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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 3, MARCH 2007 859

Constrained Least-Squares Optimization inConformal Array Antenna Synthesis

Leo I. Vaskelainen

AbstractThe least mean square error of the constrained least-squares optimization method is studied. The phases of non-zerogoal values and the phases of non-zero constraint values can beselected by a fast iteration for optimum fit to the goal function am-plitude. This kind of modified least-squares optimization methodopens up a fast and robust synthesis method for large conformalantenna arrays. Constraints can be written also for the element ex-citation amplitudes and a phase synthesis with predefined ampli-tudes can be done. Iterative weight-correction technique is used foraccurate sidelobe control. Synthesis examples of two-dimensionalelliptical and large three-dimensional conformal antenna arraysare presented.

Index TermsAntenna arrays, conformal antennas, leastsquares methods, optimization.

I. INTRODUCTION

THE synthesis problem of an arbitrary array antenna can

be seen as a general optimization problem, where optimal

complex element excitation values are sought for. This opti-

mization can include maximizing the directivity, minimizing the

sidelobes to a certain level, design of some shaped or contoured

beam using some excitation amplitude distribution, or even op-

timization of element places.

To complete this kind of optimization global optimization

methods are usually used, such as genetic algorithm [1][3],

simulated annealing [4] or particle swarm optimization [2], [5].

A hybrid approach is also used [6]. The global optimization

methods are very flexible to use, but they need a large amount

calculation of directivity patterns.

The least-squares optimization in its basic form gives directly

a solution for the unknown excitation values. Unfortunately this

solution is for some (guessed) phase values of the goal function,

which doesnt necessarily give optimum fit for the synthesized

pattern amplitude. For finding the best amplitude fit some itera-

tive method must be used [7].

The least-squares optimization is also not directly suitable

for phase synthesis, which is a nonlinear optimizing problem.

An approximate equivalent linear problem can be written by al-

lowing the excitation changes in complex plane only to orthog-

onal directions compared to the previous excitation values [8].

This leads also to an iterative process.

In this paper, the least mean square (LMS) error of the

constrained least-squares optimization problem is studied. It is

found, that the LMS error for phases of the goal function and

Manuscript received May 31, 2006; revised September 29, 2006.Theauthor is with theTechnicalResearchCentre of Finland (VTT), FI-02044

VTT, Finland (e-mail: [email protected]).

Digital Object Identifier 10.1109/TAP.2007.891860

possible constraint values has a nonlinear, but a very simple

form. In addition, in this LMS error only the nonzero values

must be considered and the number of nonzero goal values is

usually much smaller than the total number of directions used

in the synthesis. Therefore the optimization of these phase

values is a very fast process. A simple iterative optimization

method for these phases is presented in Section II.

The fast optimization technique for the phases of constraint

values allows this method to be used also for phase synthesis.

Basically the least-squares optimization controls only the

general level of sidelobes, not the level of individual peaks.The optimization error in different directions can be controlled

by selection of weights used in the LMS error. Iterative control

of individual error values can be achieved by correction of the

weight values according to the previous synthesis error values.

II. LEAST-SQUARES OPTIMIZATION METHOD WITHPHASE-OPTIMIZATION

A. Constrained Least-Squares Optimization Problem

The least-squares optimization problem is presented in ma-

trix-form

(1)

which includes linear equations for functionswith variables

(2)

If the number of equations is greater than the number of

unknown coefficients , -values can be solved only in that

sense, that the sum of squared differences of the left and right

sides of (2) is minimized.

In the constrained least-squared optimization problem the co-

efficients are solved in this least-squares sense, but they must

also satisfy exactly extra linear equations

(3)

The solution of the constrained least-squares optimization

problem is known. Here, the guidelines of[9] are followed. The

solution of the constrained least-squares problem is also pre-

sented in [10] and [11].

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860 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 3, MARCH 2007

The LMS error is

(4)

where includes the weights for

every squared difference value in the sum. is a vector of the

so far unknown Lagrange multipliers [one for every equation in

(3)].

