Exact Bit Error Probability of Multi-hop Decode-and ... - Semantic Scholar

Exact Bit Error Probability of Multi-hop Decode-and-Forward Relaying with Selection Combining Bao Quoc Vo-Nguyen and Hyung Yun Kong Department of Electrical Engineering, University of Ulsan, San 29 of MuGeo Dong, Nam-Gu, Ulsan, 680-749 Korea {baovnq,hkong}@mail.ulsan.ac.kr http://wcomm.ulsan.ac.kr

Abstract. In this paper, an exact closed-form bit error rate expression for M -PSK is presented for multi-hop Decode-and-Forward Relaying (MDFR) scheme, in which selection combining technique is employed at each node. We have shown that the proposed protocol offers remarkable diversity advantage over direct transmission as well as the conventional decode-and-forward relaying (CDFR) scheme. Simulations are performed to confirm our theoretical analysis. Keywords: Bit Error Rate (BER), Decode-and-Forward Relaying, Rayleigh fading, Selection Combining, M -PSK, cooperative communication.

1

Introduction

Recently, relaying dual-hop transmission has gained more attention under forms of cooperative communications and it is treated as one of the candidates to overcome the channel impairment like fading, shadowing and path loss [1]. The main idea is that in a multi-user network, two or more users share their information and transmit jointly as a virtual antenna array. This enables them to obtain higher diversity than they could have individually [1-9]. In the past, relatively few contributions concerning evaluating performance of the DF relaying protocol with multi relays and maximal ratio combining (MRC) or selection combining (SC) have been published [2-9]. In particular, in [2], Jeremiah Hu and Norman C. Beaulieu derived a closed-form expression for outage probability of the CDFR networks with SC when the statistics of the channels between the source, relays, and destination are assumed to be independent and identically distributed (i.i.d.) and independent but not identically distributed (i.n.d.). In [4, 5], the performance of CDFR with maximal ratio combining at the destination in terms of outage probability and bit error probability over independent but not identically distributed channels was also examined. In [3, 6-9], a class of multi-hop cooperative scheme employing decode-and-forward relaying with MRC, called multi-hop Decode-and-Forward Relaying (MDFR) scheme, was proposed, and various performance metrics were also provided. D.-S. Huang et al. (Eds.): ICIC 2009, LNAI 5755, pp. 718–727, 2009. c Springer-Verlag Berlin Heidelberg 2009 

Exact Bit Error Probability of MDFR with Selection Combining

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However, to the best of the authors’ knowledge, there is no publication concerning the exact expression for bit error rate of the MDFR with selection combining in both i.i.d. and i.n.d. Rayleigh fading channels. In this paper, we focus on selective decode-and-forward relaying where the relay must make an independent decision on whether or not to decode and forward source information [1]. In addition, a concept of cooperative diversity protocols for multi-hop wireless networks, which allows relay nodes to exploit all information they overhear from their previous nodes along the route to the destination to increase the change of cooperation, is applied. To that effect, the receiver at each node can employ a variety of diversity combining techniques to obtain diversity from the multiple signal replicas available from its preceding relaying nodes and the source. Although optimum performance is highly desirable, practical systems often sacrifice some performance in order to reduce their complexity. Instead of using maximal ratio combining which requires exact knowledge of the channel state information, a system may use selection combining which is the simplest combining method. It only selects the best signal out of all replicas for further processing and neglects all the remaining ones. The benefit of using SC as opposed to MRC is reduced hardware complexity at each node in the network. In addition, it also reduces the computational costs and may even lead to a better performance than MRC, because in practice channels with very low SNR can not accurately estimated and contribute much noise. The contributions of this paper are as follows. We derive an exact closedform expression bit error rate for M -PSK of the MDFR scheme. In addition, the comparison between the performance of MDFR and that of CDFR [2] is performed and it confirms that the proposed protocol outperforms CDFR in all range of operating SNRs. The rest of this paper is organized as follows. In Sect. 2, we introduce the model under study and describe the proposed protocol. Section 3 shows the formulas allowing for evaluation of average BER of the system. In Sect. 4, we contrast the simulations and the results yielded by theory. Finally, the paper is closed in Sect. 5.

