A Semi-Closed-Form Solution to Optimal Distributed Beamforming for Two-Way Relay Networks

Abstract

In this correspondence, we present a computationally simple semi-closed-form solution to the problem of designing distributed beamformer for two-way (bi-directional) multi-relay networks. In such a network, the relay nodes use amplify-and-forward relaying protocol to help two transceivers exchange information in a bidirectional manner. We consider a total power minimization approach to optimally find the relay beamforming weights and the transceiver transmit powers. This approach is based on the minimization of the total transmit power, consumed in the whole network, subject to SNR constraints at the two transceivers. We show that as far as the relay beamforming weight vector is concerned, this minimization problem is equivalent to the minimization of the total transmit power for a one-way relay network where the target SNR of the receiving transceiver is equal to the sum of the target SNRs of the two transceivers in the original two-way relay network. Based on this observation, we show that the relay beamforming weight vector can be obtained in a closed from given that an intermediate parameter, namely the transmit power of the transmitter in the equivalent one-way relay network, is available. This intermediate parameter is shown to be the solution to a one-dimensional optimization problem, and thus, it can be obtained using a simple bisection method. Our semi-closed-form solution not only reveals the structure of the optimal beamforming weight vector, but also leads to a one-dimensional search regardless of the number of relays. This provides the computational advantage over the gradient based numerical method of Havary-Nassab et al., where the gradient dimension reflects the number of relays.Index Terms-bidirectional network beamforming, cooperative network, distributed beamforming, power control, relay selection, SNR balancing, two-way communications, two-way relaying.

Jointly optimal distributed beamforming and power control in asynchronous two-way relay networks

“…In (11), using the definitions of H0(w) and H1(w) in [9] and noting that RcpH1(w) = 0N s×Nt holds true, the output of the CP removal block is written as rq(i) as…”

“…It is wellknown that the inner optimization problem in (42) is amenable to a semi-closed-form solution for the optimal wn, denoted w o n , which is given by [11] w o n =κn zn(β 1 +β 2 ) λn…”

Optimal resource sharing and network beamforming for bidirectional relay networks

“…The optimization problem (35) has been proved to have a unique minimum, hence, we can solve it using the efficient algorithm proposed in [1], [3]. We now discuss how the control channel can be employed to broadcast the information needed by the two networks nodes to calculate their design parameters of interest.…”

“…In the literature on two-way relay networks, amplify-and-forward (AF) relaying protocol has been used frequently because it can be implemented via simple analogue amplification circuits. Based on different design criteria, numerous two-way relay techniques have been studied in the literature, [1]- [5]. However, the problem of spectral, temporal and, relay resource sharing between two or multiple networks with different priorities is still at the edge of exploration.…”

Optimal power allocation and network beamforming for OFDM-based relay networks

“…In the first step, we obtain wi by solving the minimization problem (13) for a given feasible pair of r1 and r2 such that 1 T r1 = r max 1 and 1 T r2 = r max 2 . This problem is guaranteed to have a unique solution given as bellow [28] …”