The derivative vector of (4)

(5)

can be seen to be

(6)

When this is set to be the unknown values of are

(7)

By using (3) can be solved to be

(8)

If we write

(9)

(10)

and

(11)

we finally can write the solution for coefficients

(12)

In array antenna synthesis problems the phase of the goal

function (phases of ) usually is not relevant. The least-squares

problem is modified in such a way, that we have to ask, in which

way we must choose the phases of to minimize (4).

The constraint-(3) can be set for the array antenna gain values

or, as we can see later on, for the element excitation values. In

both cases we must find also the optimum values for phases of

. The constrained gain values (absolute values of ) and the

unconstrained gain values (absolute values of ) can also be

in conflict, if the values of are not carefully selected. Usually

we only want to set constraints for the shape of the gain function

or for the excitation distribution. So our goal is to choose the

relations of the absolute values of , but we can change these

values by choosing an optimum general level for -values.

B. Selection of Phase Values of and and Optimizing

Absolute Values of

If we do not use any other criteria for the selection of - and-values, we must study the LMS error (4). When (3) is used and

the solution (12) is inserted in (4), after a rather straightforward

calculation we can get

(13)

where

(14)

(15)

and

(16)

Here, one can notice that in (13) only the nonzero values of

and are relevant. So instead of , and we can use

matrices, where only the rows corresponding the nonzero values

of and are selected.

The first term of (13) doesnt depend of the phases of .

The second term can be minimized by selection of phases ofthe -values and the last term by selection of phases of the

-values. If there are both nonzero goal values and nonzero

constraint values , the sum of the thirdand fourth terms must be

considered. It affects both to the phase selections and it depends

on the relationship between the amplitudes of and .

The optimization process can now be divided into five stages.

1) If there are both nonzero -values and nonzero -values,

replace -values by -values and select the complex co-

efficient so that expression

get its minimum value.

2) If there are nonzero -values, select the phases of so

that is minimized.

3) If there are nonzero -values, select the phases of so that

is minimized.

4) Repeat steps 1)3) until the required accuracy is obtained.

5) Solve from (12).

In Section II-C simple procedures are presented for steps

1)3). Any other optimization method capable for solving a

nonlinear optimization problem (genetic algorithm, particle

swarm optimization etc.) can be used. The important result

of this section is, that the optimization of phases of and

is separated from the solution of . These phases can be

optimized even without knowing the values of .

C. Phase Optimization of Goal and Constraint ValuesIf in expression

(17)

is replaced by , it gets the form

(18)

is the phase angle of and is the phase angle of

term . The optimum value for is

(19)

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VASKELAINEN: CONSTRAINED LEAST-SQUARES OPTIMIZATION IN ANTENNA SYNTHESIS 861

For step 2) in section we can optimize the phase of one

selected term of matrix . Equation (17) can be written as in

(20)

When this is simplified, it gets the form

(21)

where

(22)

and is the phase angle of . is a constant which does not

depend on the phase of . contains all other but the :th

element of , and contains all but the :th element of the

column of

(23)

From (21) the optimum choice for the phase of can be seen

to be

(24)

Equally for one selected term of can be in the form

(25)

which also can be simplified to a form

(26)

where now is

(27)

and is the phase angle of . contains all other but the

th element of and contains all but the :th element of

column of .

Optimum choice for the phase of is

(28)

These results (19), (24), and (28) can now be repeated

for every element of and and used in steps 1) to 3) of

Section II-B. This iteration is very fast, because the matrix sizes

are small. The LMS error is always, when (19), (24), or (28) is

used, smaller or equal than the previous value, and that assures

a fast convergence to the nearest minimum. This iteration is so

fast, that it can be repeated several times with random initial

phase values for assuring that the final result is not stuck in

some local minimum.

III. ARRAY ANTENNA SYNTHESIS PROBLEM

A. Synthesis With Gain-constraints

The electric field in far-field of an arbitrary array antenna

having elements can be calculated in directions ,

, by a matrix equation

(29)

where contains the far-field values of one polarization com-

ponent in directions, contains excitation values of

elements and is a matrix, and its elements can be

calculated from (2) using .

is the field-gain of element in direction ,

is the position vector of :th element and is the direction

vector to direction

When is replaced by (29) represents the unconstrained

least-squares optimization problem (1). The phases of values

can be optimized as presented in Section II-C and the excitation

values can be solved from (12). This kind of synthesis can be

done for one polarization component. Both polarization com-

ponents can be included in the synthesis process by including

them in matrices and in (29).

