## 3 Novel Asynchronous Cooperative Diversity Scheme
### The Fast Symbol-Pair-Wise ML Decoder Searching the Global Optimum Signals
The ML decoder for signals in (5) selects the symbol matrix with respect to the transmitted signal by minimizing the log of the likelihood function:\[\hat{\mathbf{S}}_{l}=\min_{\mathbf{S}_{l}}\left|\mathbf{V}_{l}-\widetilde{\mathbf{h} }\mathbf{S}_{l}\right|^{2} \tag{6}\]  
Moreover, the quasi-orthogonality of the matrix \(\mathbf{S}_{l}\) simplified the optimum decision to express in the following sum as:  
\[\min_{s_{1,j},s_{4,j}}f(s_{1,j},s_{4,j})+\min_{s_{2,j},s_{3,j}}f(s _{2,j},s_{3,j}) \tag{7}\] \[\qquad=\min_{s_{1,j},s_{4,j}}\Biggl{\{}\Biggl{(}\sum_{n=1}^{4} \left|\widetilde{h}_{n}\right|^{2}\left|\widetilde{\left|s_{1,j}\right|^{2}} \right|+\left|s_{4,j}\right|^{2}\right)\] \[\qquad\qquad+2\,\mathrm{Re}\Bigl{\{}\bigl{(}-\widetilde{h}_{1}v _{1,j}^{*}-\widetilde{h}_{2}^{*}v_{2,j}-\widetilde{h}_{3}^{*}v_{3,j}- \widetilde{h}_{4}v_{4,j}^{*}\bigr{)}s_{1,j}\] \[\qquad\qquad+\Bigl{(}-\widetilde{h}_{1}v_{4,j}^{*}+\widetilde{h }_{2}^{*}v_{3,j}+\widetilde{h}_{3}^{*}v_{2,j}-\widetilde{h}_{4}v_{1,j}^{*} \bigr{)}s_{4,j}\Bigr{\}}\] \[\qquad\qquad+4\,\mathrm{Re}\Bigl{\{}\widetilde{h}_{1}\widetilde{ h}_{4}^{*}-\widetilde{h}_{2}^{*}\widetilde{h}_{3}\Bigr{\}}\mathrm{Re} \Bigl{\{}s_{1,j}s_{4,j}^{*}\Bigr{\}}\ \Bigr{\}}\] \[\qquad\qquad+\min_{s_{2,j},s_{3,j}}\Biggl{\{}\Biggl{(}\sum_{n=1}^{ 4}\left|\widetilde{h}_{n}\right|^{2}\left|\widetilde{\left|s_{2,j}\right|^{2} +\left|s_{3,j}\right|^{2}\right)\] \[\qquad\qquad+2\,\mathrm{Re}\Bigl{\{}\bigl{(}\widetilde{h}_{1}^{ *}v_{2,j}-\widetilde{h}_{2}v_{1,j}^{*}+\widetilde{h}_{3}v_{4,j}^{*}- \widetilde{h}_{4}^{*}v_{3,j}\bigr{)}s_{2,j}\] \[\qquad\qquad+\Bigl{(}\widetilde{h}_{1}^{*}v_{3,j}+\widetilde{h}_{2 }v_{4,j}^{*}-\widetilde{h}_{3}^{*}v_{1,j}-\widetilde{h}_{4}^{*}v_{2,j}\Bigr{)} s_{3,j}\Bigr{\}}\] \[\qquad\qquad+4\,\mathrm{Re}\Bigl{\{}-\widetilde{h}_{1}^{*} \widetilde{h}_{4}+\widetilde{h}_{2}^{*}\widetilde{h}_{3}^{*}\Bigr{\}}\mathrm{ Re}\Bigl{\{}s_{2,j}s_{3,j}^{*}\Bigr{\}}\ \Biggr{\}}\]  
This leads to the fast symbol-pair-wise ML decoder with a lower complexity but best performance. Furthermore, we can find that it is due to the fact that the source selects half of the symbols from a rotated signal constellation set that ensures the proposed scheme with \(\det\bigl{[}\left(\mathbf{S}_{l}-\mathbf{S}_{m}\right)^{H}\left(\mathbf{S}_{l }-\mathbf{S}_{m}\right)\bigr{]}\
eq 0\) for any \(0\leq l\
eq m\leq N-1\), which, based on the analysis of [9], implies its diversity gain 4(1-loglogP/P), approaching the diversity order 4 when the total power P is large. Meanwhile, if in the proposed scheme, the source randomly picks up the symbols from a signal constellation set, the resulting scheme is of QSTBC characteristic, that is to say, it cannot guarantee \(\left(\mathbf{S}_{l}-\mathbf{S}_{m}\right)\) is full rank such that its diversity order is smaller than 4. Similarly, it would be inferred that the ASTTS scheme obtains diversity order 2. Since the diversity order determines the slope of the symbol error rate (SER) curve, the performance of the RQSTBC scheme is superior to those of QSTBC scheme and ASTTS scheme, which would be verified in our simulation in next section.