## 3 Novel Asynchronous Cooperative Diversity Scheme
### The Fast Symbol-Pair-Wise ML Decoder Searching the Global Optimum Signals  
First, the destination strips off the CPs from four OFDM frames. Then, it shifts the last \(N_{cp}\) samples of the \(N\)-point vector as the first \(N_{cp}\) samples to reconstruct the signals, which were manipulated by the relays with the time-reversion. Subsequently, the received signals are performed by the \(N\)-point DFT operation. Second, due to the fact that (\(DFT(s)\))* = \(IDFT(s\)*), (\(IDFT(s)\))* = \(DFT(s\)*), \(DFT(\zeta(DFT(s)))=s\), we can deduce the sufficient statistics with respect to the transmitted signals by representing the resulting signals in terms of the (\(k\),\(l\)) th entry \(s_{kl}\) (\(1\leq k\leq 4\), \(0\leq l\leq N-1\)) of the modulated symbol matrix \(\mathbf{S}\), written by:  
\[\mathbf{V}_{i} = \left[ \begin{array}{cccc}v_{1,\ l}&v_{2,\ l}&v_{3,\ l}&v_{4, \ l}\end{array} \right]\mathbf{=\widetilde{h}S}_{l}+\mathbf{\widetilde{n}} \tag{5}\] \[= \left[\begin{array}{cccc}\widetilde{h}_{1}&\widetilde{h}_{2}& \widetilde{h}_{3}&\widetilde{h}_{4} \begin{bmatrix}S_{1,\ d}&S_{1,\ B}\\ -S_{1,\ B}^{*}&S_{1,\ d}^{*}\end{bmatrix} +\mathbf{\widetilde{n}}\right.\] \[= \left[ \begin{array}{cccc}ph_{S_{2,\ l}}h_{R_{1,\ D}}&pf_{l}^{ \delta_{1}}h_{S_{2,\ l}}^{*}h_{R_{2,\ D}}&pf_{l}^{\delta_{1}}h_{S_{2,\ l}}h_{R_{3,\ D}}&pf_{l}^{\delta_{1}}h_{S_{4,\ l}}^{*}h_{R_{4,\ l}}\begin{bmatrix}s_{1,\ l}&s_{2,\ l}&s_{3,\ l}&s_{4,\ l}\\ -s_{2,\ l}^{*}&s_{1,\ l}^{*}&-s_{4,\ l}^{*}&s_{3,\ l}^{*}\\ -s_{3,\ l}^{*}&-s_{4,\ l}^{*}&s_{1,\ l}^{*}&s_{2,\ l}^{*}\\ s_{4,\ l}&-&s_{3,\ l}&-s_{2,\ l}&s_{1,\ l}\end{array} \right]\] \[+ \left[ \begin{array}{cccc}p\left(N_{1,\ l}(I)h_{R_{1,\ D}}+N_{2,\ l}(I)f_{l}^{ \delta_{1}}h_{R_{2,\ D}}+N_{3,\ l}(I)f_{l}^{\delta_{2}}h_{R_{2,\ D}}+N_{41}(I)f _{l}^{\delta_{3}}h_{R_{4,\ D}}\right)+n_{1}^{\prime}(I)\\ p\left(N_{12}(I)h_{R_{1,\ D}}+N_{22}(I)f_{l}^{\delta_{1}}h_{R_{2,\ D}}+N_{32}(I)f _{l}^{\delta_{2}}h_{R_{3,\ D}}+N_{42}(I)f_{l}^{\delta_{3}}h_{R_{4,\ D}}\right)+n_{2 }^{\prime}(I)\\ p\left(N_{13}(I)h_{R_{1,\ D}}+N_{23}(I)f_{l}^{\delta_{1}}h_{R_{2,\ D}}+N_{33}(I)f _{l}^{\delta_{2}}h_{R_{3,\ D}}+N_{43}(I)f_{l}^{\delta_{3}}h_{R_{4,\ D}}\right)+n_{ 3}^{\prime}(I)\\ p\left(N_{14}(I)h_{R_{1,\ D}}+N_{24}(I)f_{l}^{\delta_{1}}h_{R_{2,\ D}}+N_{34}(I)f_{l}^{ \delta_{2}}h_{R_{3,\ D}}+N_{44}(I)f_{l}^{\delta_{3}}h_{R_{4,\ D}}\right)+n_{4 }^{\prime}(I)\end{array} \right]\]  
where \(S_{1,\ d}=\left[ \begin{array}{cc}s_{1,\ l}&s_{2,\ l}\\ -s_{2,\ l}^{*}&s_{1,\ l}^{*}\end{array} \right]\) and \(S_{1,\ B}=\left[ \begin{array}{cc}s_{3,\ l}&s_{4,\ l}\\ -s_{4,\ l}^{*}&s_{3,\ l}^{*}\end{array} \right]\), \(N_{j,\ k}(I)\) (\(1\leq j\leq 4\), \(1\leq k\leq 4\)) is the DFT of \(n_{jk}(I)\), \(n_{k}^{\prime}(I)\) is the AWGN at the destination with zero mean and unit variance for the \(l\)th symbol of the \(k\)th OFDM frame, \(f_{l}^{\delta_{j,-1}}=e^{-2\pi i\delta_{j,-l}/N}\) (where \(\delta_{j\ -\ 1}\) is the relative time delay of the \(j\)th relay with respect to the first relay) denotes a phase drift in the frequency domain induced by the relative time delay in the time domain among different relays. From (5), it can be seen that the matrix \(\mathbf{S}_{l}\) is of the Jafarkhani structure QSTBC code, which endows the proposed scheme with maximum rate.