Projection Matrix Method for Shaped Beam Synthesis in Phased Arrays and Reflectors

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 55, NO. 3, MARCH 2007 675

Projection Matrix Method for Shaped Beam Synthesisin Phased Arrays and Reflectors

Arun K. Bhattacharyya, Fellow, IEEE

Abstract—We present the projection matrix method for shapedbeam synthesis in a general array of different size elements. Themethod relies on orthogonal projection of the desired far-fieldintensity vector onto the space spanned by the far-field intensityvectors of the array elements. It is found that for a uniformconvergence of the solution, the far-field sample space must beextended beyond the coverage region; otherwise the projectionmatrix becomes ill conditioned. A general guideline for thefar-field sample space is provided. The method, with necessaryamendments, is then employed successfully for a reflector surfacesynthesis. The method is found to be several times faster thanthe gradient search method commonly used for beam synthesis.Numerical results for array and shaped reflector syntheses areshown and the advantages discussed.

Index Terms—Array, beam synthesis, projection matrix, shapedreflector.

I. INTRODUCTION

SHAPED beam array synthesis invites considerable atten-tions to researchers because arrays offer in-orbit recon-

figurability, which is an attractive feature for communicationand broadcasting satellites in a rapidly changing market place.Several synthesis algorithms have been proposed in the openliterature [1]–[25]. Among them, the most commonly usedalgorithms are the gradient search algorithm [2], [12], [25], theconjugate match algorithm [9], the successive projection algo-rithm [7], [10], [15], [16], and the genetic algorithm [19]–[22],[24]. The gradient search algorithm minimizes the objectivefunction by moving the search point along the steepest descentpath. For a shaped beam, the mean deviation of the achievedgain distribution from the desired gain distribution is generallyconsidered as the objective function. The conjugate matchalgorithm utilizes the principle that the gain at a far-field pointcan be enhanced if the incremental excitation vector is propor-tional to the complex conjugate of the far-field intensity vector.The successive projection algorithm essentially determinesan intersection point of a number of sets; each set consistsof a number of points satisfying the intensity at a far-fieldpoint. Under certain assumptions, these three methods becomemathematically equivalent if applied to phased array synthesis[26, ch. 11]. Generally, these algorithms are computationally

Manuscript received March 2, 2006; revised September 7, 2006.The author is with Northrop Grumman Space Technology, Redondo Beach,

CA 90278 USA (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TAP.2007.891570

efficient but may suffer from a local minimum problem if thestarting solution is not selected judicially. The genetic algo-rithm, on the other hand, represents a distinctly different classof algorithm. This algorithm relies on the principle of “survivalof the fittest.” This algorithm is very slow but does not have alocal minimum problem. The particle swarm algorithm belongsto the same class as the genetic algorithm but with a differentsearch process [23].

In this paper, we present the projection matrix algorithm(PMA) for shaped beam synthesis in arrays and reflectors.The projection matrix method exists in linear algebra [27] forobtaining the best fit solution of a system of linear equations.This method has been applied in the arena of signal processing[28] and other branches of engineering. However, a directapplication of the “as is” projection matrix method to an arraysynthesis suffers from stability and convergence problems.We investigate the root cause of these problems and suggestnecessary amendments for a successful application in beamsynthesis. We consider a general array, where the elements havedifferent radiation patterns. Using the concept of orthogonalvector space, we define the projection matrix associated withan array synthesis. It is found that the projection matrix, ingeneral, is ill conditioned with a large condition number. Weestablish that by properly defining the far-field sample space,the condition number can be reduced significantly, allowinguniform convergence of the solution. The general procedureand a flow chart of the algorithm are illustrated.

We then apply the PMA to synthesize a simple flat-top cir-cular beam. We considered two arrays of the same aperture size:one with equal size elements and the other with different size el-ements. We perform the “phase only” synthesis, keeping the am-plitude distribution unaltered, because that is very realistic foran onboard satellite array antenna. First we consider uniformamplitude excitations. The array with different size elementsshows improved performance because of its intrinsic aperturefield taper associated with the aperture size variation. The per-formance of the equal-size-element array can be improved tothe level of different-size-element array if a tapered excitationis applied. This implies that an array of different-size elementsmay be preferable for active arrays because identical solid statepower amplifiers (SSPAs) can be utilized for all the elements ofthe array.

