Patent Application: US-201314761162-A

Abstract:
a method of manufacturing an array antenna comprising : a design phase wherein an array layout of said array antenna is synthesized and radiating elements are designed to be arranged according to said array layout ; and a phase of physically making said array antenna wherein the radiating elements are arranged according to said array layout . the design phase comprises the steps of : defining a continuous reference aperture ; subdividing said continuous reference aperture into a plurality of elementary cells with assigned power levels ; determining , within each said elementary cell , a position for at least one maximum efficiency radiating element ; determining a size and an aperture field amplitude of each said maximum efficiency radiating element , such that a variation of a cumulative field distribution of the resulting array antenna aperture over each said elementary cell is substantially equal to a variation of a cumulative field distribution of said reference aperture over the same elementary cell , subject to size constraints .

Description:
the first step of the method of the invention is to define the desired radiative properties of the array to be designed . this is the same as in the prior art . usually a specified gain ( g ), a beamwidth ( bw ) and a peak sidelobe level ( sll ) are indicated . then , an aperture field distribution (“ reference aperture ”) able to guarantee these radiative characteristics is identified . the reference pattern , which represents the target of the performances of the aperiodic array , may be obtained with a number of standard techniques to design continuous apertures . as an example , taylor amplitude distribution laws for linear [ 27 ] and circular [ 28 ] apertures can be considered . according to the prior art , discrete sampling of the continuous aperture distribution may be directly used to realize a periodic array antenna with continuous amplitude tapering ( configuration a ). nevertheless , the direct implementation of the discrete field distribution with a specific dynamic range of the amplitude may be difficult , expensive and inefficient in terms of dc - to - rf power conversion at hpa level . in facts , especially when the requirements in terms of sll are very stringent , the feeding distribution of the antenna aperture presents a high variability as a function of the position . one of the strongest motivations to introduce aperiodic arrays with equi - amplitude excited elements consists in the necessity to reduce complexity and cost of an array antenna including its beamforming network and amplification section . an effective solution to the problem of approximating a desired radiation pattern of a reference linear continuous taper function with an aperiodic linear array of uniformly excited omnidirectional radiating elements has been first proposed by doyle [ 6 ] ( as reported in [ 7 ]). doyle approach consists in a weighted least - mean - square approximation of the desired radiation pattern with the array factor of the aperiodic array . the original doyle &# 39 ; s design method was first extended to different weighted least - mean - square criteria and array geometries by toso and angeletti in [ 15 ] and recently refined , improved and applied to circular arrays in a number of works [ 18 ]-[ 26 ]. the first problem that we consider is that of approximating a desired radiation pattern f 0 ( u ), generated by a linear source of continuous amplitude field i 0 ( x ), with the radiation pattern f a ( u ) of a discrete aperiodic array with amplitude field i a ( x ). in the far - field approximation , the radiation pattern of a linear continuous aperture is given by the fourier transform of the aperture field : the angle ∂, representing the observation direction , is measured with respect to the direction perpendicular to the antenna aperture . a particularly effective solution consists in exploiting a weighted mean square error ( w - mse ) minimization of the difference between the reference pattern and the unknown one . the following equality , first observed by doyle , can be demonstrated to hold true : where , the cumulative function i ( x ) of the aperture field distribution i ( x ) is introduced , and it is assumed that the amplitude fields i 0 ( x ) and i a ( x ) are both normalised such that , to derive expression ( 3 ) we can follow doyle reasoning evaluating the difference of the far - field radiation patterns : being e jk 0 ux bounded and provided that ( i 0 ( x )− i s ( x )) vanishes as x →−∞ and x →∞, the first term nullifies and we have : to obtain doyle equation ( 3 ) we now need to apply the parseval - plancherel theorem , which is derived below for sake of completeness and correctness of the multiplying constants . central to this derivation is the representation of the dirac delta function by means of a complete set of orthogonal functions [ arfken and weber pp . 88 - 89 ] which leads to the closure equation for the continuous spectrum of plane waves [ arfken and weber pp . 90 ]: by applying the parseval - plancherel theorem ( 9 ) to expression ( 8 ) we obtain doyle equality : the left term in equation ( 12 ) represents a weighted mean square error ( w - mse ) of the patterns with an inverse quadratic weighting function . the right term is a mean square error ( mse ) of the cumulative functions of the field distributions . the synthesis problem is so made equivalent to the identification of an array cumulative function i a ( x ) optimally fitting the reference cumulative function i 0 ( x ). a solution to the problem of synthesizing a linear array of omnidirectional radiating elements with assigned number of elements n and amplitude excitations { a k ; k = 1 ÷ n } was discussed in angeletti and toso [ 19 ]. the referenced work extends doyle approach [ 6 ]-[ 7 ] which is valid for uniform excitations we recall that for an array of omnidirectional radiating elements with amplitude excitations { a k ; k = 1 ÷ n } and phase centres { x k ; k = 1 ÷ n } we have : the method proposed in angeletti and toso [ 19 ] is based on the