## 4 Simulation and Discussion  
A simulation investigation is carried out to demonstrate the performance of the proposed scheme. We use four cooperative schemes, _i.e._, three asynchronous transmission schemes (RQSTBC, ASTTS, QSTBC), and one synchronous transmission scheme (decode forward quasi-orthogonal space time (DQST) [3, 7]). The simulation results are described in two aspects, _i.e._, the performance of RQSTBC over a range of the rotation angles and the SER. Let us consider that, in a Rayleigh fading channel, there is a two-hop system, shared by one source, one destination, and four relays except for ASTTS where there are only two relays. The source employs the binary phase-shift keying modulation (BPSK) and 4-ary quadrature amplitude modulation (4-QAM). Assume that \(N=64\), \(N_{\mathrm{cp}}=16\) and the delay \(\delta\) is the integer which follows the uniform distribution in interval [0, 15]. Figure 2 depicts the SERs of the RQSTBC scheme versus Eb/N0 values over a range of rotation angles, with the source using BPSK and 4-QAM, respectively. For the graphic clarification, not all of simulation results are shown. The optimal rotation angles (Here, the optimal rotation angle means that if the source selects half of the symbols from a constellation set rotated with this angle, the RQSTBC scheme would achieve the lowest SER performance bound. ), which are used for the other simulations, are 1/2 \(\pi\) for BPSK, and 1/4 \(\pi\) for 4-QAM, respectively. Moreover, let the CGD between a pair of code words \(\mathbf{S}_{l}= \begin{bmatrix}s_{1,l}&s_{2,l}&s_{3,l}&s_{4,l}\\ -s_{2,l}^{*}&s_{1,l}^{*}&-s_{4,l}^{*}&s_{3,l}^{*}\\ -s_{3,l}^{*}&-s_{4,l}^{*}&s_{1,l}^{*}&s_{2,l}^{*}\\ s_{4,l}&-s_{3,l}&-s_{2,l}&s_{1,l}\end{bmatrix} \) and \(\mathbf{S}_{k}= \begin{bmatrix}s_{1,k}&s_{2,k}&s_{3,k}&s_{4,k}\\ -s_{2,k}^{*}&s_{1,k}^{*}&-s_{4,k}^{*}&s_{3,k}^{*}\\ -s_{3,k}^{*}&-s_{4,k}^{*}&s_{1,k}^{*}&s_{2,k}^{*}\\ s_{4,k}&-s_{3,k}&-s_{2,k}&s_{1,k}\end{bmatrix} \) from the RQSTBC in (5) be expressed by [10]:  
\[ \begin{split} CGD(\mathbf{S}_{l}\,\mathbf{S}_{k})& =\det\!\left[\!\left(\mathbf{S}_{l}-\mathbf{S}_{k}\right)^{\! \prime\prime}\cdot\left(\mathbf{S}_{l}-\mathbf{S}_{k}\right)\right]\\ &=\left\|\!\left(s_{1,l}-s_{1,k}\right)-\left(s_{4,l}-s_{4,k} \right)\!\right|^{2}+\left|\!\left(s_{2,l}-s_{2,k}\right)+\left(s_{3,l}-s_{3, k}\right)\!\right|^{2}\!\right\rangle\cdot\\ &\left\|\!\left(s_{1,l}-s_{1,k}\right)+\left(s_{4,l}-s_{4,k} \right)\!\right|^{2}+\left|\!\left(s_{2,l}-s_{2,k}\right)-\left(s_{3,l}-s_{3, k}\right)\!\right|^{2}\!\right\rangle\end{split}  \tag{8}\]  
If \(CGD_{\text{min}}(\phi)\) represents the minimum CGD of the RQSTBC scheme, then, it could be found that this minimum CGD is a function of \(\phi\). Therefore, for a given constellation, to find an optima angle for any transmission scheme with QSTBC as its relay method, based on the determinant criterion, is equal to maximize \(CGD_{\text{min}}(\phi)\) among all possible rotations. Manipulated as the same way in [3], we would derive the same results from (8), which are consistent with the results in [3]. Figure 2: **(a)** The SERs of the RQSTBC scheme versus Eb/N0, for the rotation angles being equal to 1/8\(\pi\), 1/4\(\pi\), 3/8\(\pi\), and 1/2\(\pi\), with the source using BPSK (left). **(b)** The SERs of the RQSTBC scheme versus Eb/N0, for the rotation angles being equal to 1/8\(\pi\), 1/4\(\pi\), 3/8\(\pi\), and 1/2\(\pi\), with the source using 4 QAM (right). Figure 3 represents the SERs of four cooperative schemes versus Eb/N0, with the source using BPSK, the rotation angle of which for the RQSTBC scheme is 1/2\(\pi\) and 4-QAM, the rotation angle of which for the RQSTBC scheme is 1/4\(\pi\), respectively. The SER of the proposed scheme is not only a few orders of magnitude lower than those of the ASTTS and the DQST but also lower than that of the QSTBC when the source uses BPSK. However, when the source employs 4-QAM, the SER of the proposed scheme is somewhat higher than those of the DQST and QSTBC when Eb/N0 is lower.