Capacity Region of the Two-Way Multi-Antenna Relay Channel with Analog Tx-Rx Beamforming Authors: Christian Lameiro, Alfredo Nazábal, Fouad Gholam, Javier Vía and Ignacio Santamaría University of Cantabria, Santander, SPAIN
Outline 1 RF-MIMO Two-Way Relay Channel (RF-TWRC) 2 Capacity Region 3 Semidefinite Relaxation 4 Numerical Examples 5 Conclusion
Two-way relay channel Two-phase protocol......... 1 R 2 Multiple access channel (MAC).......... 1 R 2 Broadcast channel (BC). Amplify-and-forward (AF) strategy with two-phase protocol. MAC phase: the nodes transmit simultaneously to the relay. BC phase: the relay retransmits the linearly processed signal. Perfect CSI: each node is able to null out the interference.
Overview of the state-of-the-art The capacity region of the TWRC-AF when the source nodes are single antenna terminals with fixed powers has been computed (R. Zhang et al., 2009). A suboptimal algorithm has been recently proposed for the MIMO-TWRC with AF strategy (Wang and Zhang, 2010). The capacity region of the TWRC with decode-and-forward strategy has been computed (R.F. Wyrembelski et al., 2009). In this work, we consider the multi-antenna TWRC-AF when the nodes perform analog beamforming.
RF-MIMO terminals based on analog beamforming Reduction of system size, hardware cost and power consumption. Multiplexing gain equal to 1, but full array and diversity gains. Point-to-point links, BC and MAC thoroughly studied.
System model (I) In the MAC phase, each node performs transmit beamforming with {v 1,v 2 } C NS 1, respectively; and the relay receives through the beamformer u R C NR 1. In the BC phase, the relay applies the transmit beamformer v R C NR 1 to the received signal, and the nodes perform receive beamforming with {u 1,u 2 } C NS 1, respectively. The power transmitted by the relay, assuming without loss of generality v R 2 = 1, is p R = p 1 u H R H 1 v 1 2 +p2 u H R H 2 v 2 2 +σ 2 u R 2
System model (II) Assuming perfect CSI, each node removes the self-interference before decoding its desired message. Rx signal: y i = u H i HT i v Ru H R H jv j pj s j + r i, i,j = 1,2,i j H 1 and H 2 are flat fading MIMO channels, and r i is AWGN with zero mean and variance σ [1+ u 2 R 2 u H i H T i v 2] R, i = 1,2. Goal Under power constraints at the nodes and the relay, obtain u 1, u 2, u R, v 1, v 2, v R and the power allocation to operate at any point of the capacity boundary.
Outline 1 RF-MIMO Two-Way Relay Channel (RF-TWRC) 2 Capacity Region 3 Semidefinite Relaxation 4 Numerical Examples 5 Conclusion
Optimal node beamformers Maximum ratio transmission (MRT): v i = HH i u R H H i u, i = 1,2 R Maximum ratio combining (MRC): u i = HT i v R H T i v R, i = 1,2 The optimal relay beamformers lie in the subspace spanned by the columns of the channel matrices, and can be expressed as u R = Ua r H 1 = UG 1 v R = U a t H 2 = UG 2 where U contain the left eigenvectors of [H 1,H 2 ].
The achievable bidirectional rate pairs are R 12 1 2 log p ef1 G T 2 1+ 2 a t 2 σ (1+ a 2 r 2 G T 2 a 2) t R 21 1 2 log p ef2 G 2 1+ T 1 a 2 t ( σ 2 1+ a r 2 ) G T 1 a t 2 where p efi = p i G H i a r 2 is the effective power of node i. Capacity region C(P 1,P 2,P R ) {R 12,R 21 } p 1 P 1,p 2 P 2 a t 2 =1,p R(p 1,p 2,a r) P R
A WSRmax problem cannot be invoked due to its non-convexity. The rate profile method (M. Mohseni et al., 2006) can be used to efficiently characterize the boundary of the capacity region. Proposed algorithm: for a fixed 0 α 1, bisection method over γ sum, solving the following optimization problem in each step. 3 G H 2 G H 2 minimize p 1 1 a r +p2 2 a r +σ 2 a r 2 p 1,p 2,a t,a r 2.5 subject to : p 2 G H 2 a r 2 G T 1 a t 2 σ 2 ( 1+ a r 2 G T 1 at 2 ) αγsum p 1 G H 1 a r 2 G T 2 a t 2 σ 2 ( 1+ a r 2 G T 2 at 2 ) (1 α)γsum R 2 (bps/hz) 2 1.5 1 0.5 Capacity region Rate profile p 1 P 1 0 0.5 1 1.5 2 2.5 3 p 2 P 2 R 1 (bps/hz) 0
