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Analysis for Adaptive DOA Estimation with Robust Analysis for Adaptive DOA Estimation with Robust Beamforming in Smart Antenna SystemBeamforming in Smart Antenna System

指導教授：黃文傑 W.J. Huang研究生　：蔡漢成 H.C. Tsai

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OutlineOutline

• Conception of Smart Antenna• Beamforming Method• DOA Estimation• θ- LMS Algorithm (My Point)• Local Minimum problems• θ- LMS Algorithm with Robust Beamforming• Conclusion

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Conception of Smart AntennaConception of Smart Antenna

• There are constructed by some specially geometric antenna array.

• It changes the beam-pattern with some different methods.

• It increases the CINR and Capacity

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Category -1Category -1

• Switched Beam System

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Category -2Category -2

• Adaptive Beam System

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OutlineOutline

• Conception of Smart Antenna• Beamforming Method• DOA Estimation• θ- LMS Algorithm• Local Minimum problems• θ- LMS Algorithm with Robust Beamforming• Conclusion

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Beamforming MethodBeamforming Method

Summation

Beamforming Weighting

DO

A E

stim

atio

n

*

1

( ) ( ) ( )L

Hl l

l

y t w x t t

w x

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Beamforming Tech.Beamforming Tech.

Array Response y(t)*

1

( ) ( ) ( )L

Hl l

l

y t w x t t

w x

Output power P(w) 2

1

1

1( ) ( )

1( ) ( )

N

l

NH

l

H

P y tN

t tN

w

w x x w

w R w

1. Conventional Beam-former

2. Capon’s Beam-former

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Conv. Conv. BeamformingBeamforming & Steering Vector & Steering Vector

( ) ( ) ( ) ( )

( ) ( )

n s n n

n n

u a n

x n

TkdLjjkd ee ]1[)( cos)1(cos a

22

c

fk

d d

θ(M-1)d

X

Y

)()(

)(

aa

aw

H

BF

)()(

)()()(

aa

aRaH

H

BFP

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ULA

d = 0.5 λ

M = 4 、 8 、 12

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MVDR Beamforming(Capon’s)MVDR Beamforming(Capon’s)

1)( awH

)()(

)(1

1

aRa

aRw

H

CAP

)()(

1)( 1

aRa

H

CAPP

2 22 2

min{ ( )}

min ( ) ( )H

P

E s t w

w

w a w

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OutlineOutline

• Conception of Smart Antenna• Beamforming Method• DOA Estimation• θ- LMS Algorithm• Local Minimum problems• θ- LMS Algorithm with Robust Beamforming• Conclusion

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DOA EstimationDOA Estimation

• Conventional • Capon’s• Subspace

– MUSIC

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Conventional DOA EstimationConventional DOA Estimation

w(0~180)Conventional

u(n)Receiving signal

Pattern(0~180)

DOAEst.

w(n)

BeamformingWeighting

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Capon’s DOA EstimationCapon’s DOA Estimation

w(0~180)Capon’s

u(n)Receiving signal

Pattern(0~180)

DOAEst.

w(n)

BeamformingWeighting

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MUSICMUSIC((MUMUltiple ltiple SISIgnal gnal CClassificationlassification ) )

PMUSIC Pattern

u(n)Receiving signal

Eigen decompositionNoise Space

Vn

1( )

( ) ( )MUSIC H H Hn n

P

a V V a

a(0~180)

DOAEst.

w(n)Weighting Vector

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Compare the three methodsCompare the three methods

ULA

M = 4

d = 0.5λ

User’s DOA =

90°、 120 °

SNR=10dB

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OutlineOutline

• Conception of Smart Antenna• Beamforming Method• DOA Estimation• θ- LMS Algorithm (My point)• Local Minimum problems• θ- LMS Algorithm with Robust Beamforming• Conclusion

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LMS (Least Mean Square ) LMS (Least Mean Square )

1

0

( ) ( ) ( ) ( ) ( )M

Hi i

i

y n n n w n u n

w u

( 1) ( ) { ( )}n n w w J w

( ) ( ) ( ) ( )He n d n n n w u

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ww – LMS Algorithm – LMS Algorithm

d(n)

w - LMS

u(n)

w(n)

w(n+1)

y(n)

e(n)+

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θθ- LMS Algorithm- LMS Algorithm

