Bayesian Focusing Transformation for Coherent Adaptive …webee.technion.ac.il/Sites/People/IsraelCohen/Info/... · 2010. 10. 11. · 1 Bayesian Focusing Transformation for Coherent

1

Bayesian Focusing Transformation for Coherent Adaptive Wideband

Beamforming

Yaakov Buchris

Supervised by:

Prof. Israel Cohen and Dr. Miri DoronSep 2010

2

• Background

• Existing focusing approaches

• The Bayesian focusing transformation

• The focused loaded MVDR

• Performance analysis of the loaded MVDR

• Summary and Future research

Outline

3

• Background



•The focused loaded MVDR



Outline

4

• Adaptive beamforming techniques are widely used in many real-world applications, includingwireless communicationsradar/sonaracoustics and seismic sensing

• Some of these applications require wideband adaptive beamforming processing.

Background

Background

5

Some relevant applicationsUnderwater acoustic communications

The limited available bandwidth and the very low carrier frequency result in underwater communication which is inherently broad band

5

6

Some relevant applicationsUnderwater acoustic communications

• The limited available bandwidth and the relatively low carrier frequency result in underwater communication system which is inherently broad band.

From H.L.Van Trees, 2002

6

Background

7

Smart antennas for high speed wireless data communications

• Increasing the signal bandwidth along with the use of antenna arrays is an effective way to increase the data rate in future wireless communication systems.

T.D.Hong, P.Russer, 2004

Some relevant applications-cont

7

Background

8

Speech signal processing

Processing in time domain

Broadband beamforming –processing in frequency domain

Some relevant applications-cont

8

Background

9

The signal model

1( ) ( ) ( ), , 1,.....,

2 2

P

n p np np

T Tx t s t n t t n N

in the frequency domain:

1,..., , 1, ....,( ) ( ) ( ) ( ),k j j k j k j j J k Kf f f f θx A s n

Consider an arbitrary array of N sensors, sampling a wavefield generated by wideband sources, Pin a presence of additive noise:

1 2

2 ˆ ˆ( ) ( ), ( ),..., ( ) , ( ) exp (cos sin )Pj j j j mm

ff f f f f i x yc

θA a a a a r

where

arraySource

Background

10

• Time domain methods based on tapped delay line adaptive filters are often used for wideband adaptive arrays.

• Frequency domain methods in which each frequency bin is treated as a narrowband beamformer are also used– Non Coherent .

Wideband adaptive beamforming methods

10

Background

(T.S.Rappaport, 1998 )

(R.T.Compton, 1988 )

11

1( )x f

2 ( )x f

( )Nx f

1 1( )w f

2 1( )w f

1( )Nw f

1( )Jw f

2 ( )Jw f

( )N Jw f

1( )y f

( )Jy f

IFFT

wavefield

1( )fw

( )Jfw

_ ( )Non Coherenty n

FFT

FFT

FFT

Block diagram of the non-coherent adaptive beamformer

Background

12

The coherent processing

• Drawbacks of the time domain and non-coherent methods: Slow convergence rate

Computational expensive

Signal cancellation problem

• The wideband focusing approach for adaptive beamforming based on the concept of signal subspace alignment was originally proposed by Wang & Kaveh in the 80’s– Coherent.

12

Background

13

( )jfTIt is required to find which satisfies:

0( ) ( ) ( )j jf f fθ θT A A

( )JfT

1 ( )x f

2 ( )x f

( )Nx f

1( )fT1( )k fx

( )k Jfx

IFFT 0( )fw

0( ) ( ) ( )f n n A s nwavefield( )Coherenty n

FFT

FFT

FFT

Block diagram of the coherent adaptive beamformer

( )jfT is the focusing matrix

Background

)(nky

14

The focusing operation

1,..., , 1,....,( ) ( ) ( ) ( ),k j j k j k j j J k Kf f f f θx A s n

1

01 1

0 01

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( )

s j

s j s j

s j

JinT f

k j k jj

J JinT f inT f

k j j k jj j

jinT f

k j k k kJ

y n f f e

f f e f f e

f f e n f n n

θ

θ θ

T x

A s T n

A s n A s n

The received vector is

Focus the frequency dependent steering matrices into fixed matrix matched to 0f

back14

15

Motivation for coherent processingComparison between the Array Gain (AG) of non – coherent widebandBF to AG of coherent (focusing based ) wideband BF:

10 15 20 25 30 35 40 45 505

6

7

8

9

10

11

12

13

N um ber o f s naps ho ts

AG

[dB

]

A G V s . N um ber o f s naps ho ts

N on c ohe ren tC ohe ren t

OUT

IN

SINRAGSINR

Improved convergence time due to the relatively small number of adaptive weights

Background

16

• Background






Outline

17

• Background






Outline

18

Existing Focusing approaches• There are two basic approaches to design focusing

transformations

Assuming a-priori knowledge of Directions of Arrivals (DOAs) and focus at these specific directions.