We then present the necessary modifications of the PMA forsynthesizing the surface of a reflector antenna. First we dividethe projected area of the reflector into a number of small cells

0018-926X/$25.00 © 2007 IEEE

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

and treat the reflector as an array of fictitious elements occu-pying the cells at the projected plane, similar to the develop-ment in [15] and [16]. The phases of the elements are deter-mined using the PMA. Using ray optics, the shape of the re-flector surface is then determined from the phase distribution atthe projected plane. In order to achieve a smooth reflector sur-face, the phase distribution on the projected plane should be con-tinuous. This is accomplished by updating the phase distributionat each iteration. Numerical results show the validity and effec-tiveness of the approach. The method is found to be an order ofmagnitude faster compared to the conventional gradient searchapproach.

Section II presents the projection matrix method and nec-essary modifications for phased array synthesis. Section IIIpresents numerical results for contour beams produced by twodifferent arrays. Section IV is devoted to the reflector surfacesynthesis using PMA. Section V presents a general discussionon the computation efficiency of the approach.

II. THEORY

Consider a general array of elements. The elements havedifferent element patterns and/or may be nonuniformly spaced.The radiation pattern of the array can be expressed as

(1)

where with as the wavenumber in free space and as the far-field point in sphericalcoordinate system. In (1), is the complex excitation coeffi-cient, is the location, and is the radiation pat-tern of the th element. The radial variation andthe time factor are suppressed. The radiation patternof the th element can be considered as the Fourier transformof the source field distribution as

(2)

where is the source distribution in the local coordinatesystem of the th element. The normalization constant maybe approximated as

(3)

The above normalization will make a good approx-imation for the directive pattern in the bore-sight region. As acheck, observe that for , that is, for a uniform distri-bution, becomes equal to 4 ( aperturearea), which is the directivity of a uniform source in a largearray environment. However, for an accurate computation ofarray patterns in the bore-sight as well as far sidelobe regions,

should be the active element pattern [26, p. 9] of theth element.

A. Projection Matrix Method

Suppose is the desired far-field directive pattern.Our objective is to search for a set of amplitudes

that satisfies the following equation:

(4)

To solve for s, we take far-field points and set equa-tions as below

(5)

where and. The set of equations in (5) can be expressed in the

following matrix form:

(6)

where is the shorthand for . Suppose the ma-trix (6) has a unique solution set for s. A necessary conditionfor that to happen is . Also, for a solution to exist,vector on the left-hand side of (6) must lie on the space spannedby the column vectors of the matrixwhere

(7)

If is completely known (amplitude and phase) and re-alizable with the array elements, then (6) must have a so-lution. In that situation, a set of independent equations outof equations can be used. The remaining equa-tions are dependent equations. The solution for s can be de-termined using simple matrix inversion. However, in reality, thephase distribution of is not known because only thefar-field magnitude of a shaped beam is generally specified. Fur-thermore, the desired beam shape (even the magnitude part) maynot be exactly realizable using the -elements. Thus we attemptto find the best fit solution of the problem instead of an exactsolution.

BHATTACHARYYA: PROJECTION MATRIX METHOD FOR SHAPED BEAM SYNTHESIS IN PHASED ARRAYS AND REFLECTORS 677

Fig. 1. Vector space diagram of the array problem.

To obtain the best fit solution, we invoke the projection ma-trix method. It is well known that the best fit solution in theleast square sense occurs if the error vector is orthogonal to thecolumn space of the matrix as pictorially shown in Fig. 1.Thus we have

(8)

The asterisk sign denotes complex conjugate. Notice that thevector inside the parenthesis is the error vector. Equation (8)can be used for obtaining the excitation vector as

(9)

The projection matrix is defined asbecause when it operates on a

vector, the resultant vector becomes the projection on thecolumn space of . Note that the best fit solution is the exactsolution of the matrix equation .

B. Implementation

Equation (9) gives the unknown excitation vector in terms ofthe complex far-field pattern. As mentioned, the complex fieldpattern is not known, because typically the far-field magnitudepattern is only specified. To obtain the far-field phase pattern,we use an iterative approach. First we assume a trial excitationvector normalized with respect to the input power. Thenwe substitute for at the right-hand side of (6) andobtain the corresponding far-field vector, say, . Obviously,the magnitude of the elements of will differ from that of

at most of the far-field points. We compute the differencein magnitudes and set the “error pattern” as

(10a)

with

(10b)

where is a fraction, typically 1/2 or 1/4 ( should be signif-icantly smaller for a large magnitude difference between

Fig. 2. Flow chart of the PMA.

and vectors).1 In (10b), is the th element of .We now set

(11)

and obtain the vector. The refined solution for the complexexcitation vector is

(12)

This new vector must be multiplied by a constant factorsuch that the normalization condition thatis satisfied. With this new normalized excitation vector, we re-peat the whole process until we get a satisfied far-field pattern.For a “phase only” optimization, we update the phase part only,keeping the amplitude of excitation unchanged. The flow chartin Fig. 2 illustrates the iteration process.

