Joint Transmit and Receive Antenna Selection Using a Probabilistic Distribution

Learning Algorithm in MIMO SystemsLearning Algorithm in MIMO Systems

Muhammad Naeem and Daniel C. Lee

1

OutlineOutline

M ti ti d P bl F l tiMotivation and Problem Formulation

Estimation of Distribution Algorithm (EDA)

Improved EDA

Performance ComparisonPerformance Comparison

Conclusion

2

MIMO SystemMIMO System

Encode

Mobile

Radio

Channel

er

•Capacity of a MIMO system increases with the number of antennasCapacity of a MIMO system increases with the number of antennas

•Larger number of antennas results in a high hardware cost due to the large number of RF chains

3

M ti ti

Hardware cost can be significantly reduced by

Motivation

Hardware cost can be significantly reduced by selecting a subset of antennas from the set of physically available antennas and using the signalsphysically available antennas and using the signals from the selected antennas only, without sacrificing the advantage of multi antenna diversity.

We need to Choose Nt transmit antennas from NT transmit antennas and similarly Nr receive

antennas from N receive antennas4

antennas from NR receive antennas

MotivationMotivationWe denote by Φ the collection of all possible joint transmit and receive antenna selections. Then, the number of possible ways of selecting antennas is

NN TR

tr

NNNN

Φ = ×

The computational complexity of finding an optimal Joint Transmit and Receive Antenna selection by exhaustive search grows exponentially with the number of transmit and receive antennas.

5

o t a s t a d ece e a te as

Joint Antenna Selection ProblemφWe denote by in Φ a selection of transmit and receive

antennas

Joint Antenna Selection Problem

antennas.

We denote by the channels formed betweenr tN NH φ ×∈We denote by the channels formed between selected Nt transmit antennas and Nr receive antennas. The channel capacity associated with selected transmit and

i t i

H ∈

Hφ φ φρ

receive antennas is

2( ) log det ( )( )

where is the average SNR per channel use

r

HN

t

C H I H HN

φ φ φρ

ρ

= +

6

where is the average SNR per channel use. ρ

Joint Antenna Selection ProblemJoint Antenna Selection ProblemWe can model joint transmit and receive antenna selection problem as a combinatorial optimization problem

max ( ) orC H φ

φ∈Φ

2max log det ( )( )r

HNI H H

Nφ φ

φ

ρ∈Φ

+

tNφ∈Φ

7

Conventional EDAConventional EDAGenerate random population

Evaluate IndividualsSort

X1 X2 X3 … Xn Function Values 1 1 1 0 … 0 F1 2 1 0 0 … 1 F23 0 1 1 1 Fl∆

CYes

3 0 1 1 … 1 F3 ... ... ... ... … ... ...

1 0 0 … 11l

F−∆1l−∆

l

ConvergenceCriterion satisfied

No

TerminateGenerate NewIndividuals with Conditional

Prob. Vector

l lη∆ −

Update Counter l=l+1

X1 X2 X3 … Xn 1 1 1 0 … 0 2 1 0 0 … 1

SelectBest

Individuals

1lη −

1 2 1( , , , | ) n lP −Γ = θ θ θ η

8

... ... ... ... … ... 0 1 0 … 11lη −

EDAEDAEDA can be characterized by parameters and Notations

1. Is is the space of all potential solutions2. F ( ) denotes a fitness function.

Γ

3. ∆l is the set of individuals (population) at the lth iteration. 4. ηl is the set of best candidate solutions selected from set ∆l at the

lth iteration.th5. We denote βl ≡ ∆l – ηl≡ ∆l ∩ ηC l .where ηC l is the complement of ηl.6. ps is the selection probability. The EDA algorithm selects ps|∆l|

individuals from set ∆l to make up set ηl. l p ηl7. We denote by Γ the distribution estimated from ηl (the set of

selected candidate solutions) at each iteration8. ITer are the maximum number of iteration

9

Ter

Modified EDA

l∆

1lF

−∆1l−∆l lη∆ −

l lη∆ −

1lη 1lη −

1lη −

1 2 1( , , , | ) n lP −Γ = θ θ θ η

10

Generating the initial population Generating the initial population

11

Weighted EDAWeighted EDAwe propose an idea of adding some skew in estimating the probability distribution in EDAdistribution in EDA.

The skew can be added by giving more weights to the individuals in ηl-1 that have better fitness in estimating the joint probability ηl-1 g j p ydistribution

An example of Weight values is

( ) ( )log log1 2l j

jη

ξ η−

= =( ) ( )1

, 1,2,...,log logl

j l

li

jiη

ξ ηη

=

= = − ∑

12

Simulation Results

13

25

30OptimalRandomNormGorokhov Algo

20

s/s/

Hz)

DecoupledEDA

10

15

Cap

acity

(bit

5

0 5 10 15 20 250

SNR (dB)

10% O t it SNR With NT 6 Nt 3 NR 30 N 2

14

10% Outage capacity versus SNR. With NT=6, Nt=3, NR=30, Nr=2.

25 O ti l

20

OptimalRandomNormGorokhov AlgoDecoupled

15

bits

/s/H

z)

DecoupledEDA

10

Cap

acity

(b

5

0 5 10 15 20 250

SNR (dB)

10% O t it SNR With NT 6 Nt 4 NR 18 N 3

15

10% Outage capacity versus SNR. With NT=6, Nt=4, NR=18, Nr=3

19.6

19.8 EDA-AEDA-REDA-BNEDA WBN

19.4

/s/H

z)

EDA-WBN

19

19.2

Cap

acity

(bits

18.8

4 6 8 10 12 14 16 18 20

18.6

EDA Interations

16

10% Outage capacity versus EDA iterations

20.5

19

19.5

20

(bits

/s/H

z)

18

18.5

Cap

acity

1520

2530

20

25

3017.5

P l i i5

1015

15Number of iterations

Population size

17

Tradeoff between population size and the number of iterations

The number of complex multiplications and additionsThe number of complex multiplications and additions

ESA 3R

TR NNN

NN

×

×

Decoupled Algorithm 3TRR

NNN

+ ×

tr NN

Decoupled Algorithm Rtr NN

33 NNNNNN +Gorokhov33rtTtrR NNNNNN +

EDA ( ) 3∆ RTerl NI

18

The number of complex multiplications and additionsThe number of complex multiplications and additions

,, , , ,R r T t TerN N N N I ∆ ESA Decoupled Algorithm

Gorokhov Algorithm EDA

[30 2 6 3 30 20] 23200 1213 1794 1600[30 , 2, 6, 3,30,20] 23200 1213 1794 1600

[16, 4, 6, 4, 16, 8] 582400 39147 5632 2730

[20 4 6 4 20 8] 1 5×106 103680 6656 3413[20, 4, 6, 4, 20,8] 1.5×106 103680 6656 3413

[20,8,10,6,20,8] 1.9×109 9.08×106 65280 11520

19

ConclusionsConclusionsExisting antenna selection schemes are computationally expensiveexpensive.

The performance of EDA algorithm is close to the optimal.The performance of EDA algorithm is close to the optimal.

EDA with Cyclic shifted initial population reduces the number of iterations to reach the optimal solution.

Th f f i ht d EDA i b tt th ll i t fThe performance of weighted EDA is better than all variants of EDA.

20

Thank You

21