Focus all angular directions.

},..,,{),()()( 210 Pjj fff θAAT θθ

},{),()()( 0 fff θjθj aaT

18

Existing Focusing approaches

19

Wang and Kaveh Focusing Transformation (WKFT) / H.Wang and M.Kaveh, 1985

• A focusing transformation which requires a-priori knowledge of the sources' DOAs vector

• WKFT attempts to find which satisfy ( ), 1,...,jf j JT

0 0( ) ( ) | ( ) ( ) | ( )j j jf f f f f θ θθ θT A B A B

( )fθB • contains auxiliary directions

The solution is

19


†

0 0( ) ( ) | ( ) ( ) | ( )j j jf f f f f θ θθ θT A B A B

20

Rotational Signal Subspace (RSS) /H.Wang & M.Kaveh,1988

• A class of unitary focusing matrices called RSS - No focusing loss.

• These matrices also focus only pre-estimate DOAs.

• The proposed focusing matrix, satisfies the minimization criterion:

1 2 1 20, ,.., , ,..,( )

( ) ( ) ( ) , 1,...,

. . ( ) ( )

min L Lj

j j Ff

Hj j

f f f j J

s t f f

TA T A

T T I

The solution:H

jjj fff )()()( UVT

where ( ), ( )j jf fV U are left and right SV of 0( ) ( )Hjf fθ θA A


21

The WINGS transformation minimizes

†0( ) ( ) ( )j jf f fT G G

The WINGS solution is

22 1 ( )j jd fN

e

Wavefield Interpolated narrowband generated subspace(WINGS)/ M.A.Doron & A.Nevet, 2008

0( ) ( ) ( ) ( ), ,j j jf f f f e a T a

where is a geometry dependent matrix called the samplingmatrix whose columns are the orthogonal decomposition coefficientsof the array manifold

)( fG

)()()( nmmn hfdf aG

where are orthonormal basis over ( )nh 2 ( )L


Wavefield modeling review

• This approach is based on the idea that the output of any array of arbitrary geometry can be written as a product of array geometry dependent part, and wavefield dependent part.

ˆ , . . [ , ], 2( , ) ( ) ,

( , ), . . [0, ], [ , ], 3ik st D

f d est D

rr

2( ) ( )L is the radiation density in the direction

Wavefield modeling review –cont.

Let ( )nf denote a complete and orthogonal basis for 2 ( )L

The equivalent basis functions ( )nh r in H are:

ˆ( , ) ( ) jkn nh f d f e

rr

*1 2 1 2 1 2, , ( ) ( )

Hd

Define H as the Hilbert space of the wavefields of the form of( , )f r , with the scalar product:

Wavefield modeling review –cont.

( , )G f x r

( , ) ( , ), ( , ), ( , )n n n nn

f h f f h f r r r r

Sampling operator:

( , )w H rThe orthogonal decomposition of is:

Define ng as:

( , )n nG h fg r

Wavefield modeling review –cont.Using the linearity of the sampling operator, the array output can be written as:

( , ) ( , )n n n nn n

G f G h f x r r g Gψ

The orthogonal decomposition of the steering vector is:

*

( ) ( ) ( )

( ) ( ) ( )

n n

n nn

f f f d

f f f

g a

a g

which can be rewritten: *( ) ( ) ; [ ] ( )n nf f f a G w w

1 0 1..., , , ,....G g g g - is a property of the array geometry only.

