Cooperative Transmission for Relay Networks Based on Second-Order Statistics of Channel State Information

Cooperative transmission in relay networks is considered, in which a source transmits to its destination with the help of a set of cooperating nodes. The source first transmits locally. The cooperating nodes that receive the source signal retransmit a weighted version of it in an amplify-and-forward (AF) fashion. Assuming knowledge of the second-order statistics of the channel state information, beamforming weights are determined so that the signal-to-noise ratio (SNR) at the destination is maximized subject to two different power constraints, i.e., a total (source and relay) power constraint, and individual relay power constraints. For the former constraint, the original problem is transformed into a problem of one variable, which can be solved via Newton's method. For the latter constraint, this problem is solved completely. It is shown that the semidefinite programming (SDP) relaxation of the original problem always has a rank one solution, and hence the original problem is equivalent to finding the rank one solution of the SDP problem. An explicit construction of such a rank one solution is also provided. Numerical results are presented to illustrate the proposed theoretical findings.

[1]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[2]  Zhi-Quan Luo,et al.  Distributed Beamforming for Relay Networks Based on Second-Order Statistics of the Channel State Information , 2008, IEEE Transactions on Signal Processing.

[3]  Babak Hossein Khalaj,et al.  Cooperative Transmission for Wireless Relay Networks Using Limited Feedback , 2010, IEEE Transactions on Signal Processing.

[4]  H. Vincent Poor,et al.  Collaborative beamforming for distributed wireless ad hoc sensor networks , 2005, IEEE Transactions on Signal Processing.

[5]  Shahram Shahbazpanahi,et al.  Optimal Distributed Beamforming for Two-Way Relay Networks , 2010, IEEE Transactions on Signal Processing.

[6]  Constantin Zalinescu.  On the Maximization of (not necessarily) Convex Functions on Convex Sets , 2006, J. Glob. Optim..

[7]  How to split the eigenvalues of a one-parameter family of matrices , 2001 .

[8]  D K Smith,et al.  Numerical Optimization , 2001, J. Oper. Res. Soc..

[9]  H. G. ter Morsche,et al.  Computation of eigenvalue and eigenvector derivatives for a general complex-valued eigensystem , 2006 .

[10]  Emre Telatar,et al.  Capacity of Multi-antenna Gaussian Channels , 1999, Eur. Trans. Telecommun..

[11]  Olvi L. Mangasarian,et al.  Nonlinear Programming , 1969 .

[12]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[13]  Carl D. Meyer,et al.  Matrix Analysis and Applied Linear Algebra , 2000 .

[14]  P. Tseng.  Convergence of a Block Coordinate Descent Method for Nondifferentiable Minimization , 2001 .

[15]  Michael L. Overton,et al.  Second Derivatives for Optimizing Eigenvalues of Symmetric Matrices , 1995, SIAM J. Matrix Anal. Appl..

[16]  Saeed Gazor,et al.  Multiple Peer-to-Peer Communications Using a Network of Relays , 2009, IEEE Transactions on Signal Processing.

[17]  H. Vincent Poor,et al.  Distributed transmit beamforming: challenges and recent progress , 2009, IEEE Communications Magazine.

[18]  Arogyaswami Paulraj,et al.  Optimal linear precoders for MIMO wireless correlated channels with nonzero mean in space-time coded systems , 2006, IEEE Transactions on Signal Processing.

[19]  A. Conn,et al.  An Efficient Method to Solve the Minimax Problem Directly , 1978 .

[20]  Shahram Shahbazpanahi,et al.  Filter-and-Forward Distributed Beamforming in Relay Networks With Frequency Selective Fading , 2010, IEEE Transactions on Signal Processing.

[21]  H. Vincent Poor,et al.  Opportunistic collaborative beamforming with one-bit feedback , 2008, IEEE Transactions on Wireless Communications.

[22]  R. Gallager.  Principles of Digital Communication , 2008 .

[23]  Hrvoje Stančić,et al.  Principles of Digital Communication , 2011 .

[24]  Aria Nosratinia,et al.  Coded cooperation in wireless communications: space-time transmission and iterative decoding , 2004, IEEE Transactions on Signal Processing.

[25]  H. Vincent Poor,et al.  Weighted Cross-Layer Cooperative Beamforming for Wireless Networks , 2009, IEEE Transactions on Signal Processing.

[26]  Gregory W. Wornell,et al.  Cooperative diversity in wireless networks: Efficient protocols and outage behavior , 2004, IEEE Transactions on Information Theory.

[27]  Jan de Leeuw,et al.  Block-relaxation Algorithms in Statistics , 1994 .

[28]  Shuzhong Zhang,et al.  Complex Matrix Decomposition and Quadratic Programming , 2007, Math. Oper. Res..

[29]  I. Stancu-Minasian.  Nonlinear Fractional Programming , 1997 .

[30]  H. Vincent Poor,et al.  Opportunistic collaborative beamforming with one-bit feedback , 2008, SPAWC 2008.

[31]  Michael Gastpar,et al.  Cooperative strategies and capacity theorems for relay networks , 2005, IEEE Transactions on Information Theory.

[32]  Y. Danieli.  Guide , 2005 .

[33]  Reuven Chen.  Solution of minimax problems using equivalent differentiable functions , 1985 .

[34]  K. Schittkowski,et al.  NONLINEAR PROGRAMMING , 2022 .

[35]  Yindi Jing,et al.  Network Beamforming Using Relays With Perfect Channel Information , 2007, IEEE Transactions on Information Theory.

[36]  Zhu Han,et al.  Improving Wireless Physical Layer Security via Cooperating Relays , 2010, IEEE Transactions on Signal Processing.

[37]  Don H. Johnson,et al.  Signal-to-noise ratio , 2006, Scholarpedia.

[38]  Yindi Jing,et al.  Distributed beamforming in wireless relay networks with quantized feedback , 2008, IEEE Journal on Selected Areas in Communications.

[39]  Björn E. Ottersten,et al.  Collaborative-Relay Beamforming With Perfect CSI: Optimum and Distributed Implementation , 2009, IEEE Signal Processing Letters.