Space-time block codes (STBCs) that are single-symbol decodable (SSD) in a co-located multiple antenna setting need not be SSD in a distributed cooperative communication setting. A relay network with N relays and a single source-destination pair is called a partially-coherent relay channel (PCRC) if the destination has perfect channel state information (CSI) of all the channels and the relays have only the phase information of the source-to-relay channels. In this paper, first, a new set of necessary and sufficient conditions for a STBC to be SSD for co-located multiple antenna communication is obtained. Then, this is extended to a set of necessary and sufficient conditions for a distributed STBC (DSTBC) to be SSD for a PCRC, by identifying the additional conditions. Using this, several SSD DSTBCs for PCRC are identified among the known classes of STBCs. It is proved that even if a SSD STBC for a co-located MIMO channel does not satisfy the additional conditions for the code to be SSD for a PCRC, single-symbol decoding of it in a PCRC gives full-diversity and only coding gain is lost. It is shown that when a DSTBC is SSD for a PCRC, then arbitrary coordinate interleaving of the in-phase and quadrature-phase components of the variables does not disturb its SSD property for PCRC. Finally, it is shown that the possibility of {\em channel phase compensation} operation at the relay nodes using partial CSI at the relays increases the possible rate of SSD DSTBCs from $\frac{2}{N}$ when the relays do not have CSI to 1/2, which is independent of N.
Deep Dive into Single-Symbol ML Decodable Distributed STBCs for Partially-Coherent Cooperative Networks.
Space-time block codes (STBCs) that are single-symbol decodable (SSD) in a co-located multiple antenna setting need not be SSD in a distributed cooperative communication setting. A relay network with N relays and a single source-destination pair is called a partially-coherent relay channel (PCRC) if the destination has perfect channel state information (CSI) of all the channels and the relays have only the phase information of the source-to-relay channels. In this paper, first, a new set of necessary and sufficient conditions for a STBC to be SSD for co-located multiple antenna communication is obtained. Then, this is extended to a set of necessary and sufficient conditions for a distributed STBC (DSTBC) to be SSD for a PCRC, by identifying the additional conditions. Using this, several SSD DSTBCs for PCRC are identified among the known classes of STBCs. It is proved that even if a SSD STBC for a co-located MIMO channel does not satisfy the additional conditions for the code to be SSD
The problem of fading and the ways to combat it through spatial diversity techniques have been an active area of research. Multiple-input multiple-output (MIMO) techniques have become popular in realizing spatial diversity and high data rates through the use of multiple transmit antennas. For such co-located multiple transmit antenna systems low maximumlikelihood (ML) decoding complexity space-time block codes (STBCs) have been studied by several researchers [1]- [10] which include the well known complex orthogonal designs (CODs) and their generalizations. Recent research has shown that the advantages of spatial diversity could be realized in single-antenna user nodes through user cooperation [11], [12] via relaying.
A simple wireless relay network of N + 2 nodes consists of a single source-destination pair with N relays. For such relay channels, use of CODs [1], [2] has been studied in [13]. CODs are attractive for cooperative communications for the following reasons: i) they offer full diversity gain and coding gain, ii) they are ‘scale free’ in the sense that deleting some rows does not affect the orthogonality, iii) entries are linear combination of the information symbols and their conjugates which means only linear processing is required at the relays, and iv) they admit very fast ML decoding (single-symbol decoding (SSD)). However, it should be noted that the last property applies only to the decode-and-forward (DF) policy at the relay node.
In a scenario where the relays amplify and forward (AF) the signal, it is known that the orthogonality is lost, and hence the destination has to use a complex multi-symbol ML decoding or sphere decoding [13], [14]. It should be noted that the AF policy is attractive for two reasons: i) the complexity at the relay is greatly reduced, and ii) the restrictions on the rate because the relay has to decode is avoided [15]. In order to avoid the complex ML decoding at the destination, in [16], the authors propose an alternative code design strategy and propose a SSD code for 2 and 4 relays. For arbitrary number of relays, recently in [17], distributed orthogonal STBCs (DOSTBCs) have been studied and it is shown that if the destination has the complete channel state information (CSI) of all the source-to-relay channels and the relay-to-destination channels, then the maximum possible rate is upper bounded by 2 N complex symbols per channel use for N relays. Towards improving the rate of transmission and achieving simultaneously both fulldiversity as well as SSD at the destination, in this paper, we study relay channels with the assumption that the relays have the phase information of the source-to-relay channels and the destination has the CSI of all the channels. Coding for partially-coherent relay channel (PCRC, Section 2.2) has been studied in [18], where a sufficient condition for SSD has been presented.
The contributions of this paper can be summarized as follows:
• First, a new set of necessary and sufficient conditions for a STBC to be SSD for colocated multiple antenna communication is obtained. The known set of necessary and sufficient conditions in [8] is in terms of the dispersion matrices (weight matrices) of the code, whereas our new set of conditions is in terms of the column vector representation matrices [5] of the code and is a generalization of the conditions given in [5] in terms of column vector representation matrices for CODs.
• A set of necessary and sufficient conditions for a distributed STBC (DSTBC) to be SSD for a PCRC is obtained by identifying the additional conditions. Using this, several SSD DSTBCs for PCRC are identified among the known classes of STBCs for co-located multiple antenna system.
• It is proved that even if a SSD STBC for a co-located MIMO channel does not satisfy the additional conditions for the code to be SSD for a PCRC, single-symbol decoding of it in a PCRC gives full-diversity and only coding gain is lost.
• It is shown that when a DSTBC is SSD for a PCRC, then arbitrary coordinate interleaving of the in-phase and quadrature-phase components of the variables does not disturb its SSD property for PCRC.
• It is shown that the possibility of channel phase compensation operation at the relay nodes using partial CSI at the relays increases the possible rate of SSD DSTBCs from 2 N when the relays do not have CSI to 1 2 , which is independent of N.
• Extensive simulation results are presented to illustrate the above contributions.
The remaining part of the paper is organized as follows: In Section 2, the signal model for a PCRC is developed. Using this model, in Section 3, a new set of necessary and sufficient conditions for a STBC to be SSD in a co-located MIMO is presented.
Consider a wireless network with N + 2 nodes consisting of a source, a destination, and N relays1 , as shown in Fig. 1. All nodes are half-duplex nodes, i.e., a node can either transmit or receive at a time on a specific frequency. We consider
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