- is the coefficient vector of the orthogonal( , )f r

1 0 1[...., , , ,....] ψdecomposition of

Focusing Matrices – WINGS

It is desired to find

0( ) ( ) ( ) ( ),j j jf f f f a T a e

( )jfe can be written as:

0 0( ) ( ) ( ) ( ) [ ( ) ( ) ( )]j j j j jf f f f f f f e G w T G w G T G w

Define 2j as the normalized integral of the squared error over all

possible directions, and using Parseval’s identity:

2 220

1 1( ) ( ) ( ) ( )j j j j Fd f f f f

N N

e G T G

LS solution:†

0( ) ( ) ( )j jf f fT G G

( )jfT that minimizes :( )jfe

Back

27

• Focusing on preselected DOAs Low focusing error× High sensitivity to DOAs uncertainties

• Focusing in all angular directions High focusing errorNo sensitivity to DOAs uncertainties

• We proposed a Bayesian approach which takes into account the probability densities functions of the DOAs vector

• The Bayesian approach can handle the trade-off between focusing error and DOAs uncertainties, yields an optimal MMSE focusingtransformation

Drawbacks of existing methodsExisting Focusing approaches

28

• Background






Outline

29

• Background






Outline

30

Model assumptions:

1

Pi i

• - The DOAs are independent RVs with pdfs 1

( )i

P

i ig

• A new focusing method based on a Bayesian approach and

utilizing the weighted WINGS transformation is proposed.

• It enables to use statistical knowledge of the DOAs.

• This approach yields an optimal MMSE focusing transformation.

The proposed focusing transformation:A Bayesian approach

31

)( jfTFind optimal in MMSE sense:

2

0( )

( ) ( ) ( ) ( )Ej

MMSE j j j Fff f f fargmin θ θ

TT A T Aθ

The goal:

The Bayesian Focusing Transformation (BFT)

Model assumptions:

1

Pi i

• - The DOAs are independent RVs with pdfs 1

( )i

P

ig

The BFT

32

In order to solve the last problem, we derive a solution to the following minimization integral (weighted version of the WINGS):

22

( )

1min ( ) ( )j

j jfd f

N

Te

†0( ) ( ) ( )j jf f fT C C

[ ( )] ( )[ ( )] ( )mn m nf d f h

C a

yielding:

where are orthonormal basis over ( )nh 2 ( )L

Developing the weighted WINGS

The BFT

22 1 ( )j jd fN

eWINGS

33

2 2

0 01,

( ) ( ) ( ) ( ) ( ) ( ) ( )Ei

P

j j j jFi

f f f d f f f g

θ θA T A a T aθ

It can be shown that:

defining:

substituting the above definition to the upper integral, yields:

2

0)(

)()()()(argmin)(

jjf

jMMSE fffdfj

aTaTT

2

1( ) ( )

i

P

ig

2

01 ( ) ( ( ) ( ) ( ))j jd f f fN

a T aWhich has exactly the form of the term

minimized by the weighted WINGS:

The BFT– contThe BFT

Proof:

,)()()()(

)()()()()(

)()()()()..(..

)()()(E

)()()(E

1

2

0

2

01 1

1

1

2

011

1

2

0

2

0

1

P

ijj

jjii

P

i

P

ikk

kk

P

ijjpp

P

ijj

Fjj

i

iiik

iip

ii

gfffd

fffgdgd

fffggdd

fff

fff

aTa

aTa

aTa

aTa

ATA

θ

θθθ

back

35

[ ( )] ( )[ ( )] ( )mn m nf d f h

C a

†0( ) ( , ( )) ( , ( ))MMSE j jf f f T C C

The LS solution to the minimization problem is:

where:

This image cannot currently be displayed.

The choice 2

1

( ) ( )i

P

i

g

yields an optimal

in the MMSE sense.( )jfT

The BFT– contThe BFT

Estimation of for focusing 1

( )i

N

ig

In order to develop a time progressing algorithm, the following

step is proposed:

11 1( ) ( | )i i

PP Kk ki i

g g

y

21

1

( ) ( | )i

PK

k ki

g

y

36

The BFT

Estimation of the a posteriori pdfs - Gaussian model assumption

• Assume independent RVs

_1ˆE( | )K

i i k i MMSEk

y

_i MMSE• Instead of calculate _i DF which is the output of direction

finding (DF) algorithm, e.g. MUSIC

•

1

2_|

ˆ~ ( , )Ki k k

i DF ig N

y

•

2| ~ ( , )i k i iN y

is taken as a quarter of the 3dB of an array at . i2 1.5i

37

The BFT

Estimation of the aposteriori pdfs-method 2: direct calculation

In Yariv’s paper an expression to the aposteriori pdf of the DOA ofthe desired signal was derived:

