Classifications

• • H—ELECTRICITY
  • H01—ELECTRIC ELEMENTS
  • H01Q—ANTENNAS, i.e. RADIO AERIALS
  • H01Q21/00—Antenna arrays or systems
  • H01Q21/0087—Apparatus or processes specially adapted for manufacturing antenna arrays
• • G—PHYSICS
  • G05—CONTROLLING; REGULATING
  • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
  • G05B19/00—Programme-control systems
  • G05B19/02—Programme-control systems electric
  • G05B19/18—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form
  • G05B19/4097—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by using design data to control NC machines, e.g. CAD/CAM
• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06F—ELECTRIC DIGITAL DATA PROCESSING
  • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
  • G06F17/10—Complex mathematical operations
  • G06F17/14—Fourier, Walsh or analogous domain transformations, e.g. Laplace, Hilbert, Karhunen-Loeve, transforms
• • H—ELECTRICITY
  • H01—ELECTRIC ELEMENTS
  • H01Q—ANTENNAS, i.e. RADIO AERIALS
  • H01Q21/00—Antenna arrays or systems
  • H01Q21/06—Arrays of individually energised antenna units similarly polarised and spaced apart
  • H01Q21/20—Arrays of individually energised antenna units similarly polarised and spaced apart the units being spaced along or adjacent to a curvilinear path
• • H—ELECTRICITY
  • H01—ELECTRIC ELEMENTS
  • H01Q—ANTENNAS, i.e. RADIO AERIALS
  • H01Q21/00—Antenna arrays or systems
  • H01Q21/06—Arrays of individually energised antenna units similarly polarised and spaced apart
  • H01Q21/22—Antenna units of the array energised non-uniformly in amplitude or phase, e.g. tapered array or binomial array

Definitions

• Array antennas can provide significant advantages in flexibility and versatility with respect to more conventional antennas e.g.; the array pattern may be completely reconfigured changing the complex feeding excitations (in amplitude and phase), the array excitation may be closely controlled to generate low sideiobe patterns; etc.
• Aperiodic arrays with equiamp!itude elements represent an interesting solution especially when the array has to generate pencil beams with a limited scanning range.
• the variable spacing between the elements can be used as an additional degree of freedom [2]- [22].
• a "virtual tapering" is realized piaying on the elements' positions rather than on their excitation amplitudes, i.e. by using a "density tapering" of the array elements.
• Using uniform (amplitude) excitation is very advantageous in active array antennas, especially in transmit because it allows operating the power amplifiers feeding the array elements at their point of maximum efficiency. Moreover, using uniform excitation drastically simplifies the beam- forming network and reduces the corresponding losses.
• An object of the present invention is to provide a simple, yet effective, method for designing and manufacturing array antennas by exploiting all the available degrees of freedom of the array (i.e. element positions, element dimensions, and number of elements).
• the invention allows using the element positions together with the element dimensions permitting to drastically reduce the costs and the complexity of the array antenna while maximizing the aperture filling factor and the associated aperture efficiency and directivity. Furthermore the design method of the invention is fully deterministic and analytical.
• the resulting antenna architecture is considered extremely promising for the design of arrays generating a multibeam coverage on the Earth from a geostationary satellite.
• the present invention regards the architecture and the design methodology of an innovative active array.
• the novelty of the architecture relates to the optimal joint use of elements' positions and elements' dimensions in conjunction to identical Power Amplifiers and a method to optimize it in such a way that the resulting active array satisfies the required radiative performances while guarantying the optimal efficiency of the Power Amplifiers.
• the proposed architecture realizes all the benefit of aperiodic array configurations superseding limitations of existing designs and drastically improving efficiency performances.
• the invention applies in particular to the fields of satellite communications, remote sensing and global navigation systems.
• the invention describes the design of the radiating part of an array, nevertheless it applicable to all known active or passive antenna architectures comprising an array as a sub-system of the antenna system (such as discrete lens antennas, array fed reflectors, array fed dielectric fens antennas etc.).
• the transmit and receive pattern of a radiating structure are proportional to each other. Therefore, according to the invention, an array antenna to be used in reception can by designed starting from its equivalent transmit pattern. What changes is the beam-forming network, where high-power amplifiers used in transmission are replaced by receive amplifiers.
• the "power levels" determined when synthesizing a transmit antenna correspond to the power amplification levels when the antenna is used in reception.