observation that the k - th element of the array , positioned in x k , defines an elementary cell ( t k - 1 , t k ), with x k ε ( t k - 1 , t k ), such that , i a ( t k )− i a ( t k - 1 )= a k = i 0 ( t k )− i 0 ( t k - 1 ) ( 17 ) equations ( 15 ) and ( 16 ) provide a direct mean to derive the boundaries of the elementary cells by inversion of the reference cumulative curve i 0 ( x ): the element position can be then determined inverting the following equation [ 6 ][ 7 ][ 18 ] an alternative element position determination , based on the centroid of the field distribution function for each elementary cell , is discussed in [ 18 ] and provides similar performances . the solution of the linear array synthesis with omnidirectional radiating elements , often analysed from an integral point of view , offer also a differential interpretation that will allow a broader generalization . under the assumption of a high number of elements , the dimensions of the elementary cells δt k = t k − t k - 1 will reduce to the limit that δt k → 0 for n →∞. equation ( 17 ) can be reinterpreted as , this linear differential form can be considered as a particular form of the general expression , and the boundaries t k of the elementary cells can be determined by inversion of the primitive function e ( x ) of e ( x ): it is worth noting that the primitive function e ( x ) acts like a “ grading function ”, i . e . the ordinate values determines by inversion the searched abscissa values t k . similarly , the values a k ( or equivalently the steps a k ) act like a “ grading scale ” that determine the ordinate values to be inverted by means of e ( x ) in the following we will exploit this generalized differential form of the cumulative approximation condition to solve the problem of synthesizing element positions and element dimensions of aperiodic arrays with maximum efficiency radiating elements . it will be understood that maximum efficiency condition refers to the almost constancy , on the radiating element aperture , of the electromagnetic field generated exciting the radiating element itself . synthesis of a linear aperiodic array with maximum efficiency radiating elements ( invention ) we now introduce the synthesis of a linear aperiodic array with linear radiating elements of maximum efficiency . assuming an assigned number of elements n , and assigned power excitations { p k ; k = 1 ÷ n } we want to determine the element positions , x k , and element aperture δx k such that a desired radiation pattern f 0 ( u ), generated by a linear source of continuous amplitude field i 0 ( x ), is approximated with the radiation pattern f a ( u ) of a discrete aperiodic array with amplitude field i s ( x ) where , i k ( x ) is the aperture field the of the k - th radiating element referred to its phase center , x k . the element radiation pattern referred to the element phase center , x k , will be indicated as g k ( u ), for the linear elements with maximum efficiency we will assume a constant aperture field on the element aperture , where u ( x ) is the unit step function , and the element radiation pattern referred to the element phase center , x k , is : the radiation power associated to the k - th radiation element can be determined , apart inessential impedence factors , integrating the square of the aperture field , in case the power of the k - th radiation element is pre - assigned , the following relationship between the aperture field amplitude and linear radiating element dimension must be respected : we now turn our attention to cumulative approximation condition in its differential form ( 20 ), that mutatis mutandis becomes : i 0 ( x k ) δ t k = b k δx k ( 32 ) under the additional hypothesis that the k - th radiation element has a linear dimension δx k almost equal to its elementary cell , substituting ( 33 ) and ( 31 ) in ( 32 ) we obtain the following optimality condition , i 0 ( x k ) δ t k =√{ square root over ( p k δt k )} ( 34 ) this expression can be reduced to a linear differential form in δt k by simple manipulations , i 0 2 ( x k ) δ t k = p k ( 35 ) it can be observed that the resulting optimality condition correspond to a partitioning of the reference aperture field in elementary cells with assigned power levels , nevertheless , the coincidence of the field approximation with the power approximation is strictly linked to the use of maximal efficiency linear elements and is not generally true with other elements ( e . g . with omnidirectional radiating elements ). from equation ( 35 ) it can be derived that i 0 2 ( x ) acts as derivative of the “ grading function ” and p k as “ grading scale ”. the synthesis can be performed determining , in a first step , the boundaries t k of the elementary cells by inversion of the “ grading function ” e ( x )=∫ i 0 2 ( x ) dx : in a second step the k - th element positions , x k , can be determined accordingly either to a doyle - like optimality condition ( 19 ) i . e . x k is such that , or simply considering x k the centre of the elementary cell , in a third step , the k - th element dimensions , δx k , can be determined such to contemporarily fulfil the cumulative field distribution approximation condition , δ x k ≦ δx k max 2min (( x k − t k - 1 ),( t k − x k )) ( 42 ) the aperture field amplitude coefficients b k can be derived from equation ( 40 ) using δx k from equation ( 43 ), we now turn our attention to the problem of approximating a desired radiation pattern f 0 ( θ , φ ), generated by an aperture with circular symmetric continuous amplitude field i 0 ( ρ ), with the radiation pattern f a ( θ , φ ) of a planar aperiodic array with elements disposed on circular rings of non - commensurable radius ( aperiodic ring array ). the far field radiation pattern generated by a circular symmetric field distribution i ( ρ ) varying only in a radial direction ρ is given by a hankel transform of order zero of the field distribution i ( ρ ): in the following we will use the definition of the hankel transform of n - th order accordingly to the well - known hankel integral formula [ watson , to . 