Gain constraints in some directions can be written by using

(29) and (3). Null-constraints can be used in the sidelobe area

and in some cases the main-lobe shape can be de fined by using

gain constraints.

B. Phase Synthesis

The most interesting way to use the constraint-equations is to

write constraint-equations for the phase synthesis problem. This

is done by writing the obvious matrix equation

(30)

There is the unity-matrix and includes the desired ab-

solute values for the element excitations. When we insert

and in (3), we can optimize the phases of as pre-

sented in Section II-C and the amplitudes of are forced to bethe amplitudes of .

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IV. CONTROL OF SYNTHESIS ERRORS BY WEIGHT

CORRECTION TECHNIQUE

In the least-squares optimization the synthesis errors can only

be controlled by a good choice of the goal values and by the

selection of the weights in (4). The sum of all squared errors

multiplied by these weights is minimized as presented in (4).

The resulting error distribution in the main-lobe or the sidelobedistribution may not be optimal for the final antenna design. A

better control of synthesis errors can be obtained by repeating

the synthesis described in Section II and correcting the values

of using the previous synthesis error values. Several methods

for this -correction can be used.

In Section V the -correction is done by the following

procedure.

1) The beam is synthesized as described in Sections II-B and

C.

2) In each direction in the main-lobe a reference error value

is defined (values corresponding deviation are

used).3) In each direction in the sidelobe directions another refer-

ence error value is defined (values corresponding

sidelobe level are used).

4) synthesis error divided by the reference error is

used to correct each value of . The previous value is mul-

tiplied by , where is changed from 1/4 to 1/12

during iterations.

5) Steps 1) to 4) are repeated 25 to 50 times and the best result

(the maximum main-lobe error and the maximum sidelobe

directivity can be used as the selection criteria) is selected.

This -correction is basically same kind as presented in [4].

The weights are increased (carefully) in directions where the

error is greater than the reference error and decreased where itis less than the reference error. The quality of the synthesized

beam is not getting better linearly during this process.

Particularly in this phase synthesis jumps may happen to both

directions when new weights force thesolution out of some local

minima, and therefore the best solution must be maintained. The

reference error values are good to be near or slightly below the

values finally attained in the synthesis, but more important is the

relationship of these values in different directions. The relative

errors of the beam shape tend to balance during this iteration.

V. SYNTHESIS EXAMPLES

A. 2-D Elliptic Array

In the elliptic 2-D sector array there are 36 elements on an el-

lipse with a minor axis and a major axis (eccen-

tricity 0.3). The distance between elements is measured

along the circumference of the ellipse. The array is symmetric

with respect of the minor axis. The elements are pointing in a

60 sector around positive -axis and the minor axis of the el-

lipse is on -axis.

The element directivity function of the element model is pre-

sented in Fig. 1. It is an approximate model of an

aperture radiator in -mode and the details of this model

are not presented here. This same element model is used both in2-D and in 3-D arrays.

Fig. 1. Directivity function of the element model. -component, -componentis zero.

Fig. 2. Top of the 2-D shaped beam, (B) phases of goal function optimized,

(D) phases of goal function optimized and -correction is used. Straight line isthe goal-function.

The synthesis is done by using directions

and . The goal-function

is zero when or and the top of the goal

function is sloped while .

In Figs. 2 and 3, the effect of the phase optimization and the

use of some constraints are studied. In these figures the shape

of the main-beam is presented. In Figs. 4 and 5, the slope of the

main-beam and the nearest sidelobes are studied, when different

optimization methods are used.

By using gain constraints in directions andand null-constraints in directions and the

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Fig. 3. Top of the 2-D shaped beam, (C) constrained slope-width and phasesof goal function and constraint values optimized, (E) the same synthesis with

-correction. Straight line is the goal-function.

