A New Approach to Beamformer Design
for Massive MIMO Systems Based on k-regularity
Gilwon Lee
Dept. of Electrical Engineering
KAIST
Joint work with Juho Park, Youngchul Sung and Junyeong Seo
GLOBECOM 2012 Workshop LTE-B4G, Dec. 3, 2012
Massive MIMO Systems
MIMO
Massive MIMO is an emerging technology,
which scales up MIMO by an order of magnitude.
Antenna arrays with a few hundred elements.
Massive MIMO
• Rate↑
• Transmission reliability↑
120°
• Energy efficiency↑
Internet
Internet
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
BS
MS
RF chain
RF chain
Assumptions
(1)
The size of antenna array at the MS is limited
(2)
(3)
due to hardware constraint.
The Conventional Method: Antenna Selection
At transmitter
RF chain
RF chain
RF chain
Antenna selection: M RF chains select M different antennas
out of the NT available transmit antennas.
Hardware
Complexity↓
The Conventional Method: Antenna Selection
At transmitter
RF chain
RF chain
Antenna
Selection
RF chain
However, the performance of antenna selection should be far interior
to that of a method using all of transmit antennas.
Especially, the gap of performance will be increasing
as NT increases.
The Proposed Scheme: k-regular Beamformer
At transmitter
RF chain
RF chain
RF chain
RF chain
Antenna
Selection
RF chain
RF chain
k-regular
beamformer
The Proposed Scheme: k-regular Beamformer
Specifically
RF chain
RF chain
k-regular
beamformer
RF chain
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 antennas
and 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.
⇒ k-regularity
But, how to design the matrix V?
or k-sparse constraint
Problem Formulation
Data
streams
k-regular
beamformer
Channel
k-regularity
Problem)
k-regular constraint
power constraint
Assumptions
M independent data stream transmission
with equal power for each stream
There is no power amp in k-regular
beamformer
Observations
In combinatorial approach, (brute search)
Impossible to implement
should be required to find optimum V
Need an algorithm to reduce complexity!
Observations
Without k-regular constraint,
the optimal transmit beamforming matrix V is given by
where
(SVD)
is i-th column of
The matrix is called eigen beamforming matrix
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:
to approximate the eigen beamforming matrix of H under k-regular constraint
Maximum correlation method (MCM)
⇒
Pick k largest absolute values in v
and let other values be zeroes.
After then, normalize it
Very simple, Systematic ⇒
Heuristic
⇒
Possible to analyze
Performance loss
The Relaxed Problem
Original Problem
-norm relaxation of k-regular constraint
⇒
where
How can we solve the relaxed problem?
Iterative Shrinkage Thresholding Algorithm
For a convex function
⇔
< Iterative Shrinkage Thresholding Algorithm (ISTA) >
Shrinkage operator
where
Here,
Gradient method
Iterative Shrinkage Thresholding Algorithm
⇔
If we directly apply ISTA to our problem
Iterative Shrinkage Thresholding Algorithm
⇔
If we directly apply ISTA to our problem without the power constraint,
Shrinkage operator
for i-th column vector where
Projected ISTA (PISTA)
With the power constraint,
Projected ISTA (PISTA)
Metric projection
of vector i-th column onto B
Projected ISTA (PISTA)
The Projected ISTA for k-regular Beamformer Design
0. (Initialization) Generate
randomly
1. (PISTA) Update
2. (Stop criterion) If
3. (Hard-thresholding) For
4. (Power adjusting) For
update,
update,
Simulation Results
Antenna selection scheme:
Parameters:
Simulation Results
k-regular beamformer scheme:
Parameters:
k-regular
gain
Antenna
Selection gain
200%
Simulation Results
k-regular beamformer scheme with varying k
Parameters:
eigen BF
gain
Antenna
Selection gain
with small k
Simulation Results
Distribution of antennas over numbers of connections
Parameters:
89%
69%
44%
28%
A large portion of antennas
are not connected to signals
for small k
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