201

ˆ( | ) g( ( ), , , ( ), )i

Ki k i s nk

f f f

y R a Rwhere:

)( 0fia

n

Hkk nn

KJ)()(1ˆ yyR

( )if - a periori pdf of DOA of is2s - power of the desired signal

- interference plus noise covariance matrix

g - a deterministic function

nR

- the steering vector back

Numerical study of the focusing error in the presence of DOA uncertainties

• Gaussian wideband sources

• Linear array of sensors

• Gaussian DOAs with mean ,and variance (half of the 3dB beamwidth)

• The bandwidth of the sources is 600Hz around

• The focusing frequency is

• WKFT, RSS, WINGS, BFT focusing methods

• Assuming Gaussian DOAs for theweighting function of the BFT

2P

20N

70 ,105 θ

1500cf Hz

0 1500f Hz

0 20 40 60 80 100 120 140 160 180 2000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

2 ( )

39

2 3

The BFT

Numerical study of the focusing error - cont

BFT provides considerable robustness to DOA uncertainties 40

The BFT

1200 1300 1400 1500 1600 1700 1800

0

0.05

0.1

0.15

0.2

0.25

0.3

Frequency [Hz]

Focu

sing

erro

r

WINGSBFTWKFTRSS

Fjjj ffffe )()()()( 0 θθ ATA

41

Numerical study of the focusing error - cont

BFT has relatively flat error around the true DOAs 41

The BFT

Fffffe )()()(),( 1101 aTa

0 20 40 60 80 100 120 140 1600

0.5

1

1.5

Angle [degree]

Focu

sing

erro

r

WINGSBFTWKFTRSS

Proposed algorithm – block diagram

42back

43

• Background



• The focused loaded MVDR



Outline

44

• Background






Outline

The Minimum Variance Distortionless Response (MVDR) focused beamformer

1

011

,0 0

ˆ ( ) 1ˆ( , ) , ( ) ( )ˆ( ) ( )K H

MVDR k k kHkk n

f n nKJf f

R aw y R y ya R a

)(}){,()(1 nns kkHMVDR yyw

Assuming )(1 ns to be the desired signal:

The focused vector ( )k ny can be constructed:

( )k nyOn the focused vector any adaptive narrowband beamformeralgorithm can be applied. For example, SMI-MVDR:

45

The focused loaded MVDR

)(~)()()()()( 01

nnfeffn kk

J

j

finTjkjk

js nsAxTy θ

46

Robust MVDR focused beamformer •The MVDR beamformer is known to have superior resolution and interference rejection capabilities

• In practice there is a performance degradation due to array calibrationerrors, and covariance estimation errors (SMI)

• The focused MVDR beamformer will exhibit an additional sensitivityto the focusing errors especially at high SNR

• Diagonal loading is a popular approach to improve the robustness (H.Cox et.al,1987)

1

1

( ) ( )( )

( ) ( ) ( )H

f ff

f f f

x

x

R I aw

a R I a

• Diagonal loading limiting the white noise output gain


22 )()( ffn nout w

47

The diagonal loading solutionIn its narrowband formalism, the diagonal loading solves

( )

0

min ( ) ( ) ( )

. .( ) ( ) 1,

( ) ( )

H

f

H

H

f f f

s tf f

f f T

xww R w

w a

w wThe solution is

1

1

( ) ( )( )

( ) ( ) ( )H

f ff

f f f

x

x

R I aw

a R I a

Diagonal loading limiting the white noise output gain 22 ( ) ( )out nn f f w


back

48

The focused loaded MVDRWe extend the diagonal loading solution to fit also for the focused beamformer. In the case of focused beamformer, the output noise power is

2 2

1

1 ( ) ( )out

JHf H fn n l l

lf f

J

w T T w

Thus, limiting the white noise gain yields the following quadratic constraint

The vector coefficients of the Q-loaded focused beamformer is given by

1

0,1

0 0

( )

( ) ( )

ff QL

H f

f

f f

x

x

R Q aw

a R Q a

J

ll

Hl

fHf ffJ

T1

0 )()(1, TTQQww


49

• Background






Outline

50

• Background






Outline

51

Simulation results for the case of DOAs uncertainties

Array Gain (AG) of the focused loaded MVDR beamformer – 2 sources

out

in

SINRAGSINR

The superiority of BFT is evident, especially at high SNR

-20 -10 0 10 20 30 40 5010

20

30

40

50

60

70

80

90

S NR [dB ]

