Adaptive Array Antenna-2008

8/12/2019 Adaptive Array Antenna-2008

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Department of E & ECE, IIT Kharagpur, India, 721 302

ADAPTIVE BEAM-FORMING FOR SATELLITE

COMMUNICATION

by

Prof. Binay K. Sarkar

ISRO Chair Professor

( Lecture prepared for delivering at the Summer School on Recent Trend in AntennaTechnology to be held at KIIT , Bhubaneswar on Aug.05,2008 )


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In the satellite communication systems, many users are active in the same time.

Since resources of the systems (the transmitting power, the bandwidth) are limited, it is

advisable to use multiple access to channel.

This generates a problem of summation and separation of signals in the transmission

and reception parts respectively.

. Currently the methods of multiple access widely used are

Time division multiple access (TDMA),

Frequency division multiple access (FDMA) and

Code division multiple access (CDMA),

as well as their combination


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The significant factor in the performance of multiple access in a satellite

communications system is interference.

It causes cross-talk, missed or dropped calls, and upsets customers.

Most importantly interference limits trafficcarrying capacity from the finite RF

spectrum.

Interference can come from

another users,

other cell sites operating on the same frequency, or

outofband RF energy leaking into the allocated spectrum.

The most usual types of interference are

cochannel interference and

adjacent channel interference.

Co-channel interference is caused by transmissions from non-adjacent cells using the

same set of frequencies, where there is minimal physical separation from neighboring

cells using the same set of frequencies..


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Adjacent channel interference is caused by RF leakage on the subscribers channel from

a neighboring cell using an adjacent frequency.

This can occur when the users signal is much weaker than that of the adjacent channel

user.

Signal to interference ratio (SIR) is an important indicator of call quality; it is a measure

of the ratio between the mobile phone signal (the carrier signal) and an interfering signal.

A higher SIR ratio means increasing overall system capacity.

Taking into account, what in the system of satellite communications every user has own

unique space position, this fact may be used for the separation of channels in the space

and as consequence to increase SIR ratio.

Such methods is named Space Division Multiple Access (SDMA) .

SDMA applies smart antenna array with different level of intelligence consisting from the

antenna array and digital processor.


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There are two different beamforming approaches in SDMA for satellite communications :

One (SwitchedBeam Antennas) is a to track each subscriber of a given cell with an

individual beam pattern as the target subscriber moves within the cell, and the

Other (Adaptive Array Antenna Systems) is to select one beam pattern for each

subscriber out of a number of preset fixed beam patterns depending on the location of

the subscriber.

This lecture deals with the adaptive beamforming arrays.


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FUNDAMENTALS OF ADAPTIVE ARRAYS

An adaptive array is a system consisting of an array of sensor elements and areal time adaptive signal receiver-processor that automatically adjusts the array

beam sensitive pattern so that a measure of the quality of the array performance

is improved

An adaptive array offers enhanced reliability compared to that of conventionalarray.

When a single sensor element in a conventional array fails, the sidelobestructure of the array sensitivity pattern may be significantly degraded because of

sidelobe increase. With an adaptive array, however, the response of the

remaining operational sensors in the array can be automatically adjusted until the

array sidelobes are reduced to an acceptable level.


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wave front

w1

w2

s(t)

wn

control algorithm

signal processor

sensor array

array

output

y(t)

1

2

N

x1 (t)

x2 (t)

xN) (t)

Principal System Elements Of Adaptive Array

ADAPTIVEPROCESSOR

Pattern forming network


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ULA Adaptive Beamformer

w1*

narrowbandsignal

(N-1)dsin0

2dsin0

dsin 0

I&Q

downconv-ert and

ADC

I&Q

downconv-

ert and

ADC

I &Q

downcon

vert and

ADC

I&Q

downconv-

ert and

ADC

wn* w3

*

w2*

y= w's

d

/2

123 2N

exp(j(2f0t-k.r)

plane

wavefront

Figure: 2 A ULA adaptive beamformer

0

sn = exp(j2(n-1)(d/)sin 0 ) ,

n= 1,2,..........N

N

output y = wn* sn = w's

n=1


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STEERING VECTORS

s1 exp(j*o)

s2 exp(j2 (d/)sin 0)s(o ) = s3 = exp( j2(2)(d/)sin0 )

..

sn exp(j2(n-1)(d/)sin0 )

and w1

w2

W = w3

.

