# Co-Efficient Vector Based Differential Distributed Quasi-Orthogonal Space Time Frequency Coding
## 3 Co-Efficient Vector Differential Procedure
### Differential Encoding Using Co-Efficient Vectors
We continue our discussion on differential encoding at the relay nodes by illustrating how the co-efficient vector design was implemented with \(M\)-\(PSK\) constellations for \(P\) relays. Note that we assumed that the relay nodes could perfectly compute the elements of the coefficient vector set. Considering an \(M\)-\(PSK\) modulation scheme with spectral efficiency \(m\), the encoder at each relay node generated \(\Gamma\) modulated signals from \(\Gamma m\) information bits. Let the set of recovered information bits at the encoder be given as \(b_{l},l=1,2,\dots,2^{2m}\), and the co-efficient vector set is computed as: \(\mathbf{v}\Big{\{}\left[v_{1,1},\dots,v_{\Gamma,1}\right]^{T},\left[v_{1,2}, \dots,v_{\Gamma,2}\right]^{T},\dots,\left[v_{1,2^{2m}},\dots,v_{\Gamma,2^{2m} }\right]^{T}\Big{\}}\). Thus, the mapping scheme \(P(. )\) for each set of information bits to a co-efficient vector set was defined by:  
\[P\{(b_{1}),(b_{2}),\dots,(b_{2^{2m}})\}=\Big{\{}\left[v_{1,1},\dots,v_{\Gamma,1}\right]^{T},\left[v_{1,2},\dots,v_{\Gamma,2}\right]^{T},\dots,\left[v_{1,2 ^{2m}},\dots,v_{\Gamma,2^{2m}}\right]^{T}\Big{\}} \tag{11}\]  
where \(X_{1}=\sum_{i=1}^{4}|\widetilde{\mathfrak{X}}_{q,l,n}(i)|^{2}\) was the signal power, and \(X_{2}=2 Re(\widetilde{\mathfrak{X}}_{q,l,n}(1)\widetilde{\mathfrak{X}}_{q,l,n}( 4)^{*}-\widetilde{\mathfrak{X}}_{q,l,n}(2)\)\(\widetilde{\mathfrak{X}}_{q,l,n}(3)^{*})\) was a self-interference parameter. Similarly, \(G_{1}=\sum_{p=1}^{4}\bigl{|}g_{q,l,n}(p)\bigr{|}^{2}\) was the channel power, and \(G_{2}=2 Re\bigl{\{}g_{q,l,n}(1)g_{q,l,n}(4)^{*}-g_{q,l,n}(2)g_{q,l,n}(3)^{*}\bigr{\}}\) was a self-interference parameter. The elements of the co-efficient vector \(\overline{\mathfrak{v}}_{k+1}=[v_{1},\ldots,v_{4}]\) were then recovered as follows:  
\[ \begin{array}{l}\boldsymbol{y}_{k+1,1}{\boldsymbol{y}_{k,1}}^{H}=( \widetilde{\overline{\mathfrak{x}}}_{k+1}{\boldsymbol{X}^{H}}){\boldsymbol{G }_{k+1}}{\boldsymbol{g}_{k,1}}^{H}+Z_{1}\\ =(\overline{\mathfrak{v}}_{k+1}{\boldsymbol{X}_{k}}{\boldsymbol{X}_{k}}^{H}) {\boldsymbol{G}_{k+1}}{\boldsymbol{g}_{k,1}}^{H}+Z_{1}\\ =v_{1}(X_{1}G_{1}+X_{2}G_{2})+v_{4}(X_{1}G_{2}+X_{2}G_{1})+Z_{1}\\ =v_{1}A+v_{4}B+Z_{1}\end{array}  \tag{14}\]  
Similarly,  
\[ \begin{array}{l}\boldsymbol{y}_{k+1,1}{\boldsymbol{y}_{k,2}}^{H}=( \widetilde{\overline{\mathfrak{x}}}_{k+1}{\boldsymbol{X}^{H}}){\boldsymbol{G }_{k+1}}{\boldsymbol{g}_{k,2}}^{H}+Z_{2}\\ =(\overline{\mathfrak{v}}_{k+1}{\boldsymbol{X}_{k}}{\boldsymbol{X}_{k}}^{H}) {\boldsymbol{G}_{k+1}}{\boldsymbol{g}_{k,2}}^{H}+Z_{2}\\ =v_{2}A-v_{3}B+Z_{2}\end{array}  \tag{15}\]  
\[ \begin{array}{l}\boldsymbol{y}_{k+1,1}{\boldsymbol{y}_{k,3}}^{H}=( \widetilde{\overline{\mathfrak{x}}}_{k+1}{\boldsymbol{X}^{H}}){\boldsymbol{G }_{k+1}}{\boldsymbol{g}_{k,3}}^{H}+Z_{3}\\ =(\overline{\mathfrak{v}}_{k+1}{\boldsymbol{X}_{k}}{\boldsymbol{X}_{k}}^{H}) {\boldsymbol{G}_{k+1}}{\boldsymbol{g}_{k,3}}^{H}+Z_{3}\\ =-v_{2}B+v_{3}A+Z_{3}\end{array}  \tag{16}\]  
\[ \begin{array}{l}\boldsymbol{y}_{k+1,1}{\boldsymbol{y}_{k,4}}^{H}=( \widetilde{\overline{\mathfrak{x}}}_{k+1}{\boldsymbol{X}^{H}}){\boldsymbol{G }_{k+1}}{\boldsymbol{g}_{k,4}}^{H}+Z_{4}\\ =(\overline{\mathfrak{v}}_{k+1}{\boldsymbol{X}_{k}}{\boldsymbol{X}_{k}}^{H}) {\boldsymbol{G}_{k+1}}{\boldsymbol{g}_{k,4}}^{H}+Z_{4}\\ =v_{1}B+v_{4}A+Z_{4}\end{array}  \tag{17}\]  
where \(Z_{n}\) was the noise, \(A=X_{1}G_{1}+X_{2}G_{2}\), and \(B=X_{1}G_{2}+X_{2}G_{1}\), and we referred to \(A\) and \(B\) as the differential decoding parameters required to recover \(\overline{\mathfrak{v}}_{k+1}\). The differential decoding parameters were computed at the destination as:  
\[