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This paper studied a power and bit-rate allocation optimization problem of a multi-carrier wireless bidirectional relaying system with the aim of minimizing the total transmit-power of the system. By using channel knowledge at the transmitters, the optimization problem was solved analytically, resulting in a joint power and bit-rate allocation algorithm. Simulation results confirmed the effectiveness of the proposed algorithm. It is shown that with the proposed power and bit-rate allocation algorithm, the total transmit-power of the system can be significantly reduced in contrast to the compared schemes. Introduction Recently, the realization of energy efficient wireless communications has attracted lots of interest in the research community. As such, the power minimization problem (i.e., saving as much energy as possible) has become a hot research topic for wireless bidirectional relaying systems (WBRSs) [1][9]. By using the channel statistics, the authors of [1] proposed a transmit-power allocation algorithm with the aim of reducing the overall transmit-power of the WBRS having asymmetric traffic requirements. By exploiting the statistic channel knowledge at transmitters, a joint power allocation and relay location algorithm was developed in [2] to minimize the power consumption of an asymmetric WBRS and meanwhile ensure the require quality-of-service of the system. In [3], a hybrid unidirectional and bidirectional relaying scheme developed in [4] was extended to an orthogonal frequency division multiplexing (OFDM) modulated WBRS. Moreover, the study of energy efficiency and spectrum efficiency trade-off for WBRSs can be found in [5] and references therein. By using the instantaneous and statistical channel knowledge, the authors of [8] developed two optimal power adaptation algorithms with the aim of achieving energy efficiency, while satisfying the individual peak-power limit on each node and the quality-of-service requirement of the system. It should be noted that most of the existing studies as mentioned earlier focus only on the singlecarrier systems. Thus, it is natural to question how to extend the proposed algorithms to multi-carrier systems? Do they need to be improved or they should be completely re-devised? So far, this issue has not been well studied and deserves a though investigation. In this paper, an OFDM modulated WBRS is Figure 1. Wireless bidirectional relaying system model. considered, where two source nodes exchange messages via a half-duplex amplify-and-forward (AF) relay. The main objective is to investigate effective methods for controlling transmit-power levels of each nodes in 163 the system, and meanwhile allocating the end-to-end rates to different sub-carriers so as to reduce the overall transmit-power of the system, while ensuring the rate requirements of the two source nodes. System Model An OFDM modulated multi-carrier WBRS is considered, where each node is equipped with a single antenna and operates in a half-duplex style. Figure 1 presents the system model under investigation, where two source nodes SA and SB would like to send messages to each other with the help of an AF relay node SR. Denote by Mi = [mi(1) , mi(2),· · ·, mi(T)] , i∈{A, B}, the OFDM symbol transmitted by node Si, where T is the number of the sub-carriers that can be used, and mi(t) is the message send by the source node Si on the t-th sub-carrier, t∈{1, 2, · · ·, T}. Denote by Pj = diag{pj(1), · · · , pj(T)}, j∈{A, B, R} the transmit-power at the source node Sj. The channel gain from node Si to node Sj is denoted by Hij= diag{hij(1) , · · · , hij(T)}, i, j∈{A, B, R} and i≠j. Here, hij(t) is modeled as  ij ij ij d g h  , where gij and dij denote the large-scale behavior and the distance, respectively, and α is the corresponding path-loss coefficient. Without loss of generality, it is assumed that gij obeys CN(0, 1), and the channels are reciprocal, namely, Hij = Hji. Denote by wi, i∈{A, B, R}, the additive white Gaussian noise (AWGN) at the node Si. It’s assumed that all the elements of the AWGNs obey i.i.d.   0,1 CN  . Figure 1 shows that one-round of communication between the two sources consists of two phases. During phase 1, the two source nodes SA and SB, respectively, send MA and MB to the relay SR. During phase 2, the relay SR first scales each sub-carrier of the received signal by a factor F [3]. Next, the relay broadcasts the resulting signal to the two sources. At the end of phase 2, after decoding, the source Si can obtain the mutual information given by [3], [9]     2 1 1 1 log 1 2 T T i i i t t I I t t             (1) where   i I t and   i t  are the mutual information and the signal-to-noise ratio (SNR) for the -th t sub-carrier. Here,   i t  can be written as [3], [9]                     2 2 2 2 1 j R AR BR i i R iR j jR p t p t h t h t t p t p t h t p t h t          (2) where t∈{1, 2, · · ·, T}. Problem Description and Solution The objective here is to develop a joint power and bit-rate allocation algorithm to reduce the total transmit-power of the system. Thus, a general optimization problem can be formulated as

Joint Power and Bit-rate Allocation for Multi-carrier Wireless Bidirectional Relaying Systems