+

d(n)

ө- LMS

u(n)

w(n)

w(n+1) y(n)

e(n)

ө(n+1)ө(n)

4 x1

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θθ- LMS Algorithm - LMS Algorithm

u(n) w(n) wH(n) u(n)-d(n)

Cost functionJ(θ)

find DOA θ0 Beamforming

Weighting by DOA θ0

w(n)Weighting

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Weighting is Instead of Weighting is Instead of θθ

( 1) ( ) { ( )}n n w w J w

2

( 1) ( ) { ( )}

( )( )

n n

e nn

J

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2

* cos ( 1) cos0

( , )( )

12 sin 0 1 ( 1) ( , )ikd i M kd

e nn

e ikd e M e nM

u

( 1) ( ) ( )n n n

• Adaptive θ(n) is defined

DefinitionDefinition

=

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ULA

M = 4

d = 0.5λ

DOA= 120 °

Initial DOA = 90 °

Step size = 0.01

SNR =20

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ULA

M = 4

d = 0.5λ

DOA= 0 ° ~180 °

Initial DOA = 90 °

Step size = 0.01

SNR =20

DOA=90*sin(0:0.01:80*pi) + 90;

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Steering Vector TrackingSteering Vector Tracking

ULA

M = 4

d = 0.5λ

DOA= 0 ° ~180 °

Initial DOA = 90 °

Step size = 0.01

“*” steering vector

“o” tracking vector

DOA=90*sin(0:0.05:80*pi) + 90;

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Beampattern TrackingBeampattern Tracking

ULA

M = 4

d = 0.5λ

DOA= 0 ° ~360 °

Initial DOA = 90 °

Step size = 0.01

“o” DOA

DOA=180*sin(0:0.05:80*pi) + 180;

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OutlineOutline

• Conception of Smart Antenna• Beamforming Method• DOA Estimation• θ- LMS Algorithm• Local Minimum problems• θ- LMS Algorithm with Robust Beamforming• Conclusion

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Converse to Error DirectionConverse to Error Direction

ULA

M = 4

d = 0.5λ

DOA= 90 °

Initial DOA = 120 °

Step size = 0.01

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Error patternError pattern

d(n )

x(n)4 x1

d(n )

x(n)

w(0~180)

e(0~180)4 x1 +

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Cost functionCost functionULA

M = 4

d = 0.5λ

DOA= 90 °

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ULA

M = 4

d = 0.5λ

DOA= 0 ° ~180 °

DOA=90 °

Error Surface (DOA )Error Surface (DOA )

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Error Surface (d )Error Surface (d )ULA

M = 4

d = 0 ~1λ

DOA= 90 °

d=0.5 λ

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Error Surface (M )Error Surface (M )ULA

M = 2 ~8

d = 0.5λ

DOA= 90 °

M=4

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2 Ants2 Antsθθ- LMS Algorithm- LMS Algorithm

d(n)

ө- LMS

u(n)

w(n)

w(n+1) y(n)

e(n)

ө(n+1)ө(n)

2 x1 +

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Error Surface Error Surface (M=2 d=0.5 )(M=2 d=0.5 )

ULA

M = 2

d = 0.5λ

DOA= 90 °

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Error Surface Error Surface (M=2 d=0.25 )(M=2 d=0.25 )

ULA

M = 2

d = 0.25λ

DOA= 90 °

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Simulation (1)Simulation (1)

ULA

M = 2

d = 0.25λ

DOA= 5 °

Initial DOA = 175 °

SNR = 30 dB

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ULA

M = 2

d = 0.25λ

DOA= 170 °

Initial DOA = 5 °

SNR = 30dB

Simulation (2)Simulation (2)

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2 – 4 “2 – 4 “θθ-LMS” -LMS”

2 antennas “θ-LMS”

4 antennas “θ-LMS”

Initial θ

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OutlineOutline

• Conception of Smart Antenna• Beamforming Method• DOA Estimation• θ- LMS Algorithm• Local Minimum problems• θ- LMS Algorithm with Robust Beamforming• Conclusion

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Noise problemNoise problem

• θ- LMS Algorithm needs high SNR level.