453 ], f ( ρ )=∫ 0 ∞ (∫ 0 ∞ f ( ρ ) ρ j n ( u ρ ) d ρ ) uj n ( u ρ ) du ( 48 ) so that , for our purposes , the hankel transform of n - th order will be defined as follows : h n ( f ( ρ )= 2π ∫ 0 ∞ f ( ρ ) ρ j n ( k 0 u ρ ) dρ ( 49 ) where j n ( x ) is the bessel function of order n . similarly to the linear case ( 6 ) we can evaluate the difference of the far - field radiation patterns : and the radial cumulative function of the circular aperture field is introduced : being j 0 ( k 0 uρ ) bounded , and provided that ( i 0 ( ρ )− i a ( ρ )) vanishes as ρ → 0 and ρ →∞, the first term of ( 51 ) nullifies and we have , to obtain an equation similar to doyle equation ( 3 ) we need to apply the parseval - plancherel theorem for hankel transforms , which , also in this case , is derived below for sake of completeness and correctness of the multiplying constants . also in this case , the central element of the derivation is the representation of the dirac delta function by means of a complete set of orthogonal functions [ arfken and weber pp . 88 - 89 ] which leads to the closure equation for the continuum of bessel functions [ arfken and weber eq . 11 - 59 , p . 696 ]: by applying the parseval - plancherel theorem for hankel transforms ( 56 ) to expression ( 55 ) we obtain an equivalent doyle equality for circular apertures : the left term in equation ( 59 ) represents a weighted mean square error of the patterns with an inverse quadratic weighting function . in a symmetric way , the right term is a weighted mean square error of the radial cumulative functions of the circular field distributions . similarly to the linear case , the synthesis problem is made equivalent to the identification of an approximating radial cumulative function i a ( ρ ) optimally fitting the reference radial cumulative function i 0 ( ρ ). we now introduce the synthesis of a stepwise radially - continuous aperture composed of a annular rings with maximum efficiency . assuming an assigned number of annular rings n , and assigned power excitations per ring { p k ; k = 1 ÷ n } we want to determine the annular ring mean radius , r k , and the annular ring radial extension δr k such that a desired radiation pattern f 0 ( u ), generated by a planar source with circularly symmetric continuous amplitude field i 0 ( ρ ), is approximated with the radiation pattern f a ( u ) and amplitude field i a ( ρ ) generated by an array of annular rings with maximum efficiency , where , i k ( ρ ) and g k ( u ) are , the circular aperture field , and the radiation pattern of the k - th annular ring , respectively . each annulus is defined by the area between two concentric circles of radii for an annular ring with maximum efficiency we will assume that the annular ring exhibits a constant aperture field i k ( ρ ) on the annulus , for the radiation pattern , g k ( u ), of the k - th annular ring with maximum efficiency , we have : where the lambda function is defined in jahnke and emde ( p . 128 [ 30 ]), the radiation power associated to the k - th annular ring can be determined , apart inessential impedence factors , integrating the square of the aperture field on the annulus , in case the power of the k - th annular ring is pre - assigned , the following relationship between the aperture field amplitude , b k , and annulus area must be respected : which intrinsically define a partitioning in elementary annuli of radial boundaries ( ρ k - 1 , ρ k ), and in its differential form ( 20 ), becomes : where ρ k is the mid radius of the elementary annulus of radial boundaries ( ρ k - 1 , ρ k ), we will now use the additional approximation that the k - th annular ring mid radius , r k , almost coincides with the mid radius of the elementary annulus , clearly this approximation will be valid at the limit of a high number of annular rings , so that the radial dimensions of the elementary annuli will reduce and the quantities coincide . with this additional approximation equation ( 69 ) becomes , this expression is fully equivalent to equation ( 32 ), which has been derived for the case of a linear aperiodic array with maximum efficiency radiating elements . under the additional hypothesis that the k - th annular ring has a radial extension δr k almost equal to its elementary annulus , substituting equations ( 73 ) and ( 67 ) in ( 72 ) we obtain the following optimality condition , this expression can be reduced to a linear differential form in δp k by simple manipulations , 2π i 0 2 ( r k ) r k δ ρk = p k ( 75 ) it is worth noting that , similarly to the case of a linear aperiodic array with maximum efficiency radiating elements , the resulting optimality condition correspond to a partitioning of the reference aperture field in elementary cells with assigned power levels , still , the coincidence of the field approximation with the power approximation is strictly linked to the use of maximal efficiency linear elements and is not generally valid for other approximating functions . from equation ( 76 ) it can be derived that ρi 0 2 ( ρ ) acts as derivative of the “ grading function ” and p k as “ grading scale ”. the synthesis can be performed determining , in a first step , the boundaries ρ h of the elementary cells by inversion of the “ grading function ” e ( ρ )=∫ ρi 0 2 ( ρ ) dρ : in a second step the k - th annular ring mid - radius , r k , can be determined accordingly either to a doyle - like optimality condition ( 19 ) i . e . r k is such that , or simply considering r k the centre of the elementary annulus , in a third step , the k - th annular ring radial dimensions , δr k , can be determined such to contemporarily fulfil the cumulative field distribution approximation condition , a x i 0 ( ρ k )− i 0 ( ρ k - 1 )= i a ( ρ k )− i a ( ρ k - 1 )= b k r k δr k ( 80 ) δ r k ≦ δr k max 2min (( r k − ρ k - 1 ),( ρ k − r k )) ( 82 ) the amplitude coefficients b k can be derived from equation ( 80 ) using δx k from equation ( 83 ), synthesis of an array of maximum efficiency annular - rings with power proportional to the ring circumference ( invention ) the general design procedure described in previous paragraph can be