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Codebook Design for Noncoherent MIMO Communications Via Reflection Matrices

Daifeng Wang and Brian L. Evans{wang, bevans}@ece.utexas.edu

Wireless Networking and Communications GroupThe University of Texas at Austin

IEEE Global Telecommunications ConferenceNovember 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 – 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

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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 M×T unitary matrix

Chordal Distance [J. H. Conway et. al. 1996]

P, Q in G(T,M)

2( , ) H

c Fd M P Q PQ

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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

1 2{ , ,..., }NS X X X

( , )

2

( , )

arg max {min ( , )}

arg min {max }

t

t

c i jG T M

Hi jG T M F

d

S

S

S X X

X X

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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

“

”, from [R. A. Horn & C. R. Johnson, 1985]

1

2

n

1 2 n[ ]

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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 => Xs => S

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

1 2, [ ]H H H H Hs s s NG = X X X = X X X

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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

Reflect by θ

Rotate by 2θ

cos sin

sin cos

11 12 11 12

21 22 21 22

cos sin cos sin

sin cos sin cos

x x y y

x x y y

11y

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t

t

tM K

M K-TT

P[1 1 ... 1] (1)

M K

P P P[0 0 ... 0 ... ] (2)

T T T

Flow Chart of Codebook Design

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Comparison with other designsAlgorithm Searching

complexityDecoding 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) T=2Mt

Orthogonal matrices [V. Tarokh & I. Kim, 2002]

O(2RTlog2Mt) ML O(Mt2Mr) T=2Mt

Mt=1,2,4,8

Training [P. Dayal et. al., 2004]

O(2RTT) MMSE O(Mt3Mr

3)

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 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

1 0 0 0.7416 0.1716 0.6485

0.3333 0.5369 0.7750 0.1266 0.5723 0.8102ˆ,0.3333 0.9396 0.0775 0.8759 0.2195 0.4296

0.3333 0.4027 0.8525 0.0077 0.9634 0.2680

s s

X X

ˆsX

1 0 0

0

0

0 0 1

s

X

ˆs sX Q X ˆ and s sX X