• High noise level brings the DOA estimation result worse.

• The DOA estimation error will cause the terrible performance

• We use the Robust Beamforming Method to conquer the estimation error problem.

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The flow chart of Robust BeamformingThe flow chart of Robust Beamforming

SOI DOATracker

Sample correlation matrix

Parametric desired correlation matrix

From average correction matrices

{y(k)}

DOA spreading matrix

Compute robustBeamformer

0(0) (0)dR

(0)yR

( )d KR

( )y KR

2max ( )K

( )r Kw

0max

max max

( )( )r H

d

kk

w e

e R e

Ref : Riba J., Goldberg J., Vazquez G., “Robust Beamforming for Interference Rejection in Mobile Communications”, Signal Processing, IEEE Transactions on

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Robust BeamformingRobust Beamforming

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BER AnalysisBER Analysis

BPSK

ULA

M = 4

d = 0.5 λ

DOA = 90°

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OutlineOutline

• Conception of Smart Antenna• Beamforming Method• DOA Estimation• θ- LMS Algorithm• Local Minimum problems• θ- LMS Algorithm with Robust Beamforming• Conclusion

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ConclusionConclusion

• DOA is an important parameter for beamforming system.

• But, the MUSIC algorithm is complex.• The new method “ө - LMS” is simpler to realized• Robust Beamforming can repair the fault of “ө - L

MS”

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Future WorkFuture Work

• Noise and channel problem • Multi-user problems• SINR Analysis• Multi-path & DOA distribution• Moving Source analysis• Performance Analysis

(User # 、 DOA 、 SNR 、 Beamforming method 、 Antenna # …etc.)

• Adaptive Analysis (Step-size Moving DOA 、 SNR 、 other adaptive structure)

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Capon’s BeamformingCapon’s Beamforming

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Step SizeStep Size

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Cost functionCost function

22( ) ( ) HE e n E d

J w w u

2

*

*

*

2

( ) ( ) ( )2

( ) ( )

2 ( )

H

e ee

d n n ne

n n

e n

w w

w u

w w

u

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New cost functionNew cost function

{J()} is defined partial J() by

2

*

*

( ) ( )2

( ) ( ) ( )2

H

e n e ne

d n n ne

w u

0 0( , ) ( ) ( )n s n u a

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cos ( )

( 1) cos ( )

1

( , ( )) 1( , ( ))

ikd n

i M kd n

en nn n

M M

e

aw

2

*0

*0

* cos ( 1) cos0

( , ) ( , )2 ( , )

1 ( , )2 ( , )

12 sin 0 1 ( 1) ( , )

H

H

ikd i M kd

e n ne n

ne n

M

e ikd e M e nM

wu

au

u

The Formula Derives The Formula Derives

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Initial = 90 ° DOA = 0 °

ULA

M = 2

d = 0.25λ

DOA= 0 °

Initial DOA = 90 °

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Initial = 90 ° DOA = 180 °

ULA

M = 2

d = 0.25λ

DOA= 180 °

Initial DOA = 90 °

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Noise problemNoise problem

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Degree spreadingDegree spreading

k = 0

2 2( ), ( (0), ( )), (0) 0m m m mk N k 2| (0), ( )m m mp k

,( ) ( ) ( )y d i nk yk E k k R R R R

Ref : Riba J., Goldberg J., Vazquez G., “Robust Beamforming for Interference Rejection in Mobile Communications”, Signal Processing, IEEE Transactions on

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Robust BeamformingRobust Beamforming

0max

max max

( )( )r H

d

kk

w e

e R e

( ) arg min ( )Hyr k k

ww w R w

0( )Hd k w R w

20 0 0 0 0 0 0( ) | (0), ( ) ( ) ( )H

d k p k d R a a

1

2 2(0) (0)

0

( ) (0), ( )N

Hy m m mm m

m

k k

R a a Q I

2 2 2(0)2 2 ( ) cos

(0), ( )( ) m m

m

d p qkm pq

e

Q

1

2 2

1

( ) | (0), ( ) ( ) ( )N

Hi n m m m m m m m

m

k p k d

R a a I

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Complex SurfaceComplex Surface