customised to different profiles of assigned power levels . an interesting profile of power level is such to have a power directly proportional to the ring diameter , which can be substituted in equation ( 75 ), resulting in the following optimality condition in differential form , i 0 2 ( r k ) δ p k = p 0 ( 86 ) from equation ( 86 ) it can be derived that , in this case , i 0 2 ( ρ ) acts as derivative of the “ grading function ” and the “ grading scale ” is uniform . in this case , to perform the synthesis , in a first step the boundaries p k of the elementary cells are determined by inversion of the “ grading function ” e ( ρ )=∫ i 0 2 ( ρ ) dρ : in a second step , the annular ring mid - radius , r k , can be determined accordingly to the second step described in previous paragraph . in a third step , the k - th annular ring radial dimensions , δr k , are determined such to contemporarily fulfil the cumulative field distribution approximation condition ( equation ( 80 )), assigned power condition ( equations ( 66 ), ( 67 ) and ( 85 )), δ r k ≦ δr k max 2min (( r k − ρ k - 1 ),( ρ k − r k )) ( 89 ) the amplitude coefficients b k can be derived from equation ( 80 ) using δr k from equation ( 90 ), synthesis of annular ring aperiodic arrays with maximum efficiency radiating elements ( invention ) finally , we are ready to tackle the problem of synthesizing an aperiodic planar arrays with maximum efficiency radiating elements arranged on concentric annular rings of incommensurable radius , element dimensions proportional to the annular ring dimensions and assigned power excitations , q k , per element of the same ring { q k ; k = 1 ÷ n }. in first approximation , each annular ring will exhibit a stepwise constant radial profile similar to an aperture with maximum efficiency annular - rings ( refer to fig9 a ) so that the results already developed will be fully applicable . in the final layout the continuous annular ring will be replaced by maximum efficiency radiating elements with radial element dimensions smaller than , and preferably equal to radial width of the corresponding annulus . according to the invention , the number of maximum efficiency radiating elements arranged on each ring is proportional to the ratio between the radius of said ring and the radial width of said maximum efficiency radiating elements . the invention will be described considering the case of circular radiating elements with element diameter proportional to the annular ring size ( refer to fig9 b ), nevertheless , according to the invention , different types of maximum efficiency radiating elements can be used ( e . g . with aperture corresponding to circular — fig3 a , square , in which case the radial width correspond to the diagonal or side length — fig3 b , annulus sector — fig3 c , etc .). different technological solutions for the radiating element can used ( e . g . horns , printed patches ), similarly the radiating element may be constituted sub - arrays of smaller radiating elements . being the power of the maximum efficiency radiating elements of k - th annular ring is pre - assigned to q k , the following relationship between the radiating element aperture field amplitude , b k , and radiating element aperture area , a k re , must be respected : circular radiating elements of diameter d k proportional to the annular ring radial dimensions , δr k , allows a minimisation of the unused area between annular rings . the area of the radiating element becomes , with the additional hypothesis that the number of elements accommodated in each annular ring , n k , is proportional to the ration between the annular ring circumference and the radiating element diameter ( i . e . the radial width of the radiating element ), we can evaluate the annular power level profile , that results being , which , substituted in equation ( 74 ), results in the following optimality condition in differential form , from equation ( 76 ) it can be derived that i 0 ( ρ ) acts as derivative of the “ grading function ” and √{ square root over ( q k )} as “ grading scale ”. the synthesis can be performed determining , in a first step , the boundaries ρ k of the elementary cells by inversion of the “ grading function ” e ( ρ )=∫ i 0 ( ρ ) dρ : in a second step the k - th annular ring mid - radius , r k can be determined accordingly either to a doyle - like optimality condition ( 19 ) i . e . r k is such that , or simply considering r k the centre of the elementary annulus , in a third step , the k - th annular ring radial dimensions , δr k , are determined such to fulfil the feasibility condition of non - overlapping annular rings , δ r k = δr k max 2min (( r k − ρ k - 1 ),( ρ k − r k )) ( 100 ) in a forth step , the number of elements per ring is defined accordingly to equation ( 94 ), where the symbols └ ┘ indicates the operator of maximum integer contained within the number in the brackets . the final fifth step consists in placing the defined integer number of elements n k on the annular rings of mid - radius , r k . the most obvious and most accurate choice is to put them at an equal angular distance . a deterministic or random rotation of the elements from annular ring to annular ring can be also employed , although the corresponding results do not change significantly especially for large arrays . in case the array is designed such to have a central element , the some adaptation in the definition of the grading scale , q 1 ′, must be considered . in particular , being for the central element ( k = 1 ), furthermore , considering that the central element will exhibit a true coverage of the assigned area while circular radiating elements disposed on the other annular ring will suffer an area filling factor of the resulting value to be used for the grading scale will be : in case beam scanning requirements in a region of interest ( roi ) are applicable , an additional constraint on the maximum element dimensions , δr max , can be imposed . the third step above must be complemented with the sub - step : the fourth and fifth steps described above apply without modification and fully complying