Codebook Design for Noncoherent MIMO Communications Via Reflection Matrices

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Codebook Design for Noncoherent MIMO
Communications Via Reflection Matrices

• Daifeng Wang and Brian L. Evans
• wang, bevans_at_ece.utexas.edu
• Wireless Networking and Communications Group
• The University of Texas at Austin
• IEEE Global Telecommunications Conference
• November 28, 2006

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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

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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

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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 MtT one transmit symbol block
• Y MrT one receive symbol block
• H Mr Mt random channel matrix
• W MrT AWGN matrix having i.i.d entries

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Grassmann Manifold

• Grassmann Manifold L. Zheng, D. Tse, 2002
• Stiefel Manifold S(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 MT unitary matrix
• Chordal Distance J. H. Conway et. al. 1996
• P, Q in G(T,M)

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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

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Theoretical Support

• Majorization
• Schur-Horn Theorem
• If ? majorizes ?, there exists a Hermitian matrix
  with diagonal elements listed by ? and
  eigenvalues listed by ?.
• ? majorizes ? gt
  , with eigenvalues of

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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 gt Xs gt S
• Given eigenvalues, how to reconstruct such a Gram
  matrix that it has identical diagonal elements?

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Reflection Matrix

• Reflection Angle ?
• Equivalent to rotate by 2
• Reflection matrix F
• Unitary matrix
• Application
• Modify the first diagonal element of a matrix
• , some value we desired

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Flow Chart of Codebook Design
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Comparison with other designs
Algorithm Searching complexity Decoding method Computational complexity Notes
DFT B. Hochwald et. al. 2000 O(2RTMt) GLRT O(2RT)
Coherent Codes I. Kammoun J. C. Belfiore, 2003 O(2RT(T-Mt)) GLRT O(2RT)
PSK V. Tarokh I. Kim, 2002 O(2RTMt) ML O(MtMr) T2Mt
Orthogonal matrices V. Tarokh I. Kim, 2002 O(2RTlog2Mt) ML O(Mt2Mr) T2Mt Mt1,2,4,8
Training P. Dayal et. al., 2004 O(2RTT) MMSE O(Mt3Mr3)
Reflection matrices O(2RTMt) GLRT O(2RT)
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.
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Simulation

• Mt1, Mr4, P4, T3
• 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)
• Mt2, Mr4, P8, T8
• , an 8 by 8
  identical matrix