• a large number of design techniques known from the prior art usually consider periodic array antennas constituted by equispaced antenna elements.
• the mathematical properties associated to the periodicity simplifies the analysis and synthesis of such antennas.
• a non-exhaustive list of the most famous synthesis techniques includes: the Fourier Transform method, the Schelkunoff method, the Woodward-Lawson synthesis, the Taylor method, the Dolph-Chebyshev synthesis, the Villeneuve synthesis, etc.
• the textbook [1] and all the related bibliography may be consulted. According to these techniques, an overall number of array element and a fixed inter-element spacing are determined, and then the most appropriate excitations, in amplitude and/or in phase, in order to guarantee the required performances are identified.
• the radiation pattern of a continuous aperture (or of periodic array) with a nonuniform excitation is approximately reproduced by an aperiodic array with uniform excitation and identical radiating elements.
• These sparse array configurations with identical elements suffer from high aperture efficiency and directivity limitations due to the reduced aperture filling factor.
• each Radiating Element is fed by a High Power Amplifier (HPA). Beam signals are distributed with appropriate amplitude and phase to each HPA.
• HPA High Power Amplifier
• the array synthesis can be performed sampling a desired continuous illumination function at the positions of the discrete array elements which are defined on a periodic lattices ⁇ e.g. triangular or square lattices) or on contiguous circular rings (Fig.l B).
• the element excitation is proportional to illumination function and the amplifiers work at very different power levels (different gray scale in Fig.l B).
• the configuration shows very poor power efficiency; radiation performances are very good.
• each radiating element is fed by a High Power Amplifier (HPA). Beam signals are distributed with appropriate phase to each HPA. All the amplifiers work at the same operating point on a power pooling base.
• HPA High Power Amplifier
• the radiation pattern of a continuous aperture is approximately reproduced by an aperiodic array with uniform excitation and identical radiating elements (e.g. Fig.1 D).
• HPAs High Power Amplifiers
• FIG.2A An exemplificative block diagram of the embodiment of configuration C is shown in Fig.2A and Fig.2B.
• the element positions are used together with the element dimensions permitting to maximize the aperture filling factor and the associated aperture efficiency and directivity.
• An exemplificative block diagram of the embodiment of configuration D is shown in Fig.2C and Fig.2D.
• the element positions are used together with the element dimensions with an additional constraint on the maximum element dimensions accordingly to the beam scanning requirements.
• the aperture filling factor and the associated aperture efficiency and directivity are optimized compatibly with the scanning requirement constraints.
• 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 side!obe level (SLL) are indicated.
• G Gain
• BW beamwidth
• SLL peak side!obe level
• the reference aperture 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.
• 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 / 0 (x) , with the radiation pattern F 3 (u) of a discrete aperiodic array with amplitude field i a (x) .
• a 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.
• W-MSE Weighted Mean Square Error
• Equation (12) ⁇ ⁇ l'° x) - ( ) l 2 dx
• W-MSE weighted mean square error
• MSE mean square error
• Equations (15) and (16) provide a direct mean to derive the boundaries of the elementary cells by inversion of the reference cumulative curve / 0 (x) :
• the element position can be then determined inverting the following equation [6][7][18]
• Equation (17) can be reinterpreted as,
• maximum efficiency condition refers to the almost constancy, on the radiating element aperture, of the eiectromagnetic field generated exciting the radiating element itself.
• 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) ,
• the synthesis can be performed determining, in a first step, the boundaries t of the elementary cells by inversion of the "grading function"
• x k in a second step the / -th element positions, x k ,can be determined accordingly either to a Doyle-like optimaiity condition (19) i.e. x k is such that, or simply considering x k the centre of the elementary cell,
• the k-t element dimensions, Ax k can be determined such to contemporarily fulfil the cumulative field distribution approximation condition,
• the aperture field amplitude coefficients b k can be derived from equation (40) using ⁇ from equation (43),
• An aperture function with circular symmetry can be expressed as:
• the far field radiation pattern generated by a circular symmetric field distribution i(p) varying only in a radial direction p is given by a Hankel transform of order zero of the field distribution i(p) :
• 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]:
• Equation (59) represents a weighted mean square error of the patterns with an inverse quadratic weighting function.
• the right term is a weighted mean square error of the radial cumulative functions of the circular field distributions.
