## 1 Introduction  
Though multiple transmit antennas and receiver antennas at nodes would lead to substantial gains in channel capacity and robust abilities to combat with different kinds of interference over a scattering-rich wireless environment, constrained by the space complexities, it is impractical for nodes in _ad hoc_ networks or distributed large scale wireless networks [1]. Meanwhile, the communication between the nodes accomplished by a single hop is severely affected by fading, path loss, _etc_. If the nodes between the source and the destination are regarded as the relay nodes that participate in the transmission, a virtual multi-antenna system, referred to as cooperative diversity, could be built, which leads the wireless _ad hoc_ networks [1] being indispensable to the next generation broadband radio access networks [2].  
The efficiency of the cooperative diversity for wireless _ad hoc_ networks [1,2] would be evaluated by three aspects: (i) whether it achieves full diversity; (ii) whether it transmits symbols with maximum rate; and (iii) whether the receiver could exploit the decoder with both good performance and a lower complexity [1,3].  
However, synchronization is essential to most of relevant researches on cooperative diversity, which violates the principle that the cooperative diversity is asynchronous since each relay is furnished with different antennas, each terminal of which operates in its own local working frequency. This in turn lets asynchronous cooperative diversity [4], an open research topic to deal with this problem, be attracted the most attention.  
Taking into account asynchronous characteristic of different mobile nodes, almost all of the distributed space time codes can achieve full diversity [5, 6, 7]. However, a few of them give attention to three all. Among them is the Alamouti space time transmission scheme (ASTTS) [7] that stems from the ideas of orthogonal frequency division multiplexing (OFDM), time-reversion and complex conjugation. Nevertheless, confined to Alamouti code structure, its diversity order is 2. Previously, based on the quasi-orthogonal code structure in the Jafarkhani scheme [8], _etc._, a rotated quasi-orthogonal space time block code (RQSTBC) [3] that provides both full diversity and maximum rate was proposed for four transmit antennas. In this paper, based on the ideas in [3,7,8], we present a RQSTBC asynchronous cooperative scheme, diversity order of which is 4. In contrast to the original one in [7], our novel contribution constructs the frequency domain transmitted signals of the relay nodes in terms of combining the quasi-orthogonal criterion originally proposed in [8] with time-reversion in [7]. The application of this idea lets the received signals at the destination turn on Jafarkhani code [8] structure and thus guarantees both the maximum symbol transmission rate being 1 and fast symbol-pair-wise ML decoder being used. Moreover, the source not only picks half of the symbols from a signal constellation set and the other half of them from that constellation which is rotated with the optimum angle and but also creates OFDM frames to make up for time delays of the signals. In this way, the received signals at the destination take on RQSTBC with the Jafarkhani code structure. It is this step that the proposed scheme can ensure full diversity.  
Theoretical analysis and simulation results show that the performance of the proposed scheme is superior to those of QSTBC scheme and ASTTS scheme. This is attributed to the higher diversity order of the new scheme.  
Throughout this paper we adopt the following notational conventions. Boldface capitals and lower-case letters symbolize matrices and vectors, respectively, while ( )\({}^{T}\) and ( )\({}^{H}\) represent the transpose and Hermitian transpose, respectively. Furthermore, \(\zeta\)( ) denotes the time-reversion of the signal, while ( )\({}^{*}\) expresses the complex conjugation of the signal. Finally, det[ ] is the determinant of a matrix.