Transmit Diversity with Channel Feedback

Krishna K. Mukkavilli, Ashutosh Sabharwal, Michael Orchard and Behnaam Aazhang

Department of Electrical and Computer Engineering

Rice University, Houston, Texas, USA

Introduction

Demand for high data rates in wireless communications

High spectral efficiency schemes

Fading phenomenon

Diversity schemes

Multiple antenna

Overcomes fading

Capacity grows linearly with min(t,r) [Telatar95]

Achieved via space-time codes [Tarokh98]

Background

Slow (block) fading channel

Feedback

Increase spectral efficiency

Decrease frame error rate

Spatial water filling [Telatar95]

Maximizes mutual information

Requires substantial channel information

No guarantees for practical low dimensional

codebooks

Our Focus

Practical management of unknown channel

condition

Limited feedback

Practical codebook design issues

Our approach

Role of feedback in codebook design

Phase information

Quantized phase information

Codebook Design with Feedback

Objective: Error minimizing codebooks

Unknown channel condition with reduced

dimension feedback

Issues: what feedback? what codebook?

Dominant spatial direction is the key parameter

Chernoff bound analysis

Comparison with mutual information solution

System and Channel Model

NHXY

Block Fading

n rx antennas

X Y

L

h1,i

hm,i

Channel realization known at the

receiver

Error free feedback channel

m tx antennas

Chernoff Bound

Minimize pairwise error probability given H

Use X = Wd x

where Wd is the eigenvector corresponding to

the maximum singular value of H, max

Observations:

Reduce dimension feedback required

Gaussian channel codebooks

Comparison with Mutual Information Solution

Achievable rates with two transmit and two receive antennas

2 4 6 8 10 12 14 16 18 202

3

4

5

6

7

8

9

10

11

12

SNR

bits

/se

c/H

z

Beamforming

Spatial water-filling

Phase Feedback

Dominant eigenvector solution

Requires phase and amplitude information

(2m – 2) real numbers

Proposed: using only phase information

Chernoff bound

Simulations

Beamforming with Phase Information

Problem : find such that

minimizes error probability where x

is the information vector

Solution: Choose whose components satisfy

})(phase{1 1

*

,,lK jm

kll

n

i liki

j ehhej

]1,..,1

2 jj eem

xX

Beamforming with Phase Information

X

X

X

xY

H

Beamforming vector

me j

me j 2

m1

]1... [1

2 jj eem

Special Cases

m =2 ,n = 1

m=2, n = n

m=m, n = 1

Solved for the case of m = 3 and any n

0 ),hx phase(h 2

*

211

0 ),hx hphase( 2

n

1i

*

i2,i1,1

0 ),hx phase(h m

*

mi i

Features of Beamforming with Phase Feedback

Less feedback information (m-1 real numbers)

No need for singular value decomposition

Performance loss compared to dominant

eigenvector solution, for n=1

Loss is 0.49 dB for 2 tx and 1 rx antenna

mm4

)1(4log10

Quantized Phase Feedback

Vector quantization design in phase space

Reduces required feedback to Log2 K bits

Case 2 tx antenna

Scalar quantization

1 or 2 bits suffice

• 0.5 dB loss for 1 bit

Simulation Results (1)

2 transmit antennas and 2 receive antennas

2 3 4 5 6 7 8 9 1010

-4

10-3

10-2

10-1

SNR (dB)

Bit

Err

or R

ate

Space time code (Alamouti)1 bit phase feedback 2 bit phase feedback complete phase feedback Generalized beamforming

Rayleigh block fading

L=130 symbols

Antipodal signaling

No channel code

Simulation Results (2)

3 transmit antennas and 2 receive antennas

2 4 6 8 10 12 14 16 1810

-6

10-5

10-4

10-3

10-2

10-1

SNR (dB)

Bit

Err

or R

ate

Generalized Beamforming Complete phase feedback 2-bit per phase feedback1-bit per phase feedback

Conclusions

Circle the point that does not fit the talk:

a. Simple design

b. Small feedback

c. Low down payment

d. Loss analysis

e. Colorful graphs