.

wn

To maximize the response of the beamformer to a plane wave arriving at anAoA of 0, we have the followingoptimization problem:

max y 2= max w's 2

{w} {w}

subject to w = constant<

where the constant gain constraint is imposed to ensure a final solution. Since w's is only an inner or dotproduct of two non zero norm vectors w and s , Schwarz's inequality can be applied , i.e.

w's 2 = w2 s2

with equality if (and only if ), the vectors are collinear.

This yields the result

w = k s


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This result is intuitive inasmuch as it states that the optimum beamformerapplies phase corrections to each channel to compensate for the time delays

associated with the plane wave travelling across the array. More specifically at

the nth channel forms the product

wn*sn e- j n* e j n= 1,thereby cancelling the phase term .

The beamformer thus coherently integrates the signal outputs from each

channel. Without this compensation, destructive interference would occur with acommensurate decrease in output signal strength.

An important and fundamental limitation of linear beamformers is that theywill also in general , respond to signals arriving from other angles.This can lead

to many practical problems, as strong unwanted signals from other directions

can interfere with the signal of interest.


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To visualize this effect, consider the response of the above beamformerto plane waves arriving from -90 to + 90 with w=ks, where sischosen to be a plane wave with AoA o. If we let xn denote theoutput of the nth receive channel ( andx be the corresponding vectorof received values), then the total beamformer output, steered , toangle o., is given by

N y = w'x= k xn exp(-j2(n-1)(d/)sin 0)

n=1

It has the form of discrete fourier transform(DFT) .

For this particular example, an analytical expression for for thenormalized beamformer response exists and is given by

sin[ N d/ (sin() sin (0))]

|y | = 1/Nsin[ d/ (sin() sin (0))]


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Beam Response for o =30

We set d/=0.5(half wavelength element spacing), N=16, o =30 , and xn = exp(j2(n-1)(d/)sin ) , n=1,......N

While varying from -90 to + 90, the following beamformer response results (with k=1).N=16 ,d/=0.5 o =30


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Example of Adaptive Antenna Processing

Let the signal arriving from the desired direction be called the "pilot" signal p(t)=Pejt, and let the noise,I(t)=Nejtbe incidentto the receiving array at /6 radians.

Both the pilot signal and the noise signal are assumed for this example to be at exactly the same frequency .At a point in space midway between the antenna array elements, the signal and noise are assumed to be in phase. In theexample shown , there are two identical omnis spaced apart .

w1+jw2 w3+jw4

p(t)=Pejt I(t)=Nejt

d=/2

array output

desired signal noise signal=/6. The array output due to the desired signal is

Pejt{[w1+w3]+j[w2+w4]}

. For this output to be equal to the desired output p(t)=Pejt.It is

necessary that

w1+w3=1 and w2+w4=0

. The incident interfering noise signal exhibits a

phase lead with respect to the array midpoint when impinging on

the element with complex weight w3+jw4 of pi/4 and phase lag

when striking the other element of value pi/4. Consequently thearray output due to the incident noise is given by

Nej( t- pi/4 ) [w1+jw2]+ Nej( t+ pi/4 ) [w1+jw2]

. For array noise response to be equal to zero, it is necessary

that w1+w2+w3-w4=0 and -w1+w2+w3+w4=0

solving simultaneously. The solution is,

w1=1/2,w2=1/2,w3=1/2,w4=-1/2


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Simulation of null steering beam former with four elements antenna array.

d

Plane wave

direction of

arrival

Figure : Four elements equally spaced linear array

Antenna

element

1 23 4

Desired signal at =/4, Iinterferers at /2, /6 and 0

Assume d =/2

Then steering vector is

Si=[exp(-j1.5kdsin (i)) , exp(-j0.5kdsin(i)) , exp(j0.5kdsin(i)) , exp(j1.5kdsin(i)) ]

Form matrix A

A= [S0,S1,S2,S3]

e1= [1, 0, 0, 0]

Weight vector W is given by

WH = inv (A)


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RESULT

L imi tations of simple null steeri ng beam-former

1) We get unity response in the look direction and nulls in the desired directions, but there may be

greater than unity response in the directions other than look and null directions, which is

undesirable.

In the transmission mode, such type of simple null steering beam-former wastes energy in the

directions other than look direction.

2) It requires knowledge of the directions of interfering sources.

3) The weights estimated by this scheme do not maximize the output SNR.

4) If small noise component presents in the direction where array response is greater than unity then

this noise component gets a gain over the look direction and will reduce the output SNR of adaptive

array.