Fig. 4. Slope of the 2-D shaped beam, (A) no phase optimization, (B)phases ofgoal function optimized, (C) constrained slope-width, phases of goal functionand constraint values optimized.

steepness on the beam slopes can be increased, but of course

with increased sidelobe level and increased ripple in the main-

beam.

The phase optimization of the goal values (curve B in Fig. 2)

gives a steeper slope and smaller sidelobes compared to the

normal least-squares solution with zero-phase goal function

(curve A in Fig. 4).

The -correction technique gives a better fit near the edges of

the main-beam (curves E and D compared to curves A and B)

or, if the gain-constraints are used, a lower maximum sidelobe

value (curve E compared to curve C).

As it was described in Section III-B, the phase optimization

of the gain constraint values can be used for phase synthesis. InFig. 6 an amplitude distribution with exact values

Fig. 5. Slope of the 2-D shaped beam, (D) phases of goal function optimized,(E) constrained slope-width and phases of goal function and constraint valuesoptimized. -correction is used in both cases.

Fig. 6. Phase synthesized 2-D beam, (A) phase synthesis with -correction

and (B) without -correction.

is selected and a phase synthesis with and without -correction

is calculated.

The -correction increases slightly the sidelobe level but

gives a more accurate result especially near the edges of the

beam.

B. Conformal 3-D Array

The conformal 3-D array geometry is generated from ellip-

tical subarrays. In the lowest subarray 16 elements are set on a

horizontal ellipse with minor axis and major axis

(eccentricity 0.3). The other subarrays have the same shape,

but they are placed on an elliptic cone, where the center of the

bottom ellipse is in point ( , 0, 0) and the tip of the cone

is in point ( , 0, ). In -direction the distance be-

tween these subarrays is . 14 subarrays are used. The

distance between elements varies from to mea-sured along circumferences of the ellipses. The center part of

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Fig. 7. Arrangement of directions and goal function values. 1 442 directionsare shown in figure, in actual synthesis 65 612 directions are used. Thick lineshows the direction (90 , 0 ).

Fig. 8. 3-D array synthesis result with no phase optimization.

this array is tilted 10 upwards and the elements are pointed in

a 60 sector. The general shape ofthe array can beseen in Fig. 8.

When a 3-D array is synthesized, the directions for synthesis

are generated by setting the vertex-points of an icosahedron on

a sphere. Each triangle on the sphere is then divided to (almost)

equal sized smaller triangles. In such a way it is possible to

generate nearly equally spaced directions in entire solid angle.

Main-lobe directions are defined to be directions, which are in-

side a super-ellipse

(31)

Fig. 9. 3-D array synthesis result with goal function phase optimization.

and define the size and the shape of the super-

ellipse.

Equally the sidelobe directions are defined to be directions

outside the super-ellipse

(32)

defines the desired slope-width of the beam.

In the sidelobe directions the goal-values are zeros. In the

main-lobe directions the goal-values are values giving a slopeof 2 dB/10 in -direction. Finally these direction-vectors with

their goal-values are rotated an angle around -axis and

around -axis (positive angle defined by right-hand thumb rule).

The principle of generating directions and goal-values can be

seen in Fig. 7.

A trapezoidal goal function with parameters ,

, , , and is

used. The weight values in the sidelobe directions are 1 and in

the main-beam directions inverse values of the goals are used.

The phases of the goal-values are zeros and phase optimiza-

tion is not used. No constraints are set.

The result of this synthesis is represented in Fig. 8.In Fig. 9 the same synthesis parameters are used but now the

phases of goal values are optimized. The contoured shape of

this result is more accurate and the slope of the beam side is

steeper than in Fig. 8. The ripple on top of the main-beam is

slightly increased.

In Fig. 10 gain-constraints are used to force the-lobe-shape to

exact values in 19 points.

The main-lobe values are constrained in 5 directions: with 7

spacing along the longer diameter of the beam.

Also two set of 7 null-constraints are used outside the beam

with 6 spacing along the line halving the beam, first null being

25 from the center of the beam. The phases of the goal-values

and the constrained gain-values are optimized as describedin Section II-B.

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Fig. 10. 3-D array synthesis result with gain-constraints, goal and constraintphases are optimized.