C. Convergence of Solution

The algorithm articulated in the previous section involves in-version of the matrix . Numerical inversion of

1For monotonic convergence, C may be varied adaptively based on the mag-nitude of the error vector.


this matrix may have stability problems for some cases becauseof the large condition number of the matrix. Notice that theelements of the matrix are primarily decided by the ele-ment locations and the far-field sample points. It is found thatif the far-field samples are considered only within the coverageregion, then the matrix is likely to have a large conditionnumber, which leads to an unstable, hence inaccurate solution.Increasing the sample space beyond the coverage region allevi-ates this problem. The increased sample space depends on theelement size of the array.

To investigate this problem, consider a linear array of equallyspaced isotropic elements. We assume that the far-field samplesare also equally spaced. These two assumptions do not satisfyin a general array but will help understand the general stabilityproblem in the matrix inversion. The matrix in this case turnsout to be (13) as shown at the bottom of the next page, where

is the element spacing and is the sampling interval in the-space. The elements of the matrix are given by

(14)

Observe that the diagonal elements are given by .Each off-diagonal element is a summation of complex ex-ponential terms. The complex exponents are in arithmetic pro-gression. The sum is zero if , that is, if

(15)

with and as the upper and lower limits of the far-field sampling region in the -space. If (15) is satisfied, thenbecomes a diagonal matrix with the perfect condition numberof unity.

For , (15) essentially turns out to be Shannon’s sam-pling theorem [26, p. 483]. The sampling interval in the -do-main for a fixed size array of length should be given by

, where . For nonuni-form elements, the term may be replaced by the array length.For , this interval becomes smaller, which is permis-sible by the sampling theorem. The relation in (15) suggests thatthe span of the sample space in the -domain shouldbe equal to the distance between the main lobe and the nearestgrating lobe as depicted in Fig. 3. The sample space may be se-lected arbitrarily but must encompass the coverage region of thebeam. For a planar array, the sample space should be equal to

Fig. 3. Far-field sample space region of a linear array of element spacing a.

the unit cell area in the -domain. Equation (15) can also beused for estimating the maximum allowable element spacing fora given coverage region.

For nonuniform element spacing and/or for nonuniform sam-pling interval, (15) is ambiguous because neither nor isuniquely defined. However, considering as the lowest ele-ment spacing, one can use (15) as a guideline for the samplespace. Despite the fact that is nondiagonal, one can conjec-ture that the diagonal elements will have maximum magnitudescompared to the off-diagonal elements, making a well-con-ditioned matrix for numerical inversion. A large deviation fromthe condition given in (15) will make an ill-conditioned ma-trix. In an extreme situation, the elements of the matrix couldbe of the same order, leading to a large condition number. Inparticular, the typical coverage region of a large array is muchsmaller than 2 . If the samples are considered only inside thecoverage region, then the far-field range will largely violate therelation in (15). Consequently, the condition number of willbe large, resulting in an unstable solution for [A].

III. NUMERICAL COMPUTATIONS

The right-hand side of (11) consists of three matrices, namely,and . The matrix (actually a column vector)

changes at each iteration, while the and ma-trices remain unchanged. Thus, and may becomputed only once and then stored in the memory. The com-putations can be reduced further for a contour beam, wherethe gain function is only specified in the coverage region andno constraint is imposed outside the region. In that case, onemay assume , where is the th elementof and is associated with the far-field sample pointsoutside the coverage region. Thus, only a few elements of thematrix are of significance and the remaining ele-ments of the product matrix do not participate in the iterationprocess. The number of multiplications gets reduced signifi-cantly. For an array of equal size, equally spaced elements, andwith uniformly spaced far-field samples, the element pattern canbe de-embedded from the desired gain pattern. In that situation,the matrix becomes diagonal and no numerical inversion is

(13)


Fig. 4. Array of 132 elements. The element areas are in 1:4:16 ratio.