with the optimality conditions in differential form as expressed by equation ( 96 ). this is basically due to the fact that the constitutive hypothesis ( 94 ) makes equation ( 96 ) independent on the element size but only dependent on the assigned power level per element of the k - th ring . nevertheless it must be understood that elements diameters comparable with the elementary annuli will improve the accuracy of the approximation of the cumulative function and in turn of the radiation pattern . the aperture filling factor and the associated aperture efficiency and directivity are so optimized compatibly with the scanning requirement constraints . the described procedure has been applied , for exemplification , to the reference aperture field of fig3 a , with an amplitude field i 0 ( x ) relevant to a linear taylor amplitude distribution law [ 27 ] obtaining a desired radiation pattern f 0 ( u ) ( shown in fig3 b ) with 20 db of side - lobe level ( n bar = 3 ; sll =− 20 ). assuming equal power per element , the boundaries t k of the elementary cells are determined by inversion of the “ grading function ” of equation ( 37 ) on a uniform scale ( due to the equi - power hypothesis ). this first step is shown in fig3 c . second step of determining the element phase center positions accordingly to the cumulative function ( 38 ) is shown in fig3 d . fig3 e and fig3 f show , respectively , the aperture fields and the cumulative functions of the linear array of maximum efficiency radiating elements ( in black ) and reference linear aperture ( in gray ). in fig3 g and fig3 h , the aperture fields and the cumulative functions , respectively , of the linear array of maximum efficiency radiating elements ( in black ) and reference linear aperture ( in gray ) are compared with the linear array of omnidirectional radiating elements ( dotted line ). the resultant radiation patterns are reported in fig3 i , which indicates the goodness of the proposed synthesis solution for linear array of maximum efficiency radiating elements . the described procedures have been applied , for exemplification , to the reference circular aperture field of fig4 ( radial cut ) with an aperture diameter d equal to 44λ ( λ being the operating wavelength of the antenna ) and with an amplitude field i 0 ( p ) relevant to a circular taylor amplitude distribution law [ 28 ] obtaining a desired radiation pattern f 0 ( u ) ( shown in fig5 b ) with 25 db of side - lobe level ( n bar = 2 ; sll =− 25 ). fig5 a and fig5 c show , respectively , a three - dimensional view and a colour plot of the aperture field of the reference circular aperture . fig5 d shows a colour plot of the desired radiation pattern of fig5 . fig5 e and fig5 f show three - dimensional views of the desired radiation pattern of fig5 . an array layout according to prior - art configuration a and relevant radiative performance are reported in fig6 and fig7 a to 7f . fig6 shows how the reference circular aperture field of fig4 can be radially sampled ( at a period of 2λ ) to derive the excitations of a ring array of equal sized elements ( of 2λ diameter ). fig7 a and fig7 c show , respectively , a three - dimensional view and a colour plot of the aperture field of the tapered annular rings array . in fig7 b , azimuthal cuts of the radiation pattern of the tapered annular rings array of fig7 a and fig7 c is compared with the desired radiation pattern of fig4 . fig7 d shows a colour plot of the radiation pattern of the tapered annular rings array of fig7 a and fig7 c ; and fig7 e ; and fig7 f show three - dimensional views of the pattern . an array layout according to prior - art configuration b and relevant radiative performance are reported in fig8 a to 8f . fig8 a and fig8 c show , respectively , a three - dimensional view and a colour plot of the aperture field of the aperiodic array with identical radiating elements disposed on concentric circular rings . in fig8 b , azimuthal cuts of the radiation pattern of the aperiodic array with identical radiating elements of fig8 a and fig8 c is compared with the desired radiation pattern of fig4 . fig8 d shows a colour plot of the radiation pattern of the aperiodic array with identical radiating elements of fig8 a and fig8 c ; and fig8 e ; and fig8 . f show three - dimensional views of the pattern . annular ring aperiodic arrays with maximum efficiency radiating elements ( configurations c and d ) [ invention ] all the following design examples refers to the preferred assumption of equal power per element . fig1 a to fig1 f and fig1 to fig1 f relates to the synthesis of an array according to configuration c , i . e . without any constraint on the maximum radiating element dimensions . fig1 to fig2 f and fig2 to fig2 f relates to the synthesis of an array according to configuration d , i . e . with a constraint on the maximum radiating element dimensions accordingly to scan requirements . for each design example , we will report results related , first to the approximation with continuous maximum efficiency annular - rings ( fig9 a ), and then to the design with maximum efficiency circular radiating elements with element dimensions proportional to the annular ring size ( fig9 b ). fig1 a to fig1 f relates to the design choice of the annulus mid - radius accordingly to the doyle - like optimality condition on the cumulative function ( 98 ). assuming equal power per element , the boundaries p k of the elementary annuli are determined by inversion of the “ grading function ” of equation ( 97 ) on a uniform scale ( apart the first element in case of presence of the central element ). this first step is shown in fig1 a . second step of determining the maximum efficiency annuli radial centers accordingly to the cumulative function ( 98 ) is shown in fig1 b . fig1 and fig1 show , respectively , the aperture fields and the cumulative functions of the array of maximum efficiency annular rings ( in black ) and reference circular aperture ( in gray ). fig1 a shows a three - dimensional view of the aperture field of the array of maximum efficiency annular - rings . in fig1 b , azimuthal