Codebook Design for Noncoherent MIMO Communications Via Reflection Matrices

1. Codebook Design for Noncoherent MIMO Communications Via Reflection Matrices Daifeng Wang and Brian L. Evans {wang, bevans}@ece.utexas.edu Wireless Networking and Communications Group The University of Texas at Austin IEEE Global Telecommunications Conference November 28, 2006 1

2. Problem Statement • What problem I have solved? • Design an optimal codebook for noncoherent MIMO communications. • What mathematical model I have formulated? • Inverse Eigenvalues Problem • What approach I have taken? • Using Reflection matrices • What goal I have achieved? • Low searching complexity without any limitation 2

3. Noncoherent Communications • Unknown Channel State Information (CSI) at the receiver • Fast Fading channel • e.g. wireless IP mobile systems • No enough time to obtain CSI probably • Difficult to decode without CSI 3

4. Noncoherent MIMO Channel Model • Noncoherent block fading model [Marzetta and Hochwald, 1999] • Channel remains constant over just one block • Mt transmit antennas, Mr receive antennas, T symbol times/block • T ≥ 2 Mt • Y = HX + W • X – Mt×T one transmit symbol block • Y – Mr×T one receive symbol block • H – Mr× Mt random channel matrix • W – Mr×T AWGN matrix having i.i.d entries 4

5. Grassmann Manifold • Grassmann Manifold [L. Zheng, D. Tse, 2002] • Stiefel ManifoldS(T,M) – the set of all M-dimensional subspaces in a T-dimensional hyberspace. • Grassmann Manifold G(T,M) – the set of all different M-dimensional subspaces in S(T,M). • X, an element in G(T,M), is an M×T unitary matrix • Chordal Distance [J. H. Conway et. al. 1996] • P, Q in G(T,M) 5

6. Codebook Model • Codebook S with N codewords • Codeword Xi is an element in G(T,Mt) • Optimal codebook S • Maximize the minimum distance in S 6

7. “ ”, from [R. A. Horn & C. R. Johnson, 1985] Theoretical Support • Majorization • Schur-Horn Theorem • If ω majorizes λ, there exists a Hermitian matrix with diagonal elements listed by ω and eigenvalues listed by λ. • ω majorizes λ => , with eigenvalues of 7

8. Optimal Codebook Design • Gram Matrix G of Codebook S • Optimal S The diagonal elements of G are identical • Power for the entire codebook P • Allocated P/T to each codeword equally. • Nonzero eigenvalues of G = P/T • Optimal Codebook Design • G => Xs => S • Given eigenvalues, how to reconstruct such a Gram matrix that it has identical diagonal elements? 8

9. Reflect by θ Rotate by 2θ Reflection Matrix • Reflection Angle θ • Equivalent to rotate by 2 • Reflection matrix F • Unitary matrix • Application • Modify the first diagonal element of a matrix • , some value we desired 9

10. Flow Chart of Codebook Design 10

11. Comparison with other designs R: transmit data rate in units of bits/symbol period; T: coherent time of the channel in units of symbol period; Mt : number of transmit antennas; Mr: number of receive antennas. 11

12. Simulation • Mt=1, Mr=4, P=4, T=3 • is the standard code from http://www.research.att.com/~njas/grass/index.html. • , Q is a unitary matrix. Thus, are the same point in G(T,Mt) • Mt=2, Mr=4, P=8, T=8 , an 8 by 8 identical matrix 12