4 Consider one time slot. Let x = { x$_{1}$, · · · , x$_{N}$ } T be the vector representing the signals transmitted by the stations. We assume that all powers (including noise powers) are normalized with respect to the transmit power of a station. Furthermore, all stations use the same transmit power. Thus, E { x 2 $_{i}$} = 1 , ∀ i . We also assume that E { x$_{i}$ } = 0 , ∀ i , and that there is no cooperative coding among the stations so that E { x$_{i}$x$_{j}$ } = 0 , ∀ i = j . Let y = { y$_{1}$, · · · , y$_{N}$ } T be the received signals at the relay, and u = { u$_{1}$, · · · , u$_{N}$ } T be the noise vector with i.i.d. noise samples following the complex Gaussian distribution, i.e., u$_{n}$ ∼ N$_{c}$ (0 , σ 2 $_{r}$) . Then y = H$_{u}$x + u , (1) where H$_{u}$ is the uplink channel gain matrix. The relay multiplies y by a beamforming matrix G before relaying the signals. We impose a power constraint on the signals transmitted by the relay so that E {‖ Gy ‖ $^{2}$} = p. (2) Combining (1) and (2), we have E [ X $^{H}$HH $_{u}$G$^{H}$GH$_{u}$ X + u $^{H}$G$^{H}$Gu ] = p. (3) This gives Tr [ H H $_{u}$G$^{H}$GH$_{u}$ ] + Tr [ G $^{H}$G ] σ 2 r = p. (4) Let H$_{d}$ be the downlink channel matrix. Then, the received signals at the stations in vector form are r = H$_{d}$Gy + w = H$_{d}$GH$_{u}$x + H$_{d}$Gu + w , (5) B. MIMO Switching where w is the noise vector at the receiver, with the i.i.d. noise samples following the complex Gaussian distribution, i.e., w$_{n}$ ∼ N$_{c}$ (0 , σ $^{2}$) . Suppose that the purposes of G are to realize a particular permutation without diagonal elements represented by the permutation matrix P , and to provide signal amplifications for the signals coming from the stations. That is, H$_{d}$GH$_{u}$ = AP , (6)