TABLE IFAR-FIELD SAMPLE SPACE VERSUS CONDITION NUMBER OF THE [� ] MATRIX

AND THEIR EFFECTS ON THE COVERAGE GAIN. THE SIGN “**” INDICATES

UNSTABLE SOLUTION. FOR THE ARRAY OF THREE DIFFERENT ELEMENT SIZES,a REPRESENTS THE APERTURE SIZE OF THE SMALLEST ELEMENT

necessary. However, for a general array problem, this simplifi-cation is not possible.

The method is employed to synthesize contour beams with anarray of 132 elements, shown in Fig. 4. Three different sizes ofsquare elements of aperture (area) ratio 1:4:16 are used in thearray. This arrangement is useful in an active array, because thearray elements naturally provide a taper distribution even withuniform power sources, which is advantageous for low sidelobeor beam-isolation point of views. The array under considerationyields about 12 dB aperture field taper if identical power sourcesfor the radiating elements are used.

We synthesized a circular flat-top beam of beam radius0.09 rad (5.15 ). The array size was 19.2 19.2 , where

is the wavelength in free space. According to the samplingtheorem, the far-field sample interval should be less than

rad2 and the sample space shouldbe governed by (15), which yields ,where and is the element size. We consider asthe smallest element of the array and computed

as 0.625 rad for a symmetrical beam. To verify thevalidity of (15), we computed the condition numbers [29] of the

matrix for different values of . Table I shows thecondition numbers versus far-field sample space representedby . Also shown in the table are the condition numbersof a 12 12 element array of equal-size square elements. Theaperture size of the array was also taken as 19.2 19.2 .

2We express the far-field coordinate in radians instead of radians perwavelength.

As can be noticed, the condition number improves withincreasing far-field sample space. We considered symmetricalsample space, keeping the beam at the center region, so that

in (15) becomes . This makes the sample spacelength along the -axis as 2 (the sample space lengthalong -axis was also considered as 2 ). For the uniformarray, the magnitude of 2 should be 2 for the arrayof equal size elements as per the sampling theorem. Table Iessentially supports this, because the condition number of the

matrix reduces abruptly when 2 reaches near2 . Ideally at 2 , the condition number should be unity. We,however, obtained a slightly larger magnitude primarily due tothe nonisotropic element pattern. Furthermore, the minimumgain inside the coverage region converges to a reasonable valueonly when 2 approaches 2 . A larger value than 2does not change the end result significantly.

Interestingly, for the array of different size elements, the con-vergence is reached at a lower value of 2 than 2 (recallthat we consider as the dimension of the smallest element).This indicates that for such an array, one may set a smallersample space and can have a converged solution. It follows that(15) should be applied using an intermediate element size in-stead of the smallest element size.

Figs. 5–7 show the converged contour patterns of the two ar-rays. For uniform excitation (magnitude), the array of equal sizeelements shows 23.03 dBi minimum coverage gain (Fig. 5). Thegain in the coverage region improves with a tapered excitationbecause that reduces the ripples in the coverage region [26, ch.11]. The gain increases to 24.05 dBi with a Gaussian distributionof 12 dB edge taper (Fig. 6). On the other hand, the array withdifferent size elements yields 24.13 dBi coverage gain (Fig. 7)without any excitation taper (uniform excitation). The corre-sponding gain area product (GAP) is about 21 596 square-de-grees. As mentioned before, the different-size-element array ofFig. 4 inherently provides a tapered aperture field distribution,resulting in a larger coverage gain than that of the equal-size-el-ement array counterpart. This result is significant for an ac-tive array designer because one can use identical SSPAs for theelements of the array. Using SSPAs with different power rat-ings may cause implementation problems, particularly at veryhigh-frequency range.

We employed the PMA to synthesize a more complex pat-tern. The coverage region is a 180 annular ring sector of innerand outer radii as 0.06 and 0.14 rad, respectively. Fig. 8 showsthe synthesized contour. The minimum coverage gain was23.19 dBi, which corresponds to a GAP of 17 159 square-de-grees. The GAP is somewhat lower than that of the circularbeam because of the increased complexity of the contour.

