Algebraic Distributed Differential Space-Time Codes with Low Decoding Complexity

C ONSIDER the scenario of R + 1 users wanting to communicate to a single destination using cooperative strategies. For simplicity, we consider one of the R+1 users as the source and the remaining R users as relays. Predominantly there are two types of relaying strategies discussed in the literature, i.e., 1) amplify and forward and 2) decode and forward. In this work, we focus only on amplify and forward based relaying strategies wherein the relays are allowed to perform linear processing of the received signal. Amplify and forward based relaying is simpler and moreover does not require the relays to have knowledge of the codebook used by the source or the knowledge of the source to relay channel fading gains. Recently in [1], the idea of spacetime coding for collocated MIMO (Multiple Input Multiple Output) systems has been applied in the setting of cooperative wireless relay networks in the name of distributing spacetime coding, wherein coding is performed across users and time. This strategy provides each user a diversity order equal to the number of relays even though all the users are only equipped with a single antenna. The diversity thus achieved is called as cooperative diversity. However, such strategies require that the destination have complete knowledge of the fading coefficients from all the users to itself as well as that of the fading coefficients between users. Obtaining the knowledge of the fading coefficients between the users at the destination requires additional resources. To solve this problem, in [2], Kiran et al have proposed a differential encoding/decoding setup for cooperative wireless networks that does not require the knowledge of fading coefficients between the users at the destination. Such codes were named as partially coherent distributed space-time codes in [2]. In a recent work [4], it has been shown that the same strategy of [2] offers full diversity with a suboptimal receiver that does not require the knowledge of any of the fading coefficients. In an independent work [3], Jing and Jafarkhani have proposed a differential encoding/decoding setup for cooperative wireless relay networks which is more general than the setup proposed in [2], [4] and they have also provided few code constructions. We call the class of Differential Space-Time Codes (DSTCs) which can be used in a distributed manner for cooperative diversity as Distributed DSTCs (and denote by DDSTCs) to differentiate them from DSTCs for collocated MIMO systems. The problem of designing DDSTCs is more challenging than that of DSTCs for collocated MIMO systems, since in this scenario we have additional constraints to be satisfied which are due to the cooperative diversity protocol. In [5], the setup of [2], [4] has been generalized to the case when the source and destination nodes have multiple antennas. In [5], the authors propose the use of a random diagonal unitary matrix codebook in general and for certain specific number of relays (3, 6 and 9) propose the use of cyclic codes. However, a random code is not guaranteed to offer full diversity. Moreover designing good cyclic codes becomes difficult for large number of relays. In [3], the authors propose the use of Alamouti code and Sp(2) code for the specific case of 2 and 4 relays. For an arbitrary number of relays, the authors construct fully diverse DDSTCs based on circulant matrices. But, except for the Alamouti code all other DDSTC constructions in the literature [2], [3], [4], [5] have large decoding complexity. Thus, a general full diversity code construction targeting the requirements of low encoding complexity as well as low decoding complexity is not available. This issue gains significant importance especially if the number of cooperating terminals is large, which is quite expected in applications like wireless sensor networks.

In this paper we address this issue and present the following contributions:

• The notion of encoding complexity for Space-Time codes in terms of g-group encodability is introduced (Definition 1) and its significance is highlighted.

• The differential encoding/decoding setup introduced by Kiran et al [2], Oggier et al [5] and Jing et al [3] for wireless relay networks that use codebooks consisting of unitary matrices is generalized to allow codebooks consisting of scaled unitary matrices. This generalization aids in reducing the encoding/decoding complexity of DDSTCs. When the codebook of scaled unitary matrices is used in the Jing-Hassibi protocol [1] for differential encoding at the source, the extra conditions involving the interrelationship between relay matrices and the codeword matrices that need to be satisfied are identified (Eqn. ( 4)). ) is proposed which achieves full diversity with a low complexity sub-optimal receiver (Theorem 3). Explicit construction of multidimensional signal sets that lead to full diversity (Theorem 2) are also provided. To the best of our knowledge, this is the first known low decoding comple

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