# Co-Efficient Vector Based Differential Distributed Quasi-Orthogonal Space Time Frequency Coding
## 3 Co-Efficient Vector Differential Procedure
### Differential Encoding Using Co-Efficient Vectors
\begin{array}{l}\boldsymbol{y}_{k,1}{\boldsymbol{y}_{k,4}}^{H}={\boldsymbol {X}_{k}}{\boldsymbol{X}_{k}}^{H}{\boldsymbol{g}_{k,1}}{\boldsymbol{g}_{k,4}} ^{H}+\widetilde{\widetilde{Z}}_{4}=A+\widetilde{\widetilde{Z}}_{4}\\ \boldsymbol{y}_{k,1}{\boldsymbol{y}_{k,1}}^{H}={\boldsymbol{X}_{k}}{ \boldsymbol{X}_{k}}^{H}{\boldsymbol{g}_{k,1}}{\boldsymbol{g}_{k,1}}^{H}+ \widetilde{\widetilde{Z}}_{1}=B+\widetilde{\widetilde{Z}}_{1}\end{array}  \tag{18}\]  
This implied that \(\boldsymbol{y}_{k,1}{\boldsymbol{y}_{k,4}}^{H}\approx A\) and \(\boldsymbol{y}_{k,1}{\boldsymbol{y}_{k,1}}^{H}\approx B\) since \(Z_{n}\approx\widetilde{\widetilde{Z}}_{n}\). It was thus obvious from (18) that the scheme did not require CSI to recover \(\overline{\mathfrak{v}}_{k+1}\). The non-coherent recovery of \(\overline{\mathfrak{v}}_{k+1}\) rather depended on consecutively received signals in the \(k\)th block and \((k+1)\)th block. Once \(A\) and \(B\) were computed at the destination using (18), the information signals in (14) to (17) can be recovered pair-wisely. Obviously, all the decision signals were only a function of a pair of input signals, which existed with dissimilar constellation angles. This offered the possibility of decoding in pairs. We could decide for each pair of recovered symbols independently using a pair-wise least square decoder as follows:  
\[\left(\widetilde{v}_{1},\widetilde{v}_{4}\right)=\operatorname{argmin}\biggl{[} \bigl{|}{\boldsymbol{y}_{k+1,1}{\boldsymbol{y}_{k,1}}^{H}-(v_{1}A+v_{4}B)} \bigr{|}^{2}+\Bigl{|}{\boldsymbol{y}_{k+1,1}{\boldsymbol{y}_{k,4}}^{H}-(v_{1} B+v_{4}A)}\Bigr{|}^{2}\biggr{]} \tag{19}\]  
\[\left(\widetilde{v}_{2},\widetilde{v}_{3}\right)=\operatorname{argmin}\biggl{[} \bigl{|}{\boldsymbol{y}_{k+1,1}{\boldsymbol{y}_{k,2}}^{H}-(v_{2}A-v_{3}B)} \bigr{|}^{2}+\Bigl{|}{\boldsymbol{y}_{k+1,1}{\boldsymbol{y}_{k,3}}^{H}-(v_{3}A-v _{2}B)}\Bigr{|}^{2}\biggr{]} \tag{20}\]  
The pair-wise least square decoder performed an exhaustive search over all possible combination of constellation points to determine the pair of signals that minimized the terms in (19) and (20). This decoding was equivalent to finding the minimum Euclidean distance between the noisy received signals and the known constellation points. The resultant elements of the co-efficient vector in (19) and (20) could be interpreted as noisy versions of elements of the scaled co-efficient vector set. The scaling, \(A\) and \(B\), however, have a negligible effect on the geometry of the detection region. Since all the elements of the co-efficient vector set \(\mathbf{v}\) had equal lengths, the destination selected the closest co-efficient vector to \(\left[\widetilde{v}_{1},\ldots,\widetilde{v}_{T}\right]^{T}\) from \(\mathbf{v}\) as the detector output. Then, inverse mapping was applied to recover the information bits. The complexity of this process was equivalent to \(2^{M+1}\) since this was the number of constellation points to be examined. The pairwise detection algorithm described so far for \(P=4\) could be used for any cooperative network with \(u\) relays (\(2\leq u\leq P\)). The only difference in design was in the structure of the channel matrix \(\mathbf{G}_{k}\) and \(\mathbf{G}_{k+1}\). In this case, the path gain \(g_{q,t,n}(p),p>u\) between the \(p\)th relay node and the destination was set to zero. In other words, the design for a 3-relay network could be achieved from the design for a 4-relay network simply by setting the path gain \(g_{q,t,n}(4)\) between the 4 th relay and the destination to zero. Similarly, cooperative networks with five, six, and seven relays could derive their designs from the design for 8-relay networks.