IV. REFLECTOR SURFACE SYNTHESIS

The PMA is employed to synthesize the surface of a reflectorantenna producing a contour beam. As a first iteration, the aper-ture phase distribution on a projected plane is synthesized withrespect to the desired beam shape. We used “phase only” op-timization because the amplitude taper is predominantly deter-mined by the radiation pattern of the primary feed. The “pro-jected aperture” is treated as a fictitious array of elements. The


Fig. 5. Beam pattern of 12� 12 square elements array (element size= 1:6� � 1:6� ). Coverage gain = 23:03 dBi, uniform excitation, beamradius = 0:09 rad, �u=k = �v=k = 0:02 rad, u =k = 0:32 rad.

Fig. 6. Beam pattern of 12� 12 square elements array (element size= 1:6� � 1:6� ). Coverage gain = 24:03 dBi, 12 dB Gaussian taper, beamradius = 0:09 rad, �u=k = �v=k = 0:02 rad, u =k = 0:32 rad.

Fig. 7. Beam pattern of 132 square-element array of Fig. 4 with three differentsizes of elements. � = 1:25 in. The element lengths are 1, 2, and 4 in, re-spectively. Coverage gain = 24:13 dBi, beam radius = 0:09 rad, �u=k =�v=k = 0:02 rad, u =k = 0:7 rad. GAP = 21596 sq-deg.

elements, in general, may not be of equal size because the pro-jected aperture contour may not be conformal with the edge of

Fig. 8. Beam pattern of 132 square-element array of Fig. 4 with three differentsizes of elements. � = 1:25 in. The element lengths are 1, 2, and 4 in, respec-tively. Coverage gain = 23:19 dBi (coverage region is an 180 annular sectorof inner and outer radii as 0.06 and 0.14 rad), �u=k = �v=k = 0:015 rad,u =k = 0:63 rad. GAP = 17159 square-degrees.

Fig. 9. Geometry of an offset shaped reflector.

the array. The edge elements may be required to have triangu-larly shaped apertures to fit in the projected plane, particularlyfor low edge taper. For a large edge taper, an array of identicalelements may be considered by slightly deforming the projectedaperture’s edge-contour. The shape of the surface is determinedfrom the synthesized phase of the array elements using ray op-tics approximation as illustrated below.

Fig. 9 shows a geometrical sketch of a shaped reflector. Alsoshown is a reference parabolic surface. The primary feed is lo-cated at the focus of the reference parabola with as the focallength. The synthesized phase distribution at the projected aper-ture plane can be realized by perturbing the parabolic surface asshown. The perturbation is governed by the ray optics equationas follows:

(16)

where is the deviation in , is the path difference be-tween the two incident rays on the two surfaces as shown, and

is the phase of the array element at . To mini-mize thedeviation, a constant phase value (may be the meanphase) may be subtracted from the phase distribution .


The distance of a point from the focus of the parabolais ; thus for , one writes

(17)

Equations (16) and (17) can be employed to obtain the deviation. The unknown in (17) can be approximated as

. The new -coordinate for the desired surface becomes. The reflected phase of the shaped

surface at the projected plane may be checked if it agrees withthe desired value. If necessary, an error may be minimized withanother round of iteration using the refined value of .

A very important point needs to be considered while opti-mizing the phase of the fictitious array on the projected plane.In the case of a real array, generally there is no restriction onthe phase distribution because any amount of phase shift can berealized by a phase shifter. In a shaped reflector, however, thephase is realized by deforming the shape of the surface. There-fore it is critical that the phase distribution be a continuous func-tion of and ; otherwise a smooth shaped surface cannot berealized. Recall that, in the case of an array, we update the com-plex excitation vector through iterations and then extract thephase information at the end of iterations. Since the phase ofa complex variable is obtained using inverse tangent function,the phase is typically discontinuous (because the inverse func-tion is multivalued), even with a continuous complex excitationfunction over the projected aperture. To circumvent this situa-tion, we update the phase angle at each iteration by adding theincremental phase. To elaborate this important point, let us con-sider that is the complex excitation coefficient of the thfictitious array element after th iteration and is the corre-sponding phase. Suppose the above quantities are and

after ( 1) iteration. Suppose the incremental phase is. Since the magnitudes do not alter for phase only opti-

mization, we write

(18)

Assuming small, we use two-term approximation of theexponential factor and get

(19)

We consider only the real part if the right-hand side of (19) iscomplex. The phase after ( 1) iteration thus becomes

(20)

If the initial phase is a smooth function of , then this pro-cedure generally provides a smooth phase distribution at theend. Thus it is very important that the starting solution have asmooth phase distribution. A quadratic phase variation is con-sidered to be a good starting solution in most cases. Further-more, if the initial far-field gains at the sample points are verydifferent from desired gains, then becomes large; hence

Fig. 10. Phase (in rad) contour on the projected aperture of the shaped reflector.Reflector diameter = 28 in, � = 1:25 in, focal length = 30 in, vertical offset= 20 in (from reflector center). The circle with dashed line represents projectedreflector edge.