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OPTIMUM ARRAY PROCESSING FOR NARROW BAND APPLICATIONS

1. Minimum Mean square error criterion (MMSE).

2. Signal to noise ratio (SNR) criterion.

3. Minimum noise variance (MV)criterion.


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MMSE CRITERIA contd


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The Signal-to-Noise Ratio (SNR) Performance Measure

y = w's+ w'n = ys + yn

|ys |2

SNR =

E{|yn |2}

E{|yn |2} = |y|2fyn(y) dy

-

E{|yn |2} = E{|w'n|2} = w'E{|nn'|}w = w'Rw

where R = E{|nn'|} , is the covariance noise matrix(size NxN) associated with n ( assumed to be zero mean)

Case 1: Additive White Noise

|w's|2 |w's|

2

max = = max

{w} w' Rw {w} 2w2

subject to w'w= 1. For this case R = 2I, 2= kTeffB

which is mathematically identical to

w = ks


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Case 2: Additive Coloured Noise

|w's|2

max =

{w} w' Rw

subject to w'w= 1.

Noting that R1/2R-1/2= I, and applying Schwarz's inequality we have the following inequality

|w's|2 |w'R1/2R-1/2s|

2

=

w' Rw w' Rw

( w' Rw ) (s' R-1s)= (s' R-1s)

or simply

|w'R1/2R-1/2s|2

(s' R-1s)

w' Rw

Equality is achieved when R1/2w = kR-1/2s or w = k R-1s

and SINRopt= s' R-1s

R is of the form :- R= RI+ 2I: where RI is a generally positive semidefinite covariance matrix associated with a coloured noise source (i.e

clutter and orjamming) that is uncorrelated with the white noise

If there are Nj uncorrelated jammers R has the form

Nj

R= j2 sjsj' + 2Ii=1


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Minimum Noise Variance (MVDR)

where

is the observed data covariance matrix


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LIMITATIONS OF CONVENTIONAL ADAPTIVE SYSTEM

Requires large number of voltage snapshot

Computational complexity is large

Statistical Conventional methodology is required.

One needs to form the covariance matrix of the data and invert it.

Direct Data Domain least Squares Technique


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Direct Data Domain least Squares Technique

Advantages

Statistical methodology is not required

Based on single snapshot of data

Can be implemented in real time using signal processing system

Faster

Requirement

Direction of arrival of the desired signal( SOI) should be known

Disadvantage

Reduced degree of freedom


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The phasor voltage Xninduced at the nth antenna element at a particular instance of

time will then be given byp

xn= s ej2nsin

s/ + Ipe

j2nsinp/ +n

p=1

where,

S = complex amplitude of the SOI (to be determined)

s = direction of arrival of the SOI (assumed to be known)= spacing between each of the antenna elements

= wavelength of transmission (here it is assumed that we are dealing withnarrowband signals)

P = total number of interferers

p= complex amplitude of pth undesired interferer

V1 V2 V3 V4 V5 V6 V7

I1

I3S

x

z

I2

Vn

y

Figure -

Uniform

Linear array.

Di t D t D i Ad ti lli P bl


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Direct Data Domain Adaptive nulling ProblemFormulation

1 0 1 2 3 4 5 6 7 8 1[ .............. ]

n N NX X X X X X X X X X X X

0 1

1 2 1

1 ( 1) ( 1)

L

L

L L N L L

X X X

X X X

X

X X X

0 1

1 2 1

1 ( 1) ( 1)

L

L

L L N L L

s s s

s s sS

s s s

2 sin sn

j n

S e

Where

Suppose voltage snapshot is available at time is1t t

Form matrix [X] from obtained snapshot as


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Form the Matrix [U] such that

1, 1 1 11, 1

0L L L LL L

U W X S W

U X S

We obtain the following generalized eigenvalue problem:


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Forward Method

0 01,1U X S

1 11,2U X S

2exp[ sin ]s

j dZ

11,1 (1,2) Contains no component of SOIU Z U

1Same is true for 1, 2 (1,3)U Z U

1In General , ( , 1) for 1,2,..... 1 & 1,2,.....U i j Z U i j i L j L

Therefore one can form a reduced rank matrix

( 1)L L

T

generated from U such that


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1 1 1

0 1 1 2 1

1 1 1

1 2 2 3 1 2

1 1 1

1 1 1 ( ) ( 1)

L L

L L

N L N L N L N L N N L L

X Z X X Z X X Z X

X Z X X Z X X Z XT

X Z X X Z X X Z X

0

1

L

W

W = 0

W

To restore the desired signal, we fix the gain of sub-arrayformed by L+1elements in the direction of SOI.