2

System Model

We consider a wireless relay network consisting of one source, K relays and one destination operating over slow, flat, Rayleigh fading channels as illustrated in Fig. 1. The source terminal (T0 ) communicates with the destination (TK+1 ) via K relay nodes denoted as T1 , · · · , Tk , · · · , TK . Due to Rayleigh fading, the channel powers, denoted by αTi ,Tj = |hTi ,Tj |2 are independent and exponential random variables where hTi ,Tj is the fading coefficient from node Ti to node Tj with i = 0, · · · , K, j = 1, · · · , K + 1 and i < j. We define λTi ,Tj as the expected value of αTi ,Tj . The average transmit powers for the source and the relays are denoted by ρTi with i = 0, · · · , K, respectively. We further define γTi ,Tj = ρTi αTi ,Tj as the instantaneous SNR per bit for the link Ti → Tj .

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T2 T1

T0

T3

T4

Fig. 1. A MDFR system with 3 relays (K = 3)

For medium access, a time-division channel allocation scheme with K +1 time slots is occupied in order to realize orthogonal channelization, thus no inter-relay interference is considered in the signal model. According to the selective DF relaying protocol [1], the relay decides to cooperate or not with the source in its own time slot, based on the quality of its received signals. Since selection combining technique is used, the relay adaptively chooses the strongest signal (on the basic of instantaneous SNR) among available ones to demodulate and then check whether its received data are right or wrong. If they are right, that relay will cooperate with the source in its transmission time slot, otherwise, it will keep silent. We define a decoding set D(Tk ) for node Tk , k = 1, 2, · · · , K + 1, whose members are its preceding relays which decode successfully. So it is obvious that D(Tk ) is a subset of C = {T1 , T2 , · · · , TK }. In real scenario, the decoding set is determined after receiving one frame by employing cyclic-redundancy-check (CRC). However, in this paper, we assumed that the decoding set can be decided by symbol-by-symbol for mathematical tractability of BER calculation [4]. We further assume that the receivers at the destination and relays have perfect channel state information (CSI) but no transmitter CSI is available at the source and relays.

3

BER Analysis

Similarly as in [2-7], namely applying the theorem of total probability, the bit error rate of the multi-hop decode-and-forward relaying can be derived as a weighted sum of the bit error rate for SC at the destination, BD [D(TK+1 )], corresponding to each set of decoding relay D(TK+1 ). Thus the end-to-end bit error rate for M -PSK of the system Pb can be written as  Pb = Pr [D(TK+1 )] BD [D(TK+1 )] (1) D(TK+1 )∈2C

where 2C denotes the power set of C that is the set of all subsets of C.

Exact Bit Error Probability of MDFR with Selection Combining

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Since selection combining is exploited at each relay and the destination, the signal with largest SNR is always selected from the signals received from its k decoding set as well as from the source. Let us define {γi }ni=1 as the instantaneous SNR per bit of each path received by the node Tk from the set D∗ (Tk ) with their k expected values {¯ γi }ni=1 , respectively, where D∗ (Tk ) = D(Tk ) ∪ {T0 } and nk is the cardinality of the set D∗ (Tk ), i.e., nk = |D∗ (Tk )|. Under the assumption that all links are subject to independent fading, the cumulative distribution function (CDF) of βk = max ρTi αTi ,Tk = max γi ∗ i=1,...,nk

Ti ∈D (Tk )

can be determined by [10] Fβk (γ) = Pr[γ1 < γ, . . . , γi < γ, . . . , γnk < γ] =

nk  

1 − e−γ/¯γi

 (2)

i=1

Hence, the joint pdf of βk is given by differentiating (2) with respect to γ [11]. ⎡ ⎤ nk nk   ∂ ⎢ ⎥ i−1 Fβ (γ) = fβk (γ) = (3) ωi e−ωi γ ⎦ ⎣(−1) ∂γ k m ,...,m =1 i=1 1

i

m1