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Spherical Linear Interpolation forSpherical Linear Interpolation for

TransmitTransmit BeamformingBeamforming in MIMOin MIMO--OFDMOFDMSystems with Limited FeedbackSystems with Limited Feedback

Jihoon Choi and Robert W. Heath Jr.

Wireless Networking and Communications Group

The University of Texas at Austin

Abstract Presentation Next AbstractPaper

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Wireless Networking and Communication Group 2

OutlineOutline

Overview

Diversity techniques in MIMO systems

Closed-loop transmit beamforming for MIMO-OFDM

Problem Statement

Proposed Beamforming Scheme Proposed spherical interpolator

Optimization of phase rotation

Beamformer quantization + interpolation

Numerical Simulations and Discussions

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Transmit Diversity in MIMO SystemsTransmit Diversity in MIMO Systems

No channel state information (CSI) at Transmitter

Space-time codes [Tarokh et al. 1998]

Full CSI at Transmitter

Antenna selection [Wittneben et al.1994], [Heath et al. 1998] Maximum ratio transmission [Lo 1999], [Andersen 2000]

Partial CSI

Adaptive beamforming [Xia et. al. 2004]

Transmitter

MIMO channel

Receiver

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BeamformingBeamforming and Combiningand Combining

Covert MIMO to SISO

Transmit beamforming vectorw

Receive combining vectorz

Coding &

Modulation...

Hw..

Detectionand

Decodingz

Feedback Channel

( with limited data rate )

Estimation &

detection

Complex

number

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Max SNR Solution (MRT/MRC)Max SNR Solution (MRT/MRC)

SNR is proportional to

Max SNR solution is

Solution is not unique: ifw is optimal, so is wej

Other solutions possible (constrain w)

Selection diversity

Equal gain combining / transmission

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Beamforming in MIMOBeamforming in MIMO--OFDMOFDM

P/S

&

AddCP

IDFT

)1(s

)(Ns

IDFT

DFT

)1(1y

)(1 Ny

DFT

)1(rM

y

)(NyrM

M M

M

M

M

M

M

)1(1w

)(1 Nw

M)1(tMw

)(NwtM

M MM M

Feedback of Channel

State Information

M

P/S

&

AddCP

Remove

CP

&S/P

Remove

CP

&

S/P

1n

rMn

)1(1z

)(1 Nz

)1(rM

z

)(NzrM

M

)1(r

)(Nr

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Signal Model for OneSignal Model for One SubcarrierSubcarrier

Equivalent model for subcarrierk

(identical with a narrowband MIMO case))(1,1 kh

)(, kh tr MM

)(1 kw )(1 kz)(ks )(krM

M M

M

)(kwtM

)(kzrM

M

)(1 kn

)(knrM

Noise

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Problem SummaryProblem Summary

Beamforming in MIMO-OFDM requires

Feedback requirements Number of subcarriers

How can we reduce the number of vectors fed back?

Exploit correlation between vectors Send back fraction of vectors

Use smart interpolation

How can limit the feedback for each vector?

Use quantized beamforming [Love et. al. 2003]

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Correlation Between SubcarriersCorrelation Between Subcarriers

Subchannelcorrelation

Beamformer correlation

Simulations

h(d): M

t=4, M

r=1, N=64, K=8, L={6,12}

w(d): M

t=4, M

r=2, N=64, K=8, L={6,12}

subsampling rate number of channel taps

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Clustering of SubcarriersClustering of Subcarriers

Clustering (K=5)

Disadvantages

Performance degradation in cluster boundary

Cluster size (or feedback reduction) is limited

subcarriers

L

cluster 1 cluster 2 cluster 3 cluster 4

feedback L

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Proposed Beamforming MethodProposed Beamforming Method

Subsampling of beamforming vectors

LL L

Reconstruction of beamforming vectors

Transmit power constraints force

Spherical interpolation

subcarriers

L

L

feedbackL

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Conventional Spherical InterpolationConventional Spherical Interpolation

Spherical averaging [Watson 1983], [Buss et al. 2001]

Nonuniqueness of optimal beamforming vectors

Random phase rotation has significant impact on interpolations

Performance degradation of spherical interpolators

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Proposed InterpolatorProposed Interpolator

Optimization of phase rotation parameters Maximize the diversity gain

Maximize the mutual information

Spherical Linear Interpolator with phase rotation

is a parameter for phase rotation.

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Optimization of Phase RotationOptimization of Phase Rotation

Maximizing the minimum channel gain (diversity)

Difficult to get a closed-form solution

Grid search

and Pis the number of quantizedlevels

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Alternative SolutionAlternative Solution

Observation

Subcarrier (K/2+1) suffers from the largest distortion

Subcarrier (K/2+1) has the worst average channel gain

Approximation of cost function

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ClosedClosed--Form SolutionForm Solution

Differentiation with respect to

1 and 2 are complex constants satisfying

Always, there exist two solutions.

Second derivative

The solution making the second derivative be negative is the

optimal solution. closed-form solution

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Optimization in Terms of CapacityOptimization in Terms of Capacity

Maximizing the capacity

Grid search

Closed-form solution is the same as in diversity case

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Simulation EnvironmentsSimulation Environments

MIMO channels Each channel impulse response has 6 taps (L=6) with uniform

profile and follows i.i.d.CN

(0,1/L

). The channels between different transmit and receiver antennaspairs are independent.

Noise: i.i.d. with CN(0,N0)

Simulation parameters M

t=4 (# of Tx antennas), M

r=2 (# of Rx antennas)

N=64 (# of subcarriers), K=8 (subsampling rate), L=6

Basic assumptions MRC at the receiver

No delay and no transmission error in the feedback channel

QPSK, no water-filling

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Simulation ResultsSimulation Results Channel GainChannel Gain

OFDM subcarrier vs. effective channel gain

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Simulation ResultsSimulation Results UncodedUncoded BERBER

No channel coding

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Simulation ResultsSimulation Results CapacityCapacity

No power control for diversity techniques

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Simulation with QuantizationSimulation with Quantization

Quantization of beamforming vectors

By using the codebook in [Love & Heath 2003]

6 bits of feedback for each beamforming vector

Phase optimization

Grid search with P=4 (2 bits of feedback per phase)

Feedback requirements Ideal: 6N= 384 bits / OFDM symbol

Clustering: 6N/K= 48 bits / OFDM symbol

Selection: 2N= 128 bits / OFDM symbol Proposed: 8N/K = 64 bits / OFDM symbol

Orthogonal STBC (OSTBC): none

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Simulation ResultsSimulation Results -- Channel GainChannel Gain

Mt=4, Mr=2, N=64, K=8, L=6

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Simulation ResultsSimulation Results -- BERBER

BER degradation by channel prediction error

Prediction error model

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Simulation ResultsSimulation Results -- CapacityCapacity

No power control

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Simulation ResultsSimulation Results Coded BERCoded BER

1/2 convolutional code

Interleaving/deinterleaving

Frame length

30 OFDM symbols 64 bits / OFDM symbol

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ConclusionsConclusions

Summary

Interpolation for transmit beamforming in MIMO-

OFDM proposed

Relates to problem of spherical quantization

Future work

Proposed system does not obtain MIMO capacity

Develop interpolated waterfilling solutions

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