## 2 System Model  
Considering a cooperative communication system with a source \(S\), a number of potential relays \(R_{j}\) (\(j=1\),..., k,..., M = 2 K (To achieve full diversity and full symbol transmission rate, it is necessary that there are M = 4 relays in a two-hop cooperative communication system. Thus, there are only four relays in this figure.)) and a destination \(D\). Constrained by the transmitted power and the distance, the data to be transmitted from the source node to the destination node have to be with the aid of the relay nodes, the cooperative communication of which are divided into two phases. In Phase I, \(S\) broadcasts its information to potential relays, and a destination \(D\). In Phase II, \(S\) stops transmission, and potential relays start to transmit. The configuration of a cooperative communication system is depicted in Figure 1.  
In this figure, each node is equipped with a single half duplex radio and a single antenna, while \(h_{S\;R_{j}}\) and \(h_{R_{j}\;D}\) denote the channel coefficients which link the source node to the \(j\)th relay node and the \(j\)th relay node to the destination node, respectively. When all nodes work in an asynchronous cooperative communication mode, in a given time interval, D(t), the equivalent received signal is the superposition of the signals, obtained from a mapping from the M relays transmitted signals to the destination terminal, together with the white noise as expressed by:  
\[D\big{(}t\big{)}{=}\sum_{j=1}^{M}h_{{}_{S\;R_{j}}}\Big{(}f_{{}_{R_{j}}}\Big{(} h_{{}_{S\;R_{j}}}\left(S\big{(}t+\delta_{j}\big{)}\big{)}\right){+}\;n(t) \tag{1}\]  
where \(h_{{}_{S\;R_{j}}}\) and \(h_{{}_{R_{j}\;D}}\) are defined above, \(f_{{}_{R_{j}}}\) represents a mapping performed by the \(j\)th relay node, \(S\)(t) means transmitted symbol from the source node, and \(\delta\) is the relative time delay of the signals arrived at the destination node from the source node, \(n(t)\) is the additive white Gaussian noise (AWGN) stochastic process with zero mean and unit variance at time t.  
In order to obtain the most likely information transmitted from the source node, we apply the statistic features of time delay among potential paths of relays. By doing that, the constraint of synchronization is removed, and different kinds of asynchronous cooperative diversity scheme are also acquired. Though asynchronous cooperative diversity can be achieved by asynchronous cooperative transmission through not only adjusting the transmitted time delays among relay nodes to ensure that the data are sent orthogonal but also enhancing the performance of the relevant receiver, this approach can not guarantee obtain full diversity gain [4]. It is to further devise the distributed space-time coding  
Figure 1: The configuration of a cooperative communication system. schemes, based on asynchronous cooperative diversity, with both higher transmission rate and higher diversity gain that serves as a motivation to write this paper. With recent advances in space-time codes, more and more novelty achievements in this area have been applied to construct asynchronous cooperative diversity schemes to improve the performance of the distributed space-time coding schemes [5; 6; 7; 9]. In [7], a scheme of the orthogonal frequency division multiplexing transmission combined with Alamouti space-time coding is applied to achieve asynchronous cooperative diversity. However, confined by the characteristic of Alamouti space-time coding, it has to use clusters to expand scale while increasing the number of relay nodes, which adds the difficulty of coordination among relay nodes. Meanwhile, the orthogonal complex signals with both full diversity gain and full symbol transmission rate can be designed iff the number of antennas is two. Therefore, the transmission rate of the asynchronous cooperative system, based on orthogonal space-time block codes, is less than one when the number of the cluster nodes increases.