cuts of the radiation pattern of the array of maximum efficiency annular - rings of fig1 a is compared with the desired radiation pattern of fig4 . substituting the with continuous maximum efficiency annular - rings ( fig1 a ) with a set of maximum efficiency circular radiating elements with element dimensions proportional to the annular ring size we obtain the aperture field of fig1 a and the array layout of fig1 c . in fig1 b , azimuthal cuts of the radiation pattern of the annular ring arrays with maximum efficiency radiating elements of fig1 a is compared with the desired radiation pattern of fig4 ( as well as with the radiation pattern of the array of maximum efficiency annular - rings of fig1 a ). fig1 d shows a colour plot of the radiation pattern of the array of fig1 a and fig1 c ; and fig1 e ; and fig1 . f show three - dimensional views of the pattern . fig1 to fig1 f relates to the design choice of the annular ring radial centre accordingly to the elementary annulus mid - radius . fig1 and fig1 show , respectively , the aperture fields and the cumulative functions of the circular array of maximum efficiency annular rings ( in black ) and reference circular aperture ( in gray ). fig1 a shows a three - dimensional view of the aperture field of the array of maximum efficiency annular - rings . in fig1 b , azimuthal cuts of the radiation pattern of the array of maximum efficiency annular - rings of fig1 a is compared with the desired radiation pattern of fig4 . substituting the continuous maximum efficiency annular - rings ( fig1 a ) with a set of maximum efficiency circular radiating elements with element dimensions proportional to the annular ring size we obtain the aperture field of fig1 a and the array layout of fig1 c . in fig1 b , azimuthal cuts of the radiation pattern of the annular ring arrays with maximum efficiency radiating elements of fig1 a is compared with the desired radiation pattern of fig4 ( as well as with the radiation pattern of the array of maximum efficiency annular - rings of fig1 a ). fig1 d shows a colour plot of the radiation pattern of the array of fig1 a and fig1 c ; and fig1 e ; and fig1 . f show three - dimensional views of the pattern . the following design examples refers to the additional constraint of maximum element diameter of 3 . 4λ , compatible with antenna requirement of scanning of the beam over the full earth as seen from a geostationary satellite . fig1 to fig2 f relates to the design choice of the annulus mid - radius accordingly to the doyle - like optimality condition on the cumulative function ( 98 ). the first two steps of the procedure are the same of the example c1 ( refer to fig1 a and fig1 b ). imposing the additional constraint on the size of the annular ring we obtain , for the array of maximum efficiency annular rings , the aperture fields and the cumulative functions shown in fig1 and fig2 , respectively . fig2 a shows a three - dimensional view of the aperture field of the array of maximum efficiency annular - rings of example d1 . in fig2 b , azimuthal cuts of the radiation pattern of the array of maximum efficiency annular - rings of fig2 a is compared with the desired radiation pattern of fig4 . substituting the continuous maximum efficiency annular - rings ( fig2 a ) with a set of maximum efficiency circular radiating elements with element dimensions proportional to the annular ring size we obtain the aperture field of fig2 a and the array layout of fig2 c . in fig2 b , azimuthal cuts of the radiation pattern of the annular ring arrays with maximum efficiency radiating elements of fig2 a is compared with the desired radiation pattern of fig4 ( as well as with the radiation pattern of the array of maximum efficiency annular - rings of fig2 a ). fig2 d shows a colour plot of the radiation pattern of the array of fig2 a and fig2 c ; and fig2 e ; and fig2 . f show three - dimensional views of the pattern . fig2 to fig2 f relates to the design choice of the annulus mid - radius accordingly to the doyle - like optimality condition on the cumulative function ( 98 ). the first two steps of the procedure are the same of the example c2 ( refer to fig1 a for the first step ). imposing the additional constraint on the size of the annular ring we obtain , for the array of maximum efficiency annular rings , the aperture fields and the cumulative functions shown in fig2 and fig2 , respectively . fig2 a shows a three - dimensional view of the aperture field of the array of maximum efficiency annular - rings of example d2 . in fig2 b , azimuthal cuts of the radiation pattern of the array of maximum efficiency annular - rings of fig2 a is compared with the desired radiation pattern of fig4 . substituting the continuous maximum efficiency annular - rings ( fig2 a ) with a set of maximum efficiency circular radiating elements with element dimensions proportional to the annular ring size we obtain the aperture field of fig2 a and the array layout of fig2 c . in fig2 b , azimuthal cuts of the radiation pattern of the annular ring arrays with maximum efficiency radiating elements of fig2 a is compared with the desired radiation pattern of fig4 ( as well as with the radiation pattern of the array of maximum efficiency annular - rings of fig2 a ). fig2 d shows a colour plot of the radiation pattern of the array of fig2 a and fig2 c ; and fig2 e ; and fig2 . f show three - dimensional views of the pattern . summary table in fig2 compares array configurations known in prior art ( configuration a and b ) with the array configurations disclosed in the present invention declaration ( configuration c and d ). the obtained improvements can be quantified in the following key figure of merit : 20 - 40 % reduction in number of radiating elements ( res )/ power amplifiers ( pas ), 1 . 5 - 2 db improvement in directivity and aperture efficiency , the achieved improvements constitute a key competitive advantage of the proposed configurations in terms both of performance and complexity reduction . based on the described design methods , a preliminary design of an active array antenna has been performed . first , the geometry of an aperiodic and sparse front array able to radiate the required spot beam with a beam diameter of 1 . 