Recently Li and Xia have proposed a transmission scheme for wireless relay networks based on the Alamouti space time code and orthogonal frequency division multiplexing to combat the effect of timing errors at the relay nodes. This transmission scheme is amazingly simple and achieves a diversity order of two for any number of relays. Motivated by its simplicity, this scheme is extended to a more general transmission scheme that can achieve full cooperative diversity for any number of relays. The conditions on the distributed space time code (DSTC) structure that admit its application in the proposed transmission scheme are identified and it is pointed out that the recently proposed full diversity four group decodable DSTCs from precoded co-ordinate interleaved orthogonal designs and extended Clifford algebras satisfy these conditions. It is then shown how differential encoding at the source can be combined with the proposed transmission scheme to arrive at a new transmission scheme that can achieve full cooperative diversity in asynchronous wireless relay networks with no channel information and also no timing error knowledge at the destination node. Finally, four group decodable distributed differential space time codes applicable in this new transmission scheme for power of two number of relays are also provided.
Coding for cooperative wireless relay networks has attracted considerable attention recently. Distributed space time coding was proposed as a coding strategy to achieve full cooperative diversity in [1] assuming that the signals from all the relay nodes arrive at the destination at the same time. But this assumption is not close to practicality since the relay nodes are geographically distributed. In [3], a transmission scheme based on orthogonal frequency division multiplexing (OFDM) at the relay nodes was proposed to combat the timing errors at the relays and a high rate space time code (STC) construction was also provided. However, the maximum likelihood (ML) decoding complexity for this scheme is prohibitively high especially for the case of large number of relays. Several other works in the literature propose methods to combat the timing offsets but most of them are based on decode and forward at the relay node and moreover fail to address the decoding complexity issue. In [2], a simple transmission scheme to combat timing errors at the relay nodes was proposed. This scheme is particularly interesting because of its associated low ML decoding complexity. In this scheme, OFDM is implemented at the source node and time reversal/conjugation is performed at the relay nodes on the received OFDM symbols from the source node. The received signals at the destination after OFDM demodulation are shown to have the Alamouti code structure and hence single symbol maximum likelihood (ML) decoding can be performed. However, the Alamouti code is applicable only for the case of two relay nodes and for larger number of relays, the authors of [2] propose to cluster the relay nodes and employ Alamouti code in each cluster. But this clustering technique provides diversity order of only two and fails to exploit the full cooperative diversity equal to the number of relay nodes.
The main contributions of this report are as follows.
• The Li-Xia transmission scheme is extended to a more general transmission scheme that can achieve full asynchronous cooperative diversity for any number of relays.
• The conditions on the STC structure that admit its application in the proposed transmission scheme are identified. The recently proposed full diversity four group decodable distributed STCs in [4,5,6] for synchronous wireless relay networks are found to satisfy the required conditions for application in the proposed transmission scheme.
• It is shown how differential encoding at the source node can be combined with the proposed transmission scheme to arrive at a transmission scheme that can achieve full asynchronous cooperative diversity in the absence of channel knowledge and in the absence of knowledge of the timing errors of the relay nodes. Moreover, an existing class of four group decodable distributed differential STCs [7] for synchronous relay networks with power of two number of relays is shown to be applicable in this setting as well.
In Section 2, the basic assumptions on the relay network model are given and the Li-Xia transmission scheme is briefly described. Section 3 describes the transmission scheme proposed in this report and also provides four group decodable codes for any number of relays. Section 4 briefly explains how differential encoding at the source node can be combined with the proposed transmission scheme and four group decodable distributed differential STCs applicable in this scenario are also proposed. Simulation results and discussion on further work comprise Sections 5 and 6 respectively.
Notation: Vectors and matrices are denoted by lowercase and uppercase bold letters respectively. I m denotes an m × m identity matrix and 0 denotes an all zero matrix of appropriate size. For a set A, the cardinality of A is denoted by |A|. A null set is denoted by φ. For a matrix, (.) T , (.) * and (.) H denote transposition, conjugation and conjugate transpose operations respectively. For a complex number, (.) I and (.) Q denote its in-phase and quadrature-phase parts respectively.
2 Relay network model assumptions and the Li-
In this section, the basic relay network model assumptions are given and the Li-Xia transmission scheme in [2] is briefly described. The transmission scheme in [2] is based on the use of OFDM at the source node and the Alamouti code implemented in a distributed fashion for a 2 relay system. Essentially, the transmission scheme in [2] is applicable mainly for the case of 2 relays but by forming clusters of two relay nodes, it can be extended to more number of relays at the cost of sacrificing diversity benefits.
Consider a network with one source node, one destination node and R relay nodes U 1 , U 2 , . . . , U R . This is depicted in Fig. 1. Every node is assumed to have only a single antenna and is half duplex constrained. The channel gain between the source and the i-th relay f i and that between the j-th relay and the destination g j are assumed to be quasi-static, flat fading and
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