12 that is The relay transmission power can be rewritten as Problem Definition 3 : Given H$_{u}$ , H$_{d}$ , p, σ $^{2}$, σ2 r , and a desired permutation P . Solve for G , B , σ 2 e to optimize the effective throughput of the network. | a$_{j}$ | = | σ | √ σ 2 e − σ 2 r ∑ k | h ( − 1) u, ( i$_{j}$,k $_{)}$| 2 − σ 2 $_{r}$| b$_{j}$ | 2 ∑ k | h ( − 1) u, ( j,k $_{)}$| 2 , ∀ j. (22) Tr { [ H − 1 d A ( P + B )] $^{H}$H − 1 d A ( P + B ) } + σ 2 r · Tr { [ H − 1 d A ( P + B ) H − 1 u ] $^{H}$H − 1 d A ( P + B ) H − 1 u } = p. (23) Numerical Method 3 : We give a naive method here. Assume B = b I and b is a real scalar for simplicity. We set a search range for b , and for each trial of b , there are N equations in (22) and one equation in (23). Use the N + 1 equations to solve a$_{j}$ for j = 1 , · · · , N and σ 2 $_{e}$. Then find the minimum σ 2 e in among all trials within the search range of b , and the corresponding A and B as the solution. A and B are restricted to be real so far. In this section we consider complex A and B . Recall that with a given effective noise power σ 2 e we can calculate the power consumption of the relay. For complex A and B , with the extra degree of freedom of their phases, the relay power consumption can be reduced. In other words, given the transmit power constraint of the relay we can optimize the phases of A and B to reduce σ 2 $_{e}$. However, the problem is nonconvex thus intractable. V. DISCUSSION ON COMPLEX A AND B We propose two schemes to exploit the phases of A and B . The first scheme is targeted for the pairwise transmission pattern without network coding. The amplitudes of A are calculated by Numerical Method 1. Then for any pair of stations in the overall pairwise pattern, their elements in A are set to have counter phases (i.e., out of phase by π ). This setting could be proved to be optimal for two-way relay channel. To limit the scope of the current paper, this new result of ours will be published shortly elsewhere. For larger numbers of stations, this setting can still improve the throughput performance, as will be shown in Section VI. The second scheme is for the general case (pairwise or non-pairwise transmission pattern, with or without network coding). The amplitudes of A and B are calculated by Numerical Method 1 and 3 respectively. Then we divide the interval of [0 , 2 π ) equally into M bins with the values of 0 , 2 π M , · · · , 2( M − 1) π M respectively and randomly pick among them to set the phase values for the variables of A and B . We perform L trials of these random phase assignments. For each trial we could solve for