75 ° and a side - lobe level lower than − 23 db has been synthesized . this front array includes 156 high - efficiency very - compact circular hornswith apertures ranging from 30 mm to 50 mm . fig2 shows the front view of the array apertures . design of maximum efficiency radiating elements has been performed and three of the eight horns are shown in fig2 . as all apertures have to be on the same plane to simplify electrical aspects , it follows that all the horns must have the same depth for mechanical reasons . fig3 shows an artistic view of the array of fig2 . fig1 a illustrates the block diagram of a generic array with amplitude tapering known in prior - art ( configuration a ). fig1 b shows a typical layout and amplitude tapering ( in gray scale ) of a generic array with amplitude tapering known in prior - art ( configuration a ). fig1 c illustrates the block diagram of a generic aperiodic array with identical radiating elements known in prior - art ( configuration b ). fig1 d shows a typical layout of a generic aperiodic array with identical radiating elements known in prior - art ( configuration b ). fig2 a and fig2 b show exemplificative block diagram and array layout , respectively , of an array accordingly to a first embodiment of the invention ( configuration . c ). fig2 c and fig2 d show exemplificative block diagram and array layout , respectively , of an array accordingly to a second embodiment of the invention ( configuration d ). fig3 a shows an exemplificative reference linear aperture field related to a linear taylor amplitude distribution law ( n bar = 3 ; sll =− 20 ). fig3 b shows an exemplificative desired radiation pattern achievable with the reference linear aperture field of fig3 a . fig3 c shows a first step of the linear design procedure consisting in determining the elementary cells by inversion of a “ grading function ” and a “ grading scale ”. fig3 d shows a possible second step of the linear design procedure consisting in determining the element centres accordingly to the cumulative curve . fig3 e and fig3 f show , respectively , the aperture fields and the cumulative functions of the linear array of maximum efficiency radiating elements ( in black ) and reference linear aperture ( in gray ). in fig3 g and fig3 h , the aperture fields and the cumulative functions , respectively , of the linear array of maximum efficiency radiating elements ( in black ) and reference linear aperture ( in gray ) are compared with the linear array of omnidirectional radiating elements ( dotted line ). fig3 i compares the radiation patterns of the linear array of maximum efficiency radiating elements accordingly to the described design method ( in black ), the reference linear aperture ( in gray ) and the linear array of omnidirectional radiating elements accordingly to prior art ( dotted line ). fig4 shows a radial cut of an exemplificative circular reference aperture relevant to a circular taylor amplitude distribution law ( n bar = 2 ; sll =− 25 ). fig5 a shows a three - dimensional view of the aperture field of a reference circular aperture ( fig4 ). fig5 b shows an azimuthal cut of the desired radiation pattern obtained with the reference aperture field of fig4 and fig5 a . fig5 . c shows colour plot of the aperture field of the reference circular aperture of of fig4 and fig5 a . fig5 d shows a colour plot of the desired radiation pattern of fig5 . fig5 e and fig5 . f show three - dimensional views of the desired radiation pattern of fig5 . fig6 shows how the reference circular aperture field of fig4 can be radially sampled ( at a period of 2λ ) to derive the excitations of a ring array of equal sized elements ( of 2λ diameter ), according to prior art ( configuration a ). fig7 a shows a three - dimensional view of the aperture field of the tapered annular rings array according to prior art ( configuration a ). in fig7 b , azimuthal cuts of the radiation pattern of the tapered annular rings array of fig7 a is compared with the desired radiation pattern of fig4 . fig7 c shows a colour plot of the aperture field of the tapered annular rings array according to prior art ( configuration a ). fig7 d shows a colour plot of the radiation pattern of the tapered annular rings array of fig7 a and fig7 c . fig7 e ; and fig7 f show three - dimensional views of the radiation pattern of the tapered annular rings array of fig7 a and fig7 c . fig8 a shows a three - dimensional view of the aperture field of an exemplificative aperiodic array with identical radiating elements disposed on concentric circular rings according to prior art ( configuration b ). in fig8 b , azimuthal cuts of the radiation pattern of the aperiodic array with identical radiating elements of fig8 a is compared with the desired radiation pattern of fig4 . fig8 c shows a colour plot view of the aperture field of an exemplificative aperiodic array with identical radiating elements disposed on concentric circular rings according to prior art ( configuration b ). fig8 d shows a colour plot of the radiation pattern of the aperiodic array with identical radiating elements of fig8 a and fig8 c . fig8 e ; and fig8 f show three - dimensional views of the radiation pattern of the exemplificative aperiodic array with identical radiating elements of fig8 a and fig8 c . fig9 a shows the basic geometry of a maximum efficiency annular ring . fig9 b shows how the continuous annular ring of fig9 a can be replaced by maximum efficiency circular radiating elements with element dimensions proportional to the annular ring size . fig1 a to fig1 f relates to the synthesis of an array according to the first embodiment of the invention ( configuration c , i . e . without any constraint on the maximum radiating element dimensions ) together with the design choice of the annular ring radial centre accordingly to the cumulative function optimality condition ( example c1 ). fig1 a shows a first step of the circular design procedure consisting in determining the elementary annuli