In neighborhood of the constrained directions the sidelobe

level is forced to below (not very clearly seen in

Fig. 10). The constraints can be used for accurate shaping of

the beam and sidelobes, but the constraints increase the overall

sidelobe level of the beam.

In phase-synthesis of the conformal three-dimensional array

the same array geometry and the same goal function definition

is used as in the previous examples. If a low sidelobe solution

is desired the amplitude distribution must be suitably tapered.

Here, a Gaussian distribution is used, see (33) at the bottom ofthe page, where , , and are the

numbers of elements in the vertical and horizontal directions.

The shape of this distribution can be seen in Fig. 11.

The phases of the element excitations and the phases of non-

zero directivity goal values are then optimized by using con-

straint-equations , as presented in Section III. No

other constraints are used.

The synthesis result can be seen in Fig. 12. If this is com-

pared to the directivity function in Fig. 8, the sidelobe level

is increased few decibels and the shape of the beam is rather

inaccurate.

If the weight-correction technique is used in this phase syn-thesis (Fig. 13) the overall shape of the main-beam is much

closer to the goal function. The sidelobe maximum is

below the main-lobe maximum.

The phase synthesis technique allows the use of differently

shaped beams by using only the phase control in conformal ar-

rays. For example in Fig. 14 a low sidelobe pencil beam is syn-

thesized by using the same amplitude distribution of Fig. 11.

Fig. 11. Amplitude distribution for phase synthesis.

Fig. 12. Phase synthesis with amplitude distribution ofFig. 11 -correction isnot used.

Different directivity/sidelobe level values can be syn-

thesized using this same amplitude distribution, for ex-

ample 26.6 dB/ (Fig. 14), 26.9 dB/ ,

27.1 dB/ , 27.3 dB/ and 27.7/ .

Wide (90 ) fan-beams can also be synthesized for this array

with the same amplitude distribution.

VI. COMPUTATIONAL EFFICIENCY

In the example presented in Fig. 12 we have 65 612 direc-

tions, 224 elements and 694 nonzero goal values meaning, that

(33)

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Fig. 13. Phase synthesis with iterative weight correction (compare withFig. 12).

Fig. 14. Phase synthesized low sidelobe pencil-beam.

also that number of goal function and excitation phases are op-

timized. The total synthesis time was 3.89 min, from which the

calculation time of matrix in (29) was 2.58 min. A normal per-

sonal computer with Intel Pentium 4 3.00 GHz CPU and 1 GB

of RAM was used and the synthesis was programmed using

MATLAB version 7.1. The total synthesis time of the other 3-D

examples varied from 3.53 to 5.21 min. The iteration sequence

described in Section II-B has been repeated 1025 times using

random phases for goals and (in phase synthesis) excitations. If

the -correction technique was used the total synthesis timeswere several tens of minutes.

VII. DISCUSSION

A. Polarization and Frequency

Equations for any or both polarization components can be

used in (1) and (3) and linear or circular polarization compo-

nents can be selected as well. Usually it is not very useful to tryto minimize the cross-polarization components by element ex-

citation optimization, because the cross-polarization is mostly

defined by the elements. Equations for several frequencies can

be written in (1) and (3) as well, but the obtainable frequency

bandwidth broadening depends on the geometry of the array.

B. Practical Hints

The really large matrices during the optimization are and

. These matrices can be divided to several submatrices, which

can be saved in hard disk. So far and can be handled in

RAM-memory the phase optimization is fast.

When wide shaped beams are synthesized for large arrays

with the goal function phase optimization, it is very common,

that there appears a hole (very narrow null) in middle of the

beam. This kind of result really is a minimum of the LMS error.

The weight correction procedure changes the LMS error and

usually forces the result out of this kind of solution. This phe-

nomenon appears in cases, where a large array is used to realize

very steep slopes for a wide beam. In practical designs a smaller

array with less steep goal function slopes would be a more prac-

tical solution.

VIII. CONCLUSION

The least-squares optimization can be successfully used in

power pattern synthesis, where the phase of the goal function

is not fixed. The optimization of the directivity and constraint

phase values is fast, because in this optimization process the

matrix sizes are small compared to the matrix sizes used in the

final solution of element excitation values.