Fig. 11. Surface of the shaped reflector. Cell size for optimization is 0.8� 0.8in , cell size for surface interpolation is 0.4� 0.4 in .

two-term approximation used in (19) introduces error. In thatcase, vector in (10a) should be reduced by properly se-lecting the factor in (10b) to make smaller.

A reflector surface is synthesized to produce an annular sectorshaped beam as in Fig. 8. The projected area of the reflector sur-face is a circle of diameter 22.4 . For the phase synthesis atthe projected plane, we use 0.64 cell size in rectan-gular grid. The amplitude distribution of the fictitious array onthe projected plane was considered as the field distribution ofa parabolic offset reflector on its projected plane. The reflectorwas fed by a Gaussian feed of 12 dB edge taper. Fig. 10 showsthe phase contour at the projected plane. The lines are closelyspaced near the edges, signifying steeper slope near the edgeof the reflector. This is typical for a shaped reflector. From thephase distribution, the -coordinates on the reflector surface cor-responding to the cells were obtained. The -coordinates at theintermediate points were obtained using linear interpolations.Fig. 11 is the three-dimensional plot of the shaped surface and


Fig. 12. Contour pattern of shaped reflector. Reflector diameter= 28 in, � =

1:25 in, GAP = 20724 square-degrees.

TABLE IICOMPARISON BETWEEN GSA AND PMA. AN ARRAY OF ABOUT

400 ELEMENTS WAS CONSIDERED. THE DESIRED BEAM

WAS A CIRCULAR FLAT-TOP BEAM OF RADIUS 3

the constant -contours. Fig. 12 is the contour plot of the shapedbeam resulted from the reflector. The pattern of the fictitiousarray on the projected plane was also computed for examiningthe difference between the reflector pattern and its equivalentarray pattern. The difference between the two patterns is foundto be very small for the case under consideration.

V. DISCUSSIONS AND CONCLUSIONS

We have successfully developed and demonstrated the utilityof the PMA for synthesizing contour beams in phased arrays andshaped reflectors. The algorithm is found to be very fast com-pared to the GSA typically used for beam shaping. For instance,to create the beam in Fig. 8, the computation time was 38 s ona laptop computer with a 2.4 GHz Pentium 4 processor with256 MB RAM. For the shaped reflector synthesis in Fig. 11,the computation time was 152 s. The number of iterations wasabout 20 000 in both cases. For simple circularly shaped beams,the number of iterations were less that 2000 and computationtimes were much faster.

Table II presents a comparison between GSA and PMA withrespect to a circular flat-top beam synthesis of 3 beam radius.The same laptop computer described before was used for thisstudy. About 400 elements were used in both cases. For GSA,the phase distribution was approximated by a sixth order poly-nomial of element coordinates and that has 27 unknowncoefficients. The computation time increases linearly with the

number of iterations. However, unlike the GSA, the PMA re-quires a fixed amount of overhead time, which is independent ofthe number of iterations. This overhead primarily is due to thematrix inversion inherent to the PMA. Thus for a small numberof iterations, the PMA may not be very attractive because of thematrix inversion overhead, but for a larger number of iterations(which is indeed required in most cases for convergence), thePMA becomes significantly faster than the GSA. For instance,the GSA takes 170 s and the PMA takes 28 s for 500 iterations(see Table II). Furthermore, the PMA shows a faster conver-gence than GSA. It can also be noticed that the PMA convergesto a higher gain number than the GSA. This is primarily due tothe phase distribution used in GSA, which is constrained by thesixth order polynomial. The gain number can be improved byusing a higher order polynomial, but that will increase the com-putation time significantly.

For an array of identical elements and for equally spacedfar-field grid points, the matrix becomes an orthogonal ma-trix (if the element pattern is de-embedded from the desired pat-tern) that can be inverted analytically. In that situation, the com-putation time becomes significantly lower than that of the GSAeven for a small number of iterations. For 500 iterations, thecomputation time is less than 2 s with respect to the array con-sidered in Table II. For identical elements and for equally spacedfar-field grid points, the PMA becomes mathematically equiva-lent to a projection algorithm [26, ch. 11].