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1 1 1

0 1 1 2 1

1 1 1

1 2 2 3 1 2

1 1 1

1 1 1 ( 1) ( 1)

1

L

L L

L L

N L N L N L N L N N L L

Z Z

X Z X X Z X X Z X

X Z X X Z X X Z X

X Z X X Z X X Z X

0

1

L ( 1) 1(L+1) 1

W C

W 0 =

0WL

or equivalently

[A][W] = [Y]

0

1 L

i i

i

W X

C

And an estimate of desired signal is given by,


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Output of these 7 elements

is used to estimate SOI from

calculated weights

Output of these 13 elements is used to estimate

set of 7 weights

Antenna

element

0 1 2 6

11 12


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* 1 * * 1 * * 1 *1 1 2 1

* 1 * * 1 * * 1 *

1 1 1 0 ( 1) ( 1)

1 L

N N N N L L

L L L L L L

Z Z

X Z X X Z X X Z X

X Z X X Z X X Z X

'0

1

( 1) 1 ( 1) 1

0

0L L L

W C

W

W

Backward Method

The estimate of output signal is given by

*L+v L

*

i L-i+v'i=0

Z= W Xv

C

for v = 0, 1,..L


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Forward-backward method

1 1 1

0 1 1 2 1

1 1 1

1 2 2 3 1 2

1 1 1

1 1 1

* 1 * * 1 * * 1 *

1 1 2 1

* 1 * * 1 * * 1 *

1 1 1 0

1 Q

V V

V V

N Q N Q N Q N Q N N

N N N N N Q N Q

Q Q Q Q

Z Z

X Z X X Z X X Z XX Z X X Z X X Z X

X Z X X Z X X Z X

X Z X X Z X X Z X

X Z X X Z X X Z X

( 1) ( 1)Q Q

0

1

( 1) 1( 1) 1

0

0V QQ

W C

W

W

Degree of Freedom of Forward-backward

method is greater than forward andbackward method. e.g. for 13 elements

linear array.

DOF forward method = 6

DOF backward method = 6

DOF forward-backward method = 9

30Forward method(SOI=30deg and Interferers at 55,75,-25 and -50 deg)

30Backward method(SOI=30 deg and Interferers at 55,75,-25 and -50 deg)


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(a) Forward Method (b) Backward Method

c Forward-backward Method

Simulation results of DirectData Domain algorithms.

No. of elements in the array=13

Spacing between the elements=

SOI = 30 degree

Interferers= 55, 75,-25 and -50 deg.

4 interferers are 20 dB stronger thanSOI

/ 2

-100 -80 -60 -40 -20 0 20 40 60 80 100-70

-60

-50

-40

-30

-20

-10

0

10

20

30

ANGLE OF ARRIVAL IN DEGREES

BEAMP

ATTERN

(dB

)

-100 -80 -60 -40 -20 0 20 40 60 80 100-70

-60

-50

-40

-30

-20

-10

0

10

20

30


BEAMP

ATTERN

(dB

)

-100 -80 -60 -40 -20 0 20 40 60 80 100-80

-60

-40

-20

0

20

40


BEAMP

ATTERN

(dB

)

Forward-backward method(SOI=30 deg and Interferers at 55,75,-25 and -50 deg)


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Prevention of Signal Cancellation in an Adaptive Nulling algorithm

Problem arises when the actual direction of arrival of the signal is slightly off the assumed direction ofarrival

Multiple look-direction constraints can be used to prevent signal cancellation, when the direction of arrivalis not known exactly.

Multiple points are chosen on the nonadapted array pattern and a row is implemented in the matrix equations

The size of the matrix [U], for example, is established by the following parameters:

Q = number of look-direction constraints

L+1 = number of weights to be calculated

L-Q+1 = number of interferers that can be nulled

Relationship between the number of weights, number of constraints, and number of elements:

N = 2LQ


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Simulation of Forward Method With Multiple Look Direction to Prevent Signal Cancellation.

We consider an array of N+1=21 antenna elements and we employ the forward method. We apply three look directionconstraints and check the performance across the main beam. This leads to the following relationship:

N+1=21,Q=3, and so L+1=12 and 9 jammers can be cancelled.

A MATLAB code was written for SOI coming from 30 and with 3 look direction constraint placed at 26,30and34. Alsoconsidered were 9 jamming signal coming from -10,-30,-50,-70,0,10,50,60and80. In addition, we have thermal noise ateach antenna element,

If the signal changes course from 30to 26or 34it can still be detected as the algorithm prevents signal cancellation.

