Maximizing Reliability in Multi Hop Wireless Network - Paper | L A 1, Papers of Humanities

Download Maximizing Reliability in Multi Hop Wireless Network - Paper | L A 1 and more Papers Humanities in PDF only on Docsity! Maximizing Reliability In Multi-Hop Wireless Networks Rahul Vaze and Robert W. Heath Jr. The University of Texas at Austin Department of Electrical and Computer Engineering Wireless Networking and Communications Group 1 University Station C0803 Austin, TX 78712-0240 email: vaze@ece.utexas.edu, rheath@ece.utexas.edu Abstract— Distributed space-time block coding is a diversity technique to mitigate the effects of fading in multi-hop wireless networks, where multiple relay stages are used by a source to communicate with its destination. This paper proposes a new distributed space-time block code called the cascaded orthogonal space-time block code (COSTBC) for the case where the source and destination are equipped with multiple antennas and each relay stage has one or more single antenna relays. Each relay stage is assumed to have receive channel state information (CSI) for all the channels from the source to itself, while the destination is assumed to have receive CSI for all the channels. To construct COSTBC, multiple orthogonal space-time block codes are used in cascade by the source and each relay stage. COSTBC is shown to achieve the maximum diversity gain in a multi-hop wireless network with flat Rayleigh fading channels. An explicit construction of COSTBCs is also provided. It is also shown that COSTBC requires minimum decoding complexity thanks to the connection to orthogonal space-time block codes. I. INTRODUCTION It is well known that for multiple antenna point-to-point wireless channels, space-time block codes (STBC) [1], [2] can be used to improve reliability by introducing redundancy in space and time. The improved reliability is due to the increased diversity gain offered by multiple antennas, where diversity gain [2] is defined as the negative of the exponent of the signal-to-noise ratio (SNR) in the pairwise error probability expression at high SNR. Recently, the concept of STBC has also been extended to wireless networks [3]–[5] where multiple antennas of other nodes (called relays) in the network are used to construct STBC, however, in a distributed manner. Such codes are called distributed space-time block codes (DSTBC). Most of the work on the DSTBC [3]–[5] considers a two-hop wireless network, where in the first hop the source transmits the signal to all the relays and in the next hop all relays simultaneously transmit a function of the received signal to the destination. For two-hop wireless networks a DSTBC design is proposed in [4], [5] using an amplify and forward (AF) strategy, where each relay transmits a relay specific unitary transformation of the received signal. It was shown in [4], [5], that to maximize the diversity gain, the STBC transmitted by all relays using a unitary transformation should be a full-rank STBC. Algebraic constructions of maximum diversity gain achieving DSTBC for the two-hop wireless network are provided in [6]–[8]. During the preparation of this manuscript we came across three recent papers on DSTBC construction for multi-hop wireless networks [9]–[11]. In this paper we design maximum diversity gain achieving DSTBCs for multi-hop wireless networks, where multiple relay stages are used by the source to communicate with its destination. We assume that the source and the destination terminals have multiple antennas while the relays in each stage have a single antenna. We also assume that all the nodes in the network (source, relays and destination) can only work in half-duplex mode (cannot transmit and receive at the same time) and each relay and the destination has perfect receive channel state information (CSI). We propose an AF based multi-hop DSTBC, called the cascaded orthogonal space-time block code (COSTBC), where an orthogonal space-time code (OSTBC) [12] is used by the source and each relay stage to communicate with its adjacent relay stage. OSTBCs are considered because of their single symbol decodable property [1], [12], i.e. each constellation symbol of the OSTBC can be separated at the receiver with independent noise terms. With our proposed COSTBC, in the first time slot the source transmits an OSTBC to the first relay stage. Using the single symbol decodable property of the OSTBC, each relay of the first relay stage separates the different OSTBC constellation symbols from the received signal and transmits a codeword vector in the next time slot, such that the matrix obtained by stacking all the codeword vectors transmitted by the different relays of the first relay stage is an OSTBC. These operations are repeated by subsequent relay stages. We prove that COSTBC achieves the maximum diversity gain in two or more hop wireless networks when CSI is available at each relay and the destination in the receive mode. We also give an explicit construction of COSTBC for different source antennas and relay configurations. We also prove that COSTBCs have the single symbol decodable property similar to OSTBCs, and as a result COSTBCs require minimum decoding complexity. Notation: A denotes a matrix, a a vector and ai the ith element of a. det(A) and tr(A) denotes the determinant of and trace of matrix A while A 1 2 denotes the element wise square root of matrix A with all non-negative entries. The maximum eigen-value of a matrix A is denoted by λmax(A). CM×N denotes the space of M × N matrices with complex entries. |.| denotes the usual Euclidean norm of a vector. Im is an m×m identity matrix and 0m is as all zero m×m matrix. R and C denote the field of real and complex numbers. The superscripts T ,∗ ,† represent the transpose, transpose conjugate and element wise conjugate. For matrices A,B by A ≤ Stage N−1 Relay 1