AG

SIN

R[d

B]

W INGS - s im ula tionW ING S - ana lyticB F T - s im ula tionB F T - ana lyticW K F T - s im ula tionW K F T - ana lytic

51

Performance analysis of the loaded MVDR

52

Simulation results for the case of DOA uncertainties - cont

Array Gain (AG) of the focused un-loaded MVDR beamformer – 2 sources

Except the analytic BFT there is a considerable degradation

in the performance in high SNR values

Q-loading yields better performance

-20 -10 0 10 20 30 40 5010

20

30

40

50

60

70

80

S NR [d B ]

AG

SIN

R[d

B]

W ING S - s im ula tio nW ING S - a na lyticB F T - s im ula tio nB F T - a na lyticW K F T - s im ula tio nW K F T - a na lytic


53

The AG Vs. SNR for different values of number of snapshots, K

The simulative curves become closer to the analytic as the numberof snapshots increase

-20 -10 0 10 20 30 40 5010

20

30

40

50

60

70

80

90

S NR [dB ]

AG

SIN

R [d

B]

B FT - ana lyticB FT - K =46B FT - K =125B FT - K =625B FT - K =1250

Simulation results for the case of DOA uncertainties - cont


54

Simulation results for the single source case

The sensitivity of the MVDR

beamformer in high SNR values

occurs due to focusing error in

the desired source direction

With loading

Without loading

- 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 0- 5 0

- 4 0

- 3 0

- 2 0

- 1 0

0

1 0

2 0

S N R [d B ]

AG

SIN

R[d

B]

W IN G S - s im u la ti o nW IN G S - a n a ly ti cB F T - s im u la ti o nB F T - a n a ly ti cW K F T - s im u la ti o nW K F T - a n a ly ti c

-2 0 -1 0 0 1 0 2 0 3 0 4 0 5 05

6

7

8

9

1 0

1 1

1 2

1 3

S N R [d B ]

AG

SIN

R[d

B]

W IN G S - s im u la tio nW IN G S - a na lyti cB F T - s im u la tio nB F T - a na lyti cW K F T - s im u la tio nW K F T - a na lyti c


55

Performance degradation of BFT in low SNR

back

56

Simulation results for the case of single source and perfect knowledge of the DOA

We use WINGS method as a test case since it has a relatively large focusing error

Performance degradation occurs due to focusing error in the desired source direction

57

Analytical study of the focused MVDR sensitivity tofocusing errors

It can be shown that the output AG of the focused MVDR beamformer is approximately ( for the single source with single frequency)

2

2, 1,

2 2 2 22 2 2 2

1(1 )1 ( 1) 1

gM Mg

g gg g g

MAG

MM M M

where

2g - the variance of the focusing error

- the input SNR

AG decreases like 1020 log ( )

M - number of sensors

58

Robust MVDR focused beamformers for coherent wideband array processing

In order to reduce the sensitivity to focusing errors we propose two methods:

• General-Rank focused MVDR (GR-MVDR)

This method is based on modifying the MVDR beamformer by implementing a robust General-Rank (GR) beamforming scheme.

• Enhanced Focusing (EF)

This method is based on modifying the focusing transformation so that the focusing error is reduced in the direction of the desired source.

Simulation results show that the proposed methods reduce the sensitivityto focusing errors and improve the AG in high SNR values.


59

General-Rank focused MVDR (GR-MVDR)

The desired signal covariance matrix is 2

1( ) ( ) ( ) ( ) ( )

d d

Jf H Hs s j j j j j

jf f f f f

R T a a T

The structure of implies that it’s rank is higher than one.Therefore, the general rank MVDR beamformer can be used

fsR

The minimization problem is

0 0 0 0min ( ) ( ) . . ( ) ( ) 1H f H fsf f s t f f xw

w R w w R w

The solution is

1f f fGR MVDR sP

xw R R

Where denotes the principal eigenvector of a matrix.P


60

General-Rank focused MVDR (GR-MVDR) - cont

We extend this method to take into account the sensitivity to array calibration errors, SMI implementation errors and focusing errors The solution is given by the robust Q-loaded form of the GR-MVDR