We have successfully demonstrated the application of PMAtoward a reflector surface synthesis. As mentioned before, fora realizable reflector surface, the starting phase distribution onthe projected aperture should be a continuous function of theaperture coordinates. The PMA in this case updates the phasedistribution at every iteration. As a consequence, the final phasedistribution becomes a continuous function of the aperture co-ordinates in most cases. This continuous solution leads to asmooth reflector surface as evident through the numerical ex-ample. For a simple shaped beam, the grid size on the projectedplane could be of about one square wavelength. For complexbeams a smaller grid size may be necessary in order to obtain acontinuous phase distribution.

REFERENCES

[1] W. L. Stutzman, “Synthesis of shaped-beam radiation patterns usingthe iterative sampling method,” IEEE Trans. Antennas Propag., vol.AP-19, pp. 36–41, 1971.

[2] T. S. Fong and R. A. Brigenheier, “Method of conjugate gradients forantenna pattern synthesis,” Radio Sci., vol. 6, pp. 1123–1130, 1971.

[3] G. A. Deschamps and H. S. Cabayan, “Antenna synthesis and solutionof inverse problems by regularization methods,” IEEE Trans. AntennasPropag., vol. AP-20, pp. 268–274, 1972.

[4] R. Mautz and R. F. Harrington, “Computational methods for antennapattern synthesis,” IEEE Trans. Antennas Propag., vol. AP-23, pp.507–512, 1975.

[5] H. Schjaer-Jacobsen and K. Madsen, “Synthesis of nonuniformlyspaced arrays using a general nonlinear minimax optimizationmethod,” IEEE Trans. Antennas Propag., vol. AP-24, pp. 501–506,1976.

[6] R. Jorgensen, “Coverage shaping of contoured-beam antennas by aper-ture field synthesis,” Proc. Inst. Elect. Eng., vol. 127, pp. 201–208,1980.

[7] H. Elmikati and A. A. Elsohly, “Extension of projection method tononuniformly linear antenna arrays,” IEEE Trans. Antennas Propag.,vol. AP-32, pp. 507–512, May 1984.


[8] C. A. Klein, “Design of shaped-beam antennas through minimaxgain optimization,” IEEE Trans. Antennas Propag., vol. AP-32, pp.963–968, Sep. 1984.

[9] P. T. Lam, S.-W. Lee, D. C. D. Chang, and K. C. Lang, “Directivityoptimization of a reflector antenna with cluster feeds: A closed-formsolution,” IEEE Trans. Antennas Propag., vol. AP-33, pp. 1163–1174,Nov. 1985.

[10] G. T. Poulton, “Power pattern synthesis using the method of successiveprojections,” 1986 IEEE APS Symp. Dig., pp. 667–670, 1986.

[11] V. Galindo-Israel, W. A. Imbriale, and R. Mittra, “On the theory ofsynthesis of single and dual offset shaped reflector antennas,” IEEETrans. Antennas Propag., vol. AP-35, pp. 887–895, 1987.

[12] A. R. Cherrette, S. W. Lee, and R. J. Acosta, “A method for producingshaped contour radiation pattern using a single shaped reflector and asingle feed,” IEEE Trans. Antennas Propag., vol. 37, pp. 698–706, Jun.1989.

[13] R. S. Elliott and G. J. Stern, “Footprint patterns obtained by planararrays,” Proc. Inst. Elect. Eng., vol. 137, pt. H, pp. 108–112, Apr. 1990.

[14] O. M. Bucci, G. Mazzarella, and G. Panariello, “Reconfigurable arraysby phase-only control,” IEEE Trans. Antennas Propag., vol. 39, pp.919–925, Jul. 1991.

[15] G. T. Poulton and S. G. Hay, “Efficient design of shaped reflectorsusing successive projections,” Electron. Lett., vol. 27, no. 23, pp.2156–2158, Nov. 1991.

[16] H. Zhou, P. J. B. Clarricoats, G. T. Poulton, and S. G. Hay, “Powerpattern synthesis for mesh reflectors using the method of successiveprojections,” Electron. Lett., vol. 28, no. 19, pp. 1837–1839, Sep. 1992.

[17] E. Botha and D. A. McNamara, “A contoured beam synthesis techniquefor planar antenna arrays with quadrantal and centro-symmetry,” IEEETrans. Antennas Propag., vol. 41, pp. 1222–1230, Sep. 1993.