by inversion of a “ grading function ” and a “ grading scale ”. fig1 b shows a possible second step of the circular design procedure consisting in determining the annular rings radial centres accordingly to the cumulative curve . fig1 and fig1 show , respectively , the aperture fields and the cumulative functions of the array of maximum efficiency annular rings ( in black ) and reference circular aperture ( in gray ). fig1 a shows a three - dimensional view of the aperture field of the array of maximum efficiency annular - rings . in fig1 b , azimuthal cuts of the radiation pattern of the array of maximum efficiency annular - rings of fig1 a is compared with the desired radiation pattern of fig4 . fig1 a shows the aperture field of an exemplificative array according to the first embodiment ( configuration c ) obtained substituting the continuous maximum efficiency annular - rings of fig1 a with a set of maximum efficiency circular radiating elements with element dimensions proportional to the annular ring size . in fig1 b , azimuthal cuts of the radiation pattern of the annular ring arrays with maximum efficiency radiating elements of fig1 a is compared with the desired radiation pattern of fig4 ( as well as with the radiation pattern of the array of maximum efficiency annular - rings of fig1 a ). fig1 c shows the layout of an exemplificative array according to the first embodiment ( configuration c ). fig1 d shows a colour plot of the radiation pattern of the array of fig1 a and fig1 c . fig1 e ; and fig1 f show three - dimensional views of the pattern of the array of fig1 a and fig1 c . fig1 to fig1 f relates to the synthesis of an array according to the first embodiment of the invention ( configuration c , i . e . without any constraint on the maximum radiating element dimensions ) together with the design choice of annular ring radial centre accordingly to the elementary annulus mid - radius ( example c2 ). fig1 and fig1 show , respectively , the aperture fields and the cumulative functions of the circular array of maximum efficiency annular rings ( in black ) and reference circular aperture ( in gray ). fig1 a shows a three - dimensional view of the aperture field of the array of maximum efficiency annular - rings . in fig1 b , azimuthal cuts of the radiation pattern of the array of maximum efficiency annular - rings of fig1 a is compared with the desired radiation pattern of fig4 . fig1 a shows the aperture field of an exemplificative array according to the first embodiment ( configuration c ) obtained substituting the continuous maximum efficiency annular - rings of fig1 a with a set of maximum efficiency circular radiating elements with element dimensions proportional to the annular ring size . in fig1 b , azimuthal cuts of the radiation pattern of the annular ring arrays with maximum efficiency radiating elements of fig1 a is compared with the desired radiation pattern of fig4 ( as well as with the radiation pattern of the array of maximum efficiency annular - rings of fig1 a ). fig1 c shows the layout of a second exemplificative array according to the first embodiment ( configuration d ). fig1 d shows a colour plot of the radiation pattern of the array of fig1 a and fig1 c . fig1 e ; and fig1 . f show three - dimensional views of the pattern of the array of fig1 a and fig1 c . fig1 to fig2 f relates to the synthesis of an array according to the second embodiment of the invention ( configuration d , i . e . with the additional constraint of maximum element diameter : 3 . 4λ in the example ) together with the design choice of the annular ring radial centre accordingly to the cumulative function optimality condition ( example d1 ). fig1 and fig2 show , respectively , the aperture fields and the cumulative functions of the array of maximum efficiency annular rings ( in black ) and reference circular aperture ( in gray ). fig2 a shows a three - dimensional view of the aperture field of the array of maximum efficiency annular - rings . in fig2 b , azimuthal cuts of the radiation pattern of the array of maximum efficiency annular - rings of fig2 a is compared with the desired radiation pattern of fig4 . fig2 a shows the aperture field of an exemplificative array according to the second embodiment ( configuration d ) obtained substituting the continuous maximum efficiency annular - rings of fig2 a with a set of maximum efficiency circular radiating elements with element dimensions proportional to the annular ring size . in fig2 b , azimuthal cuts of the radiation pattern of the annular ring arrays with maximum efficiency radiating elements of fig2 a is compared with the desired radiation pattern of fig4 ( as well as with the radiation pattern of the array of maximum efficiency annular - rings of fig2 a ). fig2 c shows the layout of an exemplificative array according to the second embodiment ( configuration d ). fig2 d shows a colour plot of the radiation pattern of the array of fig2 a and fig2 c . fig2 e ; and fig2 f show three - dimensional views of the pattern of the array of fig2 a and fig2 c . fig2 to fig2 f relates to the synthesis of an array according to the second embodiment of the invention ( configuration d , i . e . with the additional constraint of maximum element diameter : 3 . 4λ in the example ) together with the design choice of annular ring radial centre accordingly to the elementary annulus mid - radius ( example d2 ). fig2 and fig2 show , respectively , the aperture fields and the cumulative functions of the array of maximum efficiency annular rings ( in black ) and reference circular aperture ( in gray ). fig2 a shows a three - dimensional view of the aperture field of the array of maximum efficiency annular - rings . in fig2 b , azimuthal cuts of the radiation pattern of the array of maximum efficiency annular - rings of fig2 a is compared with the desired radiation pattern of fig4 . fig2 a shows the aperture field of an exemplificative array according to the second embodiment ( configuration d ) obtained substituting the continuous maximum efficiency annular - 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