Efficient Power Minimization for MIMO Broadcast Channels with BD-GMD

Winston Ho and Ying-Chang Liang Institute for Infocomm Research (I2R) Singapore

Overview • • • • • •

Optimal Power Minimization Preliminaries: GMD and BD-GMD ZF-based Power Minimization Suboptimal Methods Simulations Conclusion

Optimal Power Minimization • Objective: ¾ To minimize the transmit power for the MIMO

broadcast channel, ¾ given user rate requirements, ¾ using Dirty Paper Coding.

• Application: ¾ Mimimize interference to neighbour cells. ¾ Users at varying distances.

• Cases: ¾ Interference-Balancing (IB). ¾ Zero-Forcing (ZF).

Optimal Power Minimization • Interference-Balancing Case ¾ IUI, noise ¾ Better performance than ZF in low SNR region ¾ Higher complexity than ZF case ¾ Many iterations ¾ Each iter. ↑ computations ¾ No. iter. random

Optimal Power Minimization • Zero-Forcing Case ¾ Lower complexity than the IB case

• User ordering affects transmission power ¾Search over K! encoding orders ♦ Limited predictable complexity

¾Suboptimal methods with much reduced

complexity ♦ close to ZF-optimal power

Overview • • • • • •

Optimal Power Minimization Preliminaries: GMD and BD-GMD ZF-based Power Minimization Suboptimal Methods Simulations Conclusion

Preliminaries Transmission Strategies for Single-User MIMO • Singular Value Decomposition (SVD)

H = USVH

¾ Different constellations for each subchannel

• Geometric Mean Decomposition (GMD)[1]

H = QRPH

¾ R is triangular, equal diagonal ¾ Same constellation for every subchannel ¾ Low complexity ¾ Good BER

[1] Y. Jiang, J. Li and W. W. Hager, “Joint Transceiver Design for MIMO Communications Using Geometric Mean Decomposition,” IEEE Trans. Signal Processing, vol. 53, no. 10, pp. 3791-3803, Oct. 2005.

Preliminaries Block-Diagonal GMD for Multi-User MIMO H = P L QH Block Diagonal & Unitary

Each Pi is unitary.

Unitary Lower Triangular

Each Li is equal diagonal.

Block-equal-diagonal

S. Lin, W. W. L. Ho, and Y.-C. Liang, “Block-diagonal Geometric Mean Decomposition (BD-GMD) for Multiuser MIMO Broadcast Channels,” Int. Symp. Personal, Indoor and Mobile Radio Commun., Helsinki, 11–14 Sep. 2006.

Overview • • • • • •

Optimal Power Minimization Preliminaries: GMD and BD-GMD ZF-based Power Minimization Suboptimal Methods Simulations Conclusion

Power Minimization Given Fixed Encoding Order • Rate requirements ' SNR requirements ¾ rate = log2(1+SNR)

• # transmit antennas = NT • # receive antennas = n1, n2, …, nK sum=NR < NT • Multiplexing: user i has ni data subchannels • SNR req’m for each subchannel = γi

Power Minimization Given Fixed Encoding Order • Optimization problem:

A

H

B

F SNR req’m:

1s

Power Minimization Given Fixed Encoding Order • Solution: ¾ BD-GMD: ¾ ¾

¾ Minimum power =

Transceiver Design A1

n a

MOD

s

F

x

H

y

MOD

z1

MOD

zK

:

B-I Channel

F

H

AK

A

Optimal User Ordering • Rearranging the Channel Matrix NT

n1

PLQH = H = N

R

: nK

to achieve minimum transmit power.

D

H

P

L Λ

QH

Optimal User Ordering • Transmit power

Λ

H

• Optimal ordering – search all K! orderings ¾ (K K!) determinant calculations

Transceiver Design • Ordered A1

n s

D

MOD

s

F

x

H

y

MOD

z1

MOD

zK

:

B-I Channel

AK

Overview • • • • • •

Optimal Power Minimization Preliminaries: GMD and BD-GMD ZF-based Power Minimization Suboptimal Methods Simulations Conclusion

Suboptimal User Ordering • Principle ¾ Successive selection of users ¾ Top down manner

• Methods ¾ 1. Successive Closest Match (SCM)

r

¾ 2. Minimize i ¾ 3. Minimize Channel Strength

Method 1: Successive Closest Match (SCM) • • • •

det(Ω2) fixed To minimize Tr(Ω2) equal elements Best case Λ = k Γ1/2 → M = desired matrix

• To minimize

Method 1: Successive Closest Match (SCM)

M

Λ

• Choose Λi closest to Mi • Min while • Min

users

Method 1: Successive Closest Match (SCM) • For user 2 onwards

• min

Method 2: Minimize ri • QR decomposition

• ri = diagonal elements of L • first element >> last element • equal SNR req’m • ↓ spread

Method 3: Minimize Channel Strength • Different channel strengths • Weakest channel, encode first • Minimize

H

DH

Complexity Optimal ZF power minimization 1. Successive Closest Match (SCM)

No. of det. calc. (K K!)

2. Minimize ri 3. Minimize Channel Strength

0

BD-GMD applied to best order: K times GMD

Overview • • • • • •

Optimal Power Minimization Preliminaries: GMD and BD-GMD ZF-based Power Minimization Suboptimal Methods Simulations Conclusion

Simulation Results

Equal SNR requirements

Unequal SNR requirements

Unequal channel strengths

Conclusion • ZF-based Power Minimization for MIMO Broadcast Channels • Block-diagonal Geometric Mean Decomposition (BD-GMD) • Equal SNR for all subchannels of each user • Optimal ordering, non-iterative • Suboptimal orderings