[18] O. M. Bucci, G. D’Elia, G. Mazzaarella, and G. Panariello, “Antennapattern synthesis: A new general approach,” Proc. IEEE, vol. 82, pp.358–371, Mar. 1994.

[19] R. L. Haupt, “An introduction to genetic algorithms for electromag-netics,” IEEE Antennas Propag. Mag., vol. 37, pp. 7–15, Apr. 1995.

[20] F. Ares, S. R. Rengarajan, E. Villanueva, E. Skochinski, and E. Moreno,“Application of genetic algorithm and simulated annealing techniquein optimizing the aperture distribution in antenna arrays,” in 1996 IEEEAPS Symp. Dig., Jul. 1996, pp. 806–809.

[21] K.-K. Yan and Y. Lu, “Sidelobe reduction in array-pattern synthesisusing genetic algorithm,” IEEE Trans. Antennas Propag., vol. 45, pp.1112–1117, Jul. 1997.

[22] J. M. Johnson and Y. Rahmat-Samii, “Genetic algorithm in engineeringelectromagnetics,” IEEE APS Mag., vol. 39, no. 4, pp. 7–25, Aug. 1997.

[23] L. I. Vaskelainen, “Phase synthesis of conformal array antennas,” IEEETrans. Antennas Propag., vol. AP-48, pp. 987–991, Jun. 2000.

[24] D. W. Boeringer and D. H. Werner, “Particle swarm optimizationversus genetic algorithms for phased array synthesis,” IEEE Trans.Antennas Propag., vol. 52, pp. 771–779, Mar. 2004.

[25] D.-W. Duan and Y. Rahmat-Samii, “A generalized diffraction syn-thesis technique for high performance reflector antennas,” IEEE Trans.Antennas Propag., vol. 43, pp. 27–40, Jan. 1995.

[26] A. K. Bhattacharyya, Phased Array Antennas—Floquet Analysis, Syn-thesis, BFNs, and Active Array Systems. New York: Wiley, 2007.

[27] G. Strang, Linear Algebra and its Applications. Philadelphia, PA:Saunders, 1988.

[28] S. Haykin, Adaptive Filter Theory, 3rd ed. Englewood Cliffs, NJ:Prentice-Hall, 1996, p. 498.

[29] E. Kreyszig, Advanced Engineering Mathematics. New York: Wiley,1988.

Arun K. Bhattacharyya (M’87–SM’91–F’02) wasborn in India in 1958. He received the B.Eng. degreein electronics and telecommunication engineeringfrom Bengal Engineering College, University ofCalcutta, India, in 1980, and the M.Tech. degree inmicrowave engineering and the Ph.D. degree fromIndian Institute of Technology, Kharagpur, in 1982and 1985, respectively.

From November 1985 to April 1987, he was withthe University of Manitoba, Winnipeg, Canada, as aPostdoctoral Fellow in the Electrical Engineering De-

partment. From May to October 1987, he was with Til-Tek Limited, Kemptville,ON, Canada as a senior antenna engineer. In October 1987, he joined the Uni-versity of Saskatchewan, Canada, as an Assistant Professor with the ElectricalEngineering Department, where he became an Associate Professor in 1990. InJuly 1991, he joined Boeing Satellite Systems (formerly Hughes Space andCommunications), Los Angeles, CA, as a Senior Staff Engineer, where he be-came Scientist and Senior Scientist in 1994 and 1998, respectively. He becamea Technical Fellow of Boeing in 2002. In September 2003, he joined NorthropGrumman Space Technology group as a Staff Scientist, Senior Grade. He is theauthor of Electromagnetic Fields in Multilayered Structures-Theory and Ap-plications (Norwood, MA: Artech House, 1994) and Phased Array Antennas(Hoboken, NJ: Wiley, 2006). He is an author of more than 80 technical papersand has received 12 patents. His technical interests include electromagnetics,printed antennas, multilayered structures, active phased arrays, and modelingof microwave components and circuits.

Dr. Bhattacharyya has received numerous awards, including the 1996 HughesTechnical Excellence Award, 2002 Boeing Special Invention Award for his in-vention of high-efficiency horns, 2003 Boeing Satellite Systems Patent Award,and 2005 Tim Hannemann Annual Quality Award, Northrop Grumman SpaceTechnology.

Documents

Projection Matrix Method for Shaped Beam Synthesis in Phased Arrays and Reflectors