1f f fGR MVDR QL sP

xw R Q R

• Note that the GR focused MVDR requires a-priori knowledge of the spectral shape of the source

• Following [8] we use a robust version combating a small signal spectrum mismatch

1

_f f fROBUST GR MVDR QL sP

xw R Q R I


61

Enhanced Focusing (EF)

• We saw that the performance degradation occurs mainly due to focusing error in the desired source direction

• Adding an additional error component in the desired source direction to the LS minimization term of the WINGS enables us to reduce the error in the source direction

220

1 ( ) ( ) ( )j j j Fw w w

N G T G

where

( ) ( ), ( )d

w w w G a G


62

Performance analysis of the robust focused MVDR beamformer

WINGS focusing method, single source


The GR-MVDR has a lower computational complexity than the EF method,since it does not require DOA dependent redesign of the focusing matrices.

63

Sensitivity to source spectrum

Performance analysis of the robust focused MVDR beamformer

We assumed the spectrum of the sources to be flat


The robust GR-MVDRCan handle spectral deviation Smaller than 1dB

64

• Background






Outline

65

• Background






Outline

66

Summary• In this work we have proposed and investigated a Bayesian approach for

focusing transformation design.

• It yields an optimal MMSE focusing transformation and consequently an

improved beamformer with better AG.

• We combat the sensitivity to modeling errors such as focusing errors by

extending the diagonal loading to the focused wideband beamformer.

• Numerical and simulative results demonstrate the superiority of the BFT and

of the Q-loaded beamformer for the multi-source case with DOA

uncertainties.

• We also investigate the sensitivity to focusing errors in high SNR and propose

two robust methods for focused MVDR aiming at reducing this sensitivity.

Summary and future research

67

Future research• Extension to the case that the signals propagate in a multipath

environment

• Optimize the BFT performance by model their PDFs in more advanced way, for example, model their variance using the CRB

• Incorporating robust extension of the WINGS method also to the BFT method aiming at increasing the robustness to the noise gain of the transformation

• Choosing the optimal focusing frequency

• Reducing the computational complexity of the BFT

Summary and future research

68

Bibliography

[1] H. L. Van-trees, \Detection, estimation and modulation theory, part iv -

optimum array processing," Wiley Interscience, 2002.

[2] T. Do-Hong and P. Russer, \Signal processing for wideband smart antenna array

applications," IEEE Microwave Mag., vol. 5, pp. 57 - 67, March. 2004.

[3] S. Ohmori, Y. Yamao, and N. Nakajima, \The future generations of mobile

communications based on broadband access technologies," IEEE Commun Mag., vol. 38,

pp. 134 - 142, Dec. 2000.

[4] H. Wang and M. Kaveh, \Coherent signal subspace processing for the detection and

estimation of angles of multiple wide band sources," IEEE Trans. on Acoustics, Speech and

Signal Processing, vol. ASSP-33, pp. 823-831, Aug.1985.

[5] H. Hung and M. Kaveh, \Focusing matrices for coherent signal subspace processing,"

IEEE Trans. on Acoustics, Speech and Signal Processing, vol. 36,pp. 1272-1281, Aug. 1988.

69

Bibliography - cont

[6] M. A. Doron and A. Nevet, \Robust wavefield interpolation for adaptive wideband

beamforming," Signal Processing, vol. 80, pp. 1579-1594, 2008.

[7] H. Cox, R. M. Zeskind, and M. M. Owen, \Robust adaptive beamforming,"

IEEE Trans. Acoust., Speech, Signal Processing, vol. 35, pp. 1365-1376, Oct.1987.

[8] S. Shahbazpanahi, A. B. Gershman, Z. Luo, and K. M. Wong, \Robust adaptive

beamforming for general-rank signal models," IEEE Trans. Signal Processing, vol. 9, pp.

2257-2269, Sep. 2003.

[9] C. C. Lee and J. H. Lee, \Robust adaptive array beamforming under steering vectors

errors," IEEE Trans. Antennas Propagat., 1997.

[10] T. S. Rappaport, \Smart antennas," IEEE Press, New York, 1998.

Documents

Bayesian Focusing Transformation for Coherent Adaptive …webee.technion.ac.il/Sites/People/IsraelCohen/Info/... · 2010. 10. 11. · 1 Bayesian Focusing Transformation for Coherent