In this work, we extend the non-orthogonal amplify-and-forward (NAF) cooperative diversity scheme to the MIMO channel. A family of space-time block codes for a half-duplex MIMO NAF fading cooperative channel with N relays is constructed. The code construction is based on the non-vanishing determinant (NVD) criterion and is shown to achieve the optimal diversity-multiplexing tradeoff (DMT) of the channel. We provide a general explicit algebraic construction, followed by some examples. In particular, in the single-relay case, it is proved that the Golden code and the 4×4 Perfect code are optimal for the single-antenna and two-antenna case, respectively. Simulation results reveal that a significant gain (up to 10 dB) can be obtained with the proposed codes, especially in the single-antenna case.

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In a recent paper, perfect (n × n) space-time codes were introduced as the class of linear dispersion space-time codes having full rate, non-vanishing determinant, a signal constellation isomorphic to either the rectangular or hexagonal lattices in 2n 2 dimensions and uniform average transmitted energy per antenna. Consequence of these conditions include optimality of perfect codes with respect to the Zheng-Tse Diversity-Multiplexing Gain tradeoff (DMT), as well as excellent low-SNR performance. Yet perfect space-time codes have been constructed only for 2, 3, 4 and 6 transmit antennas. In this paper, we construct perfect codes for all channel dimensions, present some additional attributes of this class of space-time codes and extend the notion of a perfect code to the rectangular case.

Abstract—In this paper, collocated and distributed space-time block codes (DSTBCs) which admit multigroup maximum-likelihood (ML) decoding are studied. First, the collocated case is considered and the problem of constructing space-time block codes (STBCs) which optimally tradeoff rate and ML decoding complexity is posed. Recently, sufficient conditions for multigroup ML decodability have been provided in the literature and codes meeting these sufficient conditions were called Clifford unitary weight (CUW) STBCs. An algebraic framework based on extended Clifford algebras (ECAs) is proposed to study CUW STBCs and using this framework, the optimal tradeoff between rate and ML decoding complexity of CUW STBCs is obtained for few specific cases. Code constructions meeting this tradeoff optimally are also provided. The paper then focuses on multigroup ML decodable DSTBCs for application in synchronous wireless relay networks and three constructions of four-group ML decodable DSTBCs are provided. Finally, the orthogonal frequency-division multiplexing (OFDM)-based Alamouti space-time coded scheme proposed by Li–Xia for a 2-relay asynchronous relay network is extended to a more general transmission scheme that can achieve full asynchronous cooperative diversity for arbitrary number of relays. 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. Four-group decodable DSTBCs applicable in the proposed OFDM-based transmission scheme are also given. Index Terms—Asynchronous cooperative communication, Clifford algebra, cooperative diversity, decoding complexity, distributed space-time codes, orthogonal frequency-division multiplexing (OFDM), space-time codes. I.

Abstract — We consider cooperative relay communication in a fading channel environment under the Orthogonal Amplify and Forward (OAF) and Orthogonal and Non-Orthogonal Selection Decode and Forward (OSDF and NSDF) protocols. For all these protocols, we compute the Diversity-Multiplexing Gain Tradeoff (DMT). We construct DMT optimal codes for the protocols which are sphere decodable and, in certain cases, incur minimum possible delay. Our results establish that the DMT of the OAF protocol is identical to the DMT of the Non-Orthogonal Amplify and Forward (NAF) protocol. Two variants of the NSDF protocol are considered: fixed-NSDF and variable-NSDF protocol. In the variable-NSDF protocol, the fraction of time duration for which the source alone transmits is allowed to vary with the rate of communication. Among the class of static amplify-and-forward and decode-and-forward protocols, the variable-NSDF protocol is shown to have the best known DMT for any number of relays apart from the two-relay case. When there are two relays, the variable-NSDF protocol is shown to improve on the DMT of the best previously-known protocol for higher values of the multiplexing gain. Our results also establish that the fixed-NSDF protocol has a better DMT than the NAF protocol for any number of relays. Finally, we present a DMT optimal code construction for the NAF protocol. Index Terms — cooperative diversity, distributed space-time code, orthogonal amplify and forward, non orthogonal amplify and forward, selection decode and forward, diversitymultiplexing gain tradeoff, space-time codes, cyclic division algebra codes. I.

Abstract — Design criteria and full-diversity Distributed Space Time Codes (DSTCs) for the two phase transmission based cooperative diversity protocol of Jing-Hassibi and the Generalized Nonorthogonal Amplify and Forward (GNAF) protocol are reported, when the relay nodes are assumed to have knowledge of the phase component of the source to relay channel gains. It is shown that this under this partial channel state information (CSI), several well known space time codes for the colocated MIMO (Multiple Input Multiple Output) channel become amenable for use as DSTCs. In particular, the well known complex orthogonal designs, generalized coordinate interleaved orthogonal designs (GCIODs) and unitary weight single symbol decodable (UW-SSD) codes are shown to satisfy the required design constraints for DSTCs. Exploiting the relaxed code design constraints, we propose DSTCs obtained from Clifford Algebras which have low ML decoding complexity.

Explicit codes are constructed that achieve the diversity-multiplexing gain tradeoff of the cooperative-relay channel under the dynamic decode-andforward protocol for any network size and for all numbers of transmit and receive antennas at the relays. A particularly simple code construction that makes use of the Alamouti code as a basic building block is provided for the single relay case. Along the way, we prove that space-time codes previously constructed in the literature for the block-fading and parallel channels are approximately universal, i.e., they achieve the DMT for any fading distribution. It is shown how approximate universality of these codes leads to the first DMT-optimum code construction for the general, MIMO-OFDM channel. 1

Abstract—This paper presents serially concatenated trellis coded modulations (SCTCMs) that perform consistently close to the available mutual information for periodic erasure channel (PEC), periodic fading channel (PFC) and the 2 × 2 compound matrix channel. We use both the maximum-likelihood decoding criteria and iterative decoding criteria to design universal SCTCMs for the PEC and the PFC. For the space-time channel, by de-multiplexing the symbols across the antennas, the proposed universal SCTCMs for the period-2 PFC deliver consistent performance over the eigenvalue skew of the matrix channel. Within the family of channels having the same eigenvalue skew, a time-varying linear transformation (TVLT) is used to mitigate the performance variation over different eigenvectors. The proposed space-time SCTCMs of 1.0, 2.0 and 3.0 bits per transmission require excess mutual information in the ranges 0.11-0.15, 0.23-

We consider single-source single-sink (ss-ss) multi-hop relay networks, with slow-fading links and single-antenna half-duplex relay nodes. While two-hop cooperative relay networks have been studied in great detail in terms of the diversity-multiplexing tradeoff (DMT), few results are available for more general networks. In this paper, we identify two families of networks that are multi-hop generalizations of the two-hop network: K-Parallel-Path (KPP) networks and layered networks. KPP networks, can be viewed as the union of K node-disjoint parallel relaying paths, each of length greater than one. KPP networks are then generalized to KPP(I) networks, which permit interference between paths and to KPP(D) networks, which possess a direct link from source to sink. We characterize the DMT of these families of networks completely for K> 3. Layered networks are networks comprising of layers of relays with edges existing only between adjacent layers, with more than one relay in each layer. We prove that a linear DMT between the maximum diversity dmax and the maximum multiplexing gain of 1 is achievable for single-antenna fully-connected layered networks. This is shown to be equal to the optimal DMT if the number of relaying layers is less than 4. For multiple-antenna KPP and layered networks, we provide an achievable DMT, which is significantly better than known lower bounds for half duplex networks.