-100 -80 -60 -40 -20 0 20 40 60 80 100-50

-40

-30

-20

-10

0

10

20

30

40

50


BEAMP

ATTERN

(dB

)

Forward method with 3 look directions at 26,30 and 34 deg

EFFECT OF MUTUAL COUPLING ON ADAPTIVE ARRAYS AND ITS


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COMPENSATION

Most adaptive arrays assume that the elements of the receiving are

independent isotropic point sensors that sample but do not reradiate, the

incident fields.

We further assume that the array is isolated from its surrounding.

However, in the actual scenario, each element has some physical size. Not

only do the elements spatially sample but they also reradiate the incidentfields.

The reradiated fields interact with other elements causing the elements tobe mutually coupled.

Mutual coupling not only destroys the linear wavefront assumption for thesignal of interest but also for the interferers and clutter impinging on thearray.

Thus the voltages induced at the antenna elements are required to be

corrected to compensate for the mutual coupling.

Accounting For Mutual Coupling Among An Array Of Dipoles


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g p g g y p

The assumption of a plane wave front impinging on an array implies that the

fundamental starting equation in an adaptive algorithm relating the received voltages

due to the signals incident on the individual antenna elements can be written as a sumof complex exponentials.

The presence of mutual coupling among the elements of an antenna array destroys this

plane wave assumption.

Therefore, unless the effects of mutual coupling are accounted for, adaptive algorithms

will be incapable of extracting the solution for the problem.

Mutual coupling among the elements of the antenna elements affects adversely nullingcapability of the adaptive algorithm.

Deep nulls are not formed in the direction of interferers and some times nulls are not

formed in the direction of interferers.

To illustrate the elimination of mutual coupling among the antenna elements, we


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p g g ,

consider an array of parallel thin-wire dipole antennas.

The dipoles are assumed to be z-directed of length Land radius a,and are placed along

the x-axis, separated by a distance.

The port of each antenna element is located at the center and is loaded with an

impedance of ZL ohms.

The array lies in the x-z plane as shown in the figure .

z = L

z = 0

x

x

z

LZ

LZ

LZ

Array Elements


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Compensation using open-circuit voltages


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g g

The idea of using open-circuit voltages to compensate for the effect of mutual coupling

was first proposed by Gupta and Ksinski .

They argue that due to the lack of a terminal current the reradiated fields are reducedand the open-circuit voltages are free of mutual coupling.

The principle idea is to derive the open-circuit voltages from the measured voltages for

further signal processing.

However the theory is valid only for further half wavelength dipoles with half wavelengthspacing.

For the more general case one can use the MM analysis in the analysis in conjunction

with the Thevenin and Norton equivalent circuits to obtain the open-circuit voltages.

Using the notation developed above we define a new matrix [Y] whose entries are thoserows and columns of the MM admittance matrix [Y] that correspond to the feed ports that

is,'

, 1 ,pq i lY Y p q N

1 1 / 2i p p p

1 1 / 2l q p p

Note that this matrix [Y] is defined from the admittance matrix that does not include the


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load impedances. The open-circuit voltages are then related to the short-circuit currents

as

The measured voltages are also related to the short-circuit currents through

Eliminating the short-circuit currents from the above equatins yields the open circuitvoltages as

The open-circuit voltages we refer to are obtained from the measured voltages.

1

'[ ]oc scV Y I

1

1'

Lmeas scV Y Z I

1 1' '

Loc measV Y Y Z V

Simulation Results


41/50

To show the effect of mutual coupling and its compensation, we first consider an ideal

case.

Hence, the antenna elements are considered to be point sources.

The array considered is Uniform linear array consisting of 13 elements with inter-

element spacing x = 0 .5.

We assume that no mutual coupling exists between the elements.

Figure 1:-Model of an ideal array consisting of isotropic point radiators.

x

V1 V2 V3 V4 V5 V6 V7

I1

I3S

x

z

I2

V13

y

Consider a scenario where three interfering signals are impinging on an array at the


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angles -45,20 and 55 degrees respectively.

The direction of arrival of desired signal is 0 degree.

The magnitude of SOI is 1 V/m and magnitude of interferes is 2V/m.

The model of an ideal antenna array is shown in figure (1).

The voltages measured at the ports of the antenna elements are then passed to the

signal recovery subroutine to find the weights using the direct data domain least squares

method (D3LS).