This optimization of the constraint value phases allows

the phase synthesis with predefined excitation amplitude

distribution.

Basically the least-squares optimization gives a poor control

of side lobes. In problems, where wide contoured beams aredesigned using large amount of elements, local, or even global

minimum of the LMS error can give an unsatisfying result. The

iterative correction of weights described in Section IV is a valu-

able tool for solving both these problems.

Any linear dependence, as the output voltages of a feed net-

work as function of element currents mutual couplings included,

is easy to write as part of matrix and to take into account

during the synthesis.

REFERENCES

[1] J. M. Johnson and Y. Rahmat-Samii, Genetic algorithms in engi-neering electromagnetics, IEEE Antennas Propag. Mag., vol. 39, no.4, pp. 721, Aug. 1997.

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[2] D. W. Boeringer and D. H. Werner, Particle swarm optimizationversus genetic algorithms for phased array synthesis, IEEE Trans.

Antennas Propag., vol. 52, pp. 771779, Mar. 2004.[3] F. J. Ares Pena, J. A. Rodrguez-Gonzalez, E. Villanueva-Lopez, and

S. R. Rengarajan, Genetic algorithms in the design and optimizationof antenna array patterns, IEEE Trans. Antennas Propag., vol. 47, pp.506510, Mar. 1999.

[4] J. A. Rodrguez, L. Landesa, J. L. Rodrguez Obelleiro, F. Obelleiro,

F. Ares, and A. Garca-Pino, Pattern synthesis of array antennas witharbitrary elements by simulated annealing and adaptive array theory,Microw. Opt. Technol. Lett., vol. 20, no. 1, pp. 4850, Jan. 1999.

[5] D. Gies and Y. Rahmat-Samii, Particle swarm optimization for re-configurable phase-differentiated array design, Microw. Opt. Technol.

Lett., vol. 38, no. 3, pp. 168175, Aug. 2003.[6] T. Isernia, F.J. AresPena, O. M.Bucci,M. DUrso, J.F. Gmez,and J.

A. Rodrguez, A hybrid approach for the optimal synthesis of pencilbeams through array antennas, IEEE Trans. Antennas Propag., vol.52, pp. 29122918, Nov. 2004.

[7] L. Vaskelainen, Iterative least-squares synthesis methods for con-formal array antennas with optimized polarization and frequencyproperties, IEEE Trans. Antennas Propag., vol. 45, pp. 11791185,Jul. 1997.

[8] , Phase synthesis of conformal array antennas, IEEE Trans. An-tennas Propag., vol. 48, pp. 987991, Jun. 2000.

[9] S. Haykin, Adaptive Filter Theory. Englewood Cliffs, NJ: Prentice-

Hall, 1996.

[10] P. Y. Zhou and M. A. Ingram, Pattern synthesis for arbitrary arraysusing an adaptive array method, IEEE Trans. Antennas Propag., vol.47, pp. 862869, May 1999.

[11] LiJinhua, R. Jin, and Y.Sheng, A fast synthesisalgorithm of adaptivebeams for smart antennas, Microw. Opt. Technol. Lett., vol. 36, no. 6,pp. 503507, Mar. 2003.

Leo I. Vaskelainen was born in Pieksmki, Finland,in 1944. He received the M.S. degree in technology

from Helsinki University of Technology, Espoo, Fin-land, in 1971.

From 1970 to 1975, he was an Assistant withthe Radio Laboratory of Helsinki University ofTechnology. From 1975 to 1994, he was withthe Telecommunications Laboratory, TechnicalResearch Centre of Finland (VTT), as a ResearchEngineer, Senior Research Engineer, and as a Sec-tion Leader of the Radio Section, where he was

responsible for research in the field of nuclear electromagnetic pulse (NEMP),electromagnetic compatibility of electrical equipment, antenna, microwaveand radar systems. Currently, he is a Senior Research Engineer at VTT. Hisresearch interests include mobile communications antennas, adaptive antennas,array antenna synthesis methods and radar systems. He has published several

journal and conference papers.

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