Relay 2 M 0 Relay M 1 h 1








Relay 1 Relay j RelayRelay M s M s+1 Relay 1 f ij s Relay i Source Relay 1




M N Relay M N−1 Relay p g p Destination





















































































Stage 1 1 2 Stage s Stage s+1 1 2 Fig. 1. System Diagram B,A,B ∈ Cm×m we mean xAx∗ ≤ xBx∗, ∀x ∈ C1×m. Ex(f(x)) denotes the expectation of function f with respect to x. x ∼ CN (0, σ) means x is a circularly symmetric complex Gaussian random variable with zero mean and variance σ. To define something we use the symbol :=. II. SYSTEM MODEL In this section we describe the system model considered in this paper. We consider a multi-hop wireless network where a source terminal with M0 antennas wants to communicate with a destination terminal with MN antennas via N − 1 stages of relays as shown in Fig. 1, where each relay node in any relay stage has single antenna and Mn denotes the number of relay nodes in the nth relay stage. It is assumed that all the relays do not have any data of their own and can only operate in half- duplex mode i.e. cannot transmit and receive at the same time. We also assume that any relay of relay stage n can receive signal from any relay of relay stage n − 1 only. Throughout this paper we refer to this multi-hop wireless network with N − 1 relay stages as N -hop network. As shown in Fig. 1, the channel between the source and the ith relay of the first stage of relays is denoted by hi = [h1i h2i . . . hM0i] T , i = 1, 2, . . . , M1, between the jth relay of relay stage s and the kth relay of relay stage s + 1 by fsjk, j = 1, 2, . . . , Ms k = 1, 2, . . . , Ms+1 and the channel between the lth relay of the N − 1 relay stage and the destination by gl = [gl1 gl2 . . . glMN ] T l = 1, 2, . . . , MN−1. We assume that hi ∈ CM0×1, fsjk ∈ C1×1, gl ∈ CMN×1 with independent and identically distributed (i.i.d.) CN (0, 1) entries for all i, j, k, l, s. We assume that the mth relay of nth stage knows hi, fsjk ∀i, j, k, s = 1, 2, . . . , n− 2, fn−1jm ∀j and the destination knows both hi, fsjk,gl, ∀ i, j, k, l, s. We further assume that all these channels are frequency flat and block fading, where the channel coefficients remain constant in a block of time duration Tc and change independently from block to block. A. Problem Formulation The problem we consider is in this paper is to design DSTBCs that achieve the maximum diversity gain in a N - hop network, where the diversity gain dC of a DSTBC C is defined as [2], [4] dC = − lim E→∞ log Pe (E) log E where Pe (E) is the pairwise codeword error probability of the DSTBC C at the destination and E is the sum of the power used by each node in the network. To identify the limits on the maximum possible diversity gain in a N -hop network, an upper bound on the diversity gain achievable with any DSTBC is presented next. Theorem 1: The diversity gain dC for N -hop network is upper bounded by min {MnMn+1} , n = 0, 1, . . . , N − 1. Proof: Let dC be the diversity gain of DSTBC C in a N - hop network. Let dn be the diversity gain of the best possible DSTBC Copt that can be used between relay stage n and n+1 when all the relays in relay stage n and relay stage n + 1 are allowed to collaborate, respectively, and the source message is known to all the relays of relay stage n without any error and all the relays of the relay stage n + 1 can send the received signal to the destination error free. Then, clearly, dC ≤ dn. Since the channel between the relay stage n and n + 1 is a multiple antenna channel with Mn transmit and Mn+1 receive antennas, dn ≤ MnMn+1. Hence dC ≤ MnMn+1. Since this is true for every n = 0, 1, . . . , N − 1, it follows that dC ≤ min{MnMn+1}, n = 0, 1, . . . , N − 1. III. CASCADED ORTHOGONAL SPACE-TIME BLOCK CODE In this section we propose cascaded orthogonal space- time block code (COSTBC) to construct DSTBCs for N - hop network which is shown to achieve maximum diversity gain (Theorem 1). With COSTBC in the first time slot of time duration M0, the source terminal transmits a rate-L/M0 orthogonal space-time block code (OSTBC) S0 ∈ CM0×M0 to all the relays of relay stage 1. For all the definitions and properties of OSTBC we refer the reader to [12]. The received signal r1k ∈ CM0×1 at relay k of relay stage 1 can be written as r1k = √ E0S0hk + n 1 k (1) where S0 ∈ S0 with E{tr(S∗0S0)} = M0 and E0 is the power transmitted by the source at each time instant. The noise n1k is the M0 × 1 spatio-temporal white complex Gaussian noise independent across relays with E(n1kn1∗k ) = IM0 . Since S0 is an OSTBC, with CSI the received signal r1k can be transformed