The simulation result for this ideal case is shown in Figure 2 .

The figure 2 demonstrates that in the absence of mutual coupling, the receiving

algorithm is highly accurate and can null a strong interferer.

SOI at 0 deg and interferers at -45,20,55 degree


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-100 -80 -60 -40 -20 0 20 40 60 80 100-80

-70

-60

-50

-40

-30

-20

-10

0


BEAM

PATTERN

(dB

)

g , , g

Figure 2:- Adapted beam pattern in the absence of mutual coupling (Ideal

antenna array)

In the second scenario the mutual coupling between the antenna elements is taken into


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account, as they are not isotropic elements.

Each element is of size /2 and with inter-element spacing x=0 .5.

Piecewise sinusoidal basis function is considered and a total of 7 basis function per

element is considered.

The details of the array chosen for the second scenario are presented in Table1.TABLE 1

Parameters defining the elements of the array.

Number of elements in array 13

Length of z-directed wires

Radius of wires

Spacing between wires

Loading at the center

/ 2

/ 200

/ 2

50

The receiving algorithm tries to maintain the gain of the array in the direction of SOI


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(along the direction =0) while automatically placing nulls along the directions of the

interferences.

All signals and interferers arrive from the azimuth =0.

The signal of interest and the intensities of the interferer along with their directions of

arrival are given in Table 2.

TABLE 2Complex amplitudes of the SOI and interfering signals and their directions of arrival.

Magnitude Phase(degree) DOA (degree)

SOI 1.0 V/m 0 0

Interferer 1 2.0 V/m 0 -45

Interferer 2 2.0 V/m 0 20

Interferer 3 2.0 V/m 0 55

The antenna array is analyzed using the method of moments (MM).


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The intensities of the signal and the interferences along with their directions of arrival are

used to calculate the MM voltage vector.

A MATLAB program was written to evaluate Equations. The [Z] matrix obtained is of

size (91 x 91).

Figure3 :- Adapted beam pattern in the presence of mutual coupling (No

attempt is made to compensate for mutual coupling)

-100 -80 -60 -40 -20 0 20 40 60 80 100-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0SOI=0 deg and Interferers at -45,20,55


BEAMP

ATTERN

(dB)

Figure 3 shows the beam pattern when the mutual coupling between the antenna

l t i t k i t t


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elements is taken into account.

The nulls in the antenna array pattern are shallow.

which is angle of arrival of second interferer.

The shallow nulls result in an inadequate nulling of the interference environment.

Hence the signal cannot be recovered properly.

Next, we use the open circuited voltages.

The open-circuit voltages are then passed to the D3LS algorithm, and an attempt is

made to recover the signal.

SOI=0 deg and Interferers at -45,20,55


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-100 -80 -60 -40 -20 0 20 40 60 80 100-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0


BEAM

PATTERN

(dB

)

Figure 4:- Adapted beam pattern using open-circuit voltage

compensation technique for mutual coupling

Figure 4 shows the simulation result using the open-circuit voltages.


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The figure shows that after compensation for mutual coupling the null at 20 degree is

formed more deep as compared to uncompensated case.

Also the null at 55 degree is deeper, as compared to the uncompensated case.

This example has shown that open-circuit voltages do provide some compensation for

mutual coupling.

The use of open-circuit voltages provides significantly better signal recovery than using

the measured voltages directly.

Our objective has been to compensate for the effects of mutual coupling by relating the

open-circuit voltages (voltages at the ports of the array if all were open circuited) with the

voltages measured at the ports.

The stated assumption is that the open-circuit voltages are free of mutual coupling. This

assumption is valid only in a limited sense.

The open-circuit voltages are the voltages in the presence of the other open-circuited

elements. This implies that the effects of mutual coupling have been reduced but not

eliminated.

REFERENCES


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1. Smart antennasT.K.Sarkar, M.C. Wicks, M.S.Palma, R.J. Bonneau, published by

John wiley and sons,Year 2003

2. Effect of mutual coupling on performance of adaptive arrays- By Inder J. Gupta and

A. Kseinski, IEEE Transactions On Antennas and propagation, vol. AP-31,No.5,

September 1983,pp 785-791.

3. Compensation for the effects of mutual coupling on Direct data Domain Adaptive

Algorithms-Raviraj S. Adve and T.K.Sarkar, IEEE Transactions On Antennas And

Propagation, Vol. 48, No. 1, January 2000, pp 86-94.

Adaptive Array Antenna-2008

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