Outline 1 RF-MIMO Two-Way Relay Channel (RF-TWRC) 2 Capacity Region 3 Semidefinite Relaxation 4 Numerical Examples 5 Conclusion
The initial problem is non-convex, but a solution can be found through a relaxed semidefinite programm (SDP). New optimization variables Ar = a r a H r Equivalent problem: A t = a t a H t minimize p 1 Tr(R 1 A r )+p 2 Tr(R 2 A r )+σ 2 Tr(A r ) p 1,p 2,A t,a r subject to : p 2 Tr(R 2 A r ) (1 α)γ sum σ 2 Tr(A r ) (1 α)γ sumσ 2 p 1 Tr(R 1 A r ) αγ sum σ 2 Tr(A r ) αγ sumσ 2 Tr(A t ) = 1 A t 0, rank(a t ) = 1, p 1 P 1 A r 0, rank(a r ) = 1, p 2 P 2 where R i = G i G H i, i = 1,2. Tr(R 2 A t) Tr(R 1 A t)
Analysis of the equivalent problem The equivalent problem is still non-convex due to: The cross products between the powers and the beamforming matrices. The rank-one constraints. Managing the non-convexity of the equivalent problem We can avoid the cross products by optimizing the effective powers instead, and changing the power constraints accordingly. We can find a solution of the equivalent problem relaxing the rank-one constraints, what is called a relaxed SDP.
Final optimization problem (I) Convex problem minimize p ef1,p ef2,a t,a r p ef1 +p ef2 +σ 2 Tr(A r ) subject to : p ef2 (1 α)γ sum σ 2 Tr(A r ) (1 α)γ sumσ 2 p ef1 αγ sum σ 2 Tr(A r ) αγ sumσ 2 Tr(A t ) = 1 A t 0 A r 0 p ef1 P 1 Tr(R 1 A r ) p ef2 P 2 Tr(R 2 A r ) Tr(R 2 A t) Tr(R 1 A t)
Final optimization problem (II) Key observation: if the rank of the optimal beamforming matrices is greater than one, we are able to find an optimal rank-one solution through the matrix decomposition theorem for Hermitian matrices (Y. Huang and S. Zhang, 2007). Optimal powers After solving the optimization problem, the optimal powers are given by p ( ) 1 = p ( ) 2 = ( Tr ( Tr p ( ) ef 1 R 1 A ( ) r p ( ) ef 2 R 2 A ( ) r ) )
Outline 1 RF-MIMO Two-Way Relay Channel (RF-TWRC) 2 Capacity Region 3 Semidefinite Relaxation 4 Numerical Examples 5 Conclusion
Conventional MIMO vs. analog beamforming Example scenario with single-antenna nodes, i.e., N S = 1, and fixed powers. The relay has N R = 4 antennas, and the SNR is 10 db. 1.4 1.2 Conventional MIMO RF MIMO 1.4 1.2 Conventional MIMO RF MIMO 1 1 R 2 (bps/hz) 0.8 0.6 R 2 (bps/hz) 0.8 0.6 0.4 ρ = 0.1 0.4 ρ = 0.5 0.2 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 R 1 (bps/hz) 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 R 1 (bps/hz) As ρ increases, i.e., more collinear channels, the capacity gap between analog and conventional beamforming schemes goes to 0.
Capacity region Example scenario with N S = 2 and N R = 4. The SNR is 10 db and the channels have unit variance. 3 2.5 maximum power transmission 2 R 2 (bps/hz) 1.5 1 RF TWRC SISO TWRC SISO TWRC without power optimization 0.5 0 0 0.5 1 1.5 2 2.5 3 R 1 (bps/hz) Some points of the boundary are achieved when the nodes do not transmit at maximum power.
Sum rate vs. SNR analysis The figure shows the sum-rate capacity through Monte Carlo simulations, considering N S = 1, N R = 4 and fixed powers. 7 6 RF MIMO Conventional MIMO SISO 5 Sum Rate (bps/hz) 4 3 2 1 0 0 2 4 6 8 10 12 14 16 18 20 SNR (db)
Outline 1 RF-MIMO Two-Way Relay Channel (RF-TWRC) 2 Capacity Region 3 Semidefinite Relaxation 4 Numerical Examples 5 Conclusion
Conclusion RF-MIMO wireless radios result in low-cost systems with reduced power consumption. The capacity region of the RF-TWRC has been completely characterized. The optimal beamforming vectors and the power allocation can be efficiently computed using convex optimization techniques. The capacity gap between analog and conventional beamforming schemes, when the nodes are single-antenna, goes towards 0 as the angle between the channels decreases.