mathbf{\chi}^{k}=\overline{\mathbf{X}}^{k}\mathbf{\Lambda}\mathbf{h}+\mathbf{Z}^{k}\] \[\mathbf{\chi}^{k+1}=\overline{\mathbf{X}}^{k+1}\mathbf{\Lambda}\mathbf{h}+\mathbf{Z}^ {k+1} \tag{24}\]  
where \(\mathbf{\chi}^{k}=[y_{1}^{k},\ldots,y_{\Gamma}^{k}]^{\mathcal{T}}\), \(\mathbf{y}_{i}^{k}=[y^{k}(1),\ldots,y^{k}(\Gamma)]^{\mathcal{T}}\), \(\mathbf{\chi}^{k+1}=[\mathbf{y}_{1}^{k+1},\ldots,\mathbf{y}_{\Gamma}^{k+1}]^{\mathcal{T}}\), \(\mathbf{y}_{i}^{k+1}=[y^{k+1}(1),\ldots,y^{k+1}(\Gamma)]^{\mathcal{T}}\), \(\overline{\mathbf{X}}^{k}=diag[\mathbf{x}_{k,1},\ldots,\mathbf{x}_{k,\Gamma}]\), \(\mathbf{x}_{k,i}=[x_{k,n}(1),\ldots,x_{k,n}(P)]\), \(\overline{\mathbf{X}}^{k+1}=diag[\mathbf{x}_{k+1,1},\ldots,\mathbf{x}_{k+1,\Gamma}]\), \(\mathbf{x}_{k+1,i}=[\mathbf{x}_{k+1,i}(1),\ldots,x_{k+1,i}(P)],\mathbf{\Lambda}=[\mathbf{ \Lambda}(1),\ldots,\mathbf{\Lambda}(\Gamma)]^{\mathcal{T}}\), \(\mathbf{\Lambda}(i)=\mathbf{I}_{P}\otimes\mathbf{\omega}^{T}\), \(\mathbf{\omega}=[1,e^{-2\pi n/N},\ldots,e^{-2\pi L-1n/N}]^{\mathcal{T}}\). Using the following notations:  
\[\mathbf{Y}=[{\mathbf{Y}^{k}}^{T},{\mathbf{Y}^{k+1}}^{T}]^{\mathcal{T}},{\mathbf{V}^{k+1}}=diag [\mathbf{v}_{k+1,1},\ldots,\mathbf{v}_{k+1,\Gamma}],\mathbf{v}_{k+1,i}=[v_{k+1,i}(1),\ldots,v_{k+1,j}(P)],\]  
\[\mathbf{X}=[{I_{\text{PT}}}^{T},{\mathbf{V}^{k+1}}^{T}],\mathbf{Z}=[{\mathbf{Z}^{k}}^{T},{\mathbf{ Z}^{k+1}}^{T}]^{\mathcal{T}}\text{, and the recursion }\overline{\mathbf{X}}^{k+1}=\left\{ \begin{array}{c}{\mathbf{V}^{k+1}}\overline{\mathbf{X}}^{k}, \,k\geq 1\\ {I_{\text{PT}}},k=0\end{array} \right..\]  
We could write:  
\[\mathbf{Y}=\mathbf{X}\mathbf{W}\mathbf{h}+\mathbf{Z} \tag{25}\]  
The conditional probability density function of the receive signal matrix \(\mathbf{Y}\) was:  
\[p(\mathbf{Y}|\mathbf{V}^{k+1})=\frac{\exp(-tr\{\mathbf{Y}({I_{\text{PT}}}+\text{Y}\mathbf{X} \mathbf{\Lambda}\mathbf{R}\mathbf{\Lambda}^{H}\mathbf{X}^{H})^{-1}\mathbf{Y}^{H}\})}{\pi^{\text{PT} }\text{det}({I_{\text{PT}}}+\text{Y}\mathbf{X}\mathbf{\Lambda}\mathbf{R}\mathbf{\Lambda}^{H} \mathbf{X}^{H})} \tag{26}\]  
where \(\mathbf{C}_{\text{v}}=\left({I_{\text{PT}}}+\text{Y}\mathbf{X}\mathbf{\Lambda}\mathbf{R}\mathbf{ \Lambda}^{H}\mathbf{X}^{H}\right)\) was the covariance matrix of \(\mathbf{Y}\), \(tr\) denoted the trace function, and \(\text{Y}\) was the average SNR of the received signal given as \(\text{Y}=\frac{P\text{Y}_{SR}\text{Y}_{RD}}{1+\text{Y}_{SR}+\text{PY}_{RD}}\). Thus, the non-coherent ML decoder was given by:  
\[\mathbf{\hat{V}}^{k+1}=\arg\max_{\mathbf{V}^{k+1}\in\mathbf{V}}p\left(\mathbf{Y}\middle|\mathbf{V}^{ k+1}\right) \tag{27}\]  
Substituting \(\mathbf{\chi}^{k}\) into \(\mathbf{\chi}^{k+1}\) in (24) and using \(\overline{\mathbf{X}}^{k+1}=\mathbf{V}^{k+1}\overline{\mathbf{X}}^{k}\), we had \(\mathbf{\chi}^{k+1}=\mathbf{V}^{k+1}\mathbf{\chi}^{k}+\hat{\mathbf{Z}}^{k+1}\), where \(\hat{\mathbf{Z}}^{k+1}=\mathbf{Z}^{k+1}-\mathbf{V}^{k+1}\mathbf{Z}^{k}\).