A New Approach to Beamformer Design for Massive MIMO Systems Based on k-regularity

Gilwon Lee

Dept. of Electrical Engineering

KAIST

GLOBECOM 2012 Workshop LTE-B4G, Dec. 3, 2012

Joint work with Juho Park, Youngchul Sung and Junyeong Seo

Massive MIMO Systems

Massive MIMO is an emerging technology,

which scales up MIMO by an order of magnitude.

Antenna arrays with a few hundred elements.

MIMO

Massive MIMO

• Rate↑

• Transmission reliability↑

• Energy efficiency↑Internet Internet

120°

Practical Issues on Massive MIMO

Antenna elements: cheap.

But, the multiple RF chains associated with multiple antennas are costly in terms of

size, power and hardware.

The number of RF chains is restricted in massive MIMO systems.

System Model

Single user massive MIMO

Assumptions

(1)

(3)

(2)

The size of antenna array at the MS is limited

RF chain

RF chain

due to hardware constraint.

BS MS

The Conventional Method: Antenna Selection

RF chain

RF chain

RF chain

At transmitter

Antenna selection: M RF chains select M different antennas out of the NT available transmit antennas.

HardwareComplexity↓

The Conventional Method: Antenna Selection

RF chain

RF chain

RF chain

At transmitter

However, the performance of antenna selection should be far interiorto that of a method using all of transmit antennas.

Especially, the gap of performance will be increasing as NT increases.

AntennaSelection

The Proposed Scheme: k-regular Beamformer

RF chain

At transmitter

RF chain

RF chain

RF chain

RF chain

RF chain

AntennaSelection

k-regularbeamformer

Specifically

RF chain

RF chain

RF chain

k-regularbeamformer

k-regular beamformer: Each of the M data streams is multiplied by k complex gains

and assigned to k out of the available NT transmit antennasand signals assigned to the same transmit antenna will be added to be transmitted.

The Proposed Scheme: k-regular Beamformer

For example,

Each column of V has k=2 nonzero elements.

But, how to design the matrix V?

⇒ k-regularity

or k-sparse constraint

The Proposed Scheme: k-regular Beamformer

Problem Formulation

Data streams

k-regularbeamformer Channel

M independent data stream transmissionwith equal power for each stream

Assumptions

There is no power amp in k-regularbeamformer

k-regular constraint

power constraint

k-regularity

Problem)

Observations

In combinatorial approach, (brute search)

should be required to find optimum VImpossible to implement

Need an algorithm to reduce complexity!

Observations

Without k-regular constraint,

the optimal transmit beamforming matrix V is given by

where

The matrix is called eigen beamforming matrix

(SVD)

is i-th column of

Based on this fact, we can propose a method to design k-regular beamformer.

The Maximum Correlation Method

A simple way to design k-regular BF matrix:

Maximum correlation method (MCM)

to approximate the eigen beamforming matrix of Hunder k-regular constraint

⇒ Pick k largest absolute values in v

and let other values be zeroes.

After then, normalize it

Very simple, Systematic ⇒ Possible to analyze

Heuristic ⇒ Performance loss

The Relaxed Problem

⇒

Original Problem

where

-norm relaxation of k-regular constraint

How can we solve the relaxed problem?

Iterative Shrinkage Thresholding Algorithm

For a convex function

⇔

< Iterative Shrinkage Thresholding Algorithm (ISTA) >

Gradient methodShrinkage operator

where

Here,

Iterative Shrinkage Thresholding Algorithm

⇔

If we directly apply ISTA to our problem

Iterative Shrinkage Thresholding Algorithm

⇔

If we directly apply ISTA to our problem

Shrinkage operator for i-th column vector

without the power constraint,

where

Projected ISTA (PISTA)

With the power constraint,

Metric projection of vector i-th column onto B

Projected ISTA (PISTA)

Projected ISTA (PISTA)

The Projected ISTA for k-regular Beamformer Design

0. (Initialization) Generate randomly

2. (Stop criterion) If

3. (Hard-thresholding) For update,

4. (Power adjusting) For update,

1. (PISTA) Update

Simulation Results

Antenna selection scheme:

Parameters:

Simulation Results

k-regular beamformer scheme:

Parameters:

AntennaSelection gain

k-regulargain

200%

Simulation Results

k-regular beamformer scheme with varying k

Parameters:

with small k

AntennaSelection gain

eigen BFgain

Simulation Results

Distribution of antennas over numbers of connections

Parameters:

A large portion of antennas are not connected to signals for small k

89%

69%

44%

28%

Simulation Results

Conclusion

• Proposed k-regular beamformer architecture

• Proposed PISTA and MCM to design k-regular beamforming

• Enable system designers to choose optimal trade-off their hardware constraint and required rate performance

• showed that the proposed k-regular BF significantly improves the rate gain over simple antenna selection