• 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,
• the resulting optimality condition correspond to a partitioning of the reference aperture field in elementary cells with assigned power levels, ⁇ ⁇ 2 ⁇ ) ⁇ ⁇ * 2 ⁇ ⁇ 2 ⁇ , ) ⁇ , ⁇ ⁇ (76)
• pil ⁇ p acts as derivative of the "grading function" and p k as "grading scale”.
• the annular ring mid-radius, r ⁇ .can be determined accordingly to the second step described in previous paragraph.
• the k-t annular ring radial dimensions, Ar k are determined such to contemporarily fulfil the cumulative field distribution approximation condition (equation (80)), assigned power condition (equations (66), (67) and (85)), br Ar k Po (88) and the feasibility condition,
• the amplitude coefficients b k can be derived from equation
• each annular ring will exhibit a stepwise constant radial profile similar to an aperture with maximum efficiency annular- rings (refer to Fig.9A) 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.
• 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 radiating element can be constituted sub-arrays of smaller radiating elements.
• Circular radiating elements of diameter d k proportional to the annular ring radial dimensions, Ar k , allows a minimisation of the unused area between annular rings.
• the area of the radiating element becomes,
• N k 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),
• i Q (p) acts as derivative of the "grading function" and Jq ⁇ as "grading scale".
• r k the / -th annular ring mid-radius, r k ,can be determined accordingly either to a Doyle-like optimality condition (19) i.e. r k is
• N k (101 ) where the symbols [ J 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.
• grading scale will be:
• Ar k min(Ar A max , Ar max ) (104)
• Equation (96) 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-t ring. Nevertheless it must be understood that elements diameters comparable with the elementary annu!i 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.
• Fig.3E and Fig.3F 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).
• Fig.5A and Fig.5C show, respectively, a three-dimensional view and a colour plot of the aperture field of the reference circular aperture.
• Fig.SD shows a colour plot of the desired radiation pattern
• Fig.SE and Fig.5F show three-dimensional views of the desired radiation pattern of Fig.5.
• Fig.6 shows how the reference circular aperture field of Fig.4 can be radially sampled (at a period of 2 ⁇ ) to derive the excitations of a ring array of equal sized elements (of 2 ⁇ diameter).
• Fig.7A and FigJC show, respectively, a three-dimensional view and a colour plot of the aperture field of the Tapered Annular Rings Array.
• Fig.7D shows a colour plot of the radiation pattern of the Tapered Annular Rings Array of Fig.7A and FigJC; and FigJE; and Fig.7F show three-dimensional views of the pattern.
• Aperiodic Array with Identical Radiating Elements (Configuration B) [prior art]
• Fig.8A and Fig.8C 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.
• Fig.8D shows a colour plot of the radiation pattern of the Aperiodic Array with Identical Radiating Elements of Fig.8A and Fig.8C; and Fig.8E; and Fig.8.F show three-dimensional views of the pattern.
• Annular Ring Aperiodic Arrays with Maximum Efficiency Radiating Elements (Configurations C and D) [Invention]
• Fig.lOA to Fig.14F and Fig. 5 to Fig.18F relates to the synthesis of an array according to Configuration C, i.e. without any constraint on the maximum radiating element dimensions.
• Fig.19 to Fig.22F and Fig.23 to Fig.26F 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.
• Fig.10A to Fig.14F relates to the design choice of the annulus mid-radius accordingly to the Doyle-like optimaiity condition on the cumulative function (98).
• Fig.11 and Fig. 2 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).
• Fig.13A shows a three-dimensional view of the aperture field of the Array of Maximum Efficiency Annular-Rings.
• Fig.14D shows a colour plot of the radiation pattern of the array of Fig.14A and Fig.14C; and Fig.14E; and Fig.14. F show three- dimensional views of the pattern.
• Example C2 (Unconstrained Size - Mid-Centre) [Invention] Fig.15 to Fig.18F relates to the design choice of the annular ring radial centre accordingly to the elementary annulus mid-radius.
• Fig, 15 and Fig.16 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).
• Fig.17A shows a three-dimensional view of the aperture field of the Array of Maximum Efficiency Annuiar-Rings.
• Annular Ring Arrays with Maximum Efficiency Radiating Elements of Fig.18A is compared with the desired radiation pattern of Fig.4 (as well as with the radiation pattern of the Array of Maximum Efficiency Annular-Rings of Fig.17A).
• Fig.18D shows a colour plot of the radiation pattern of the array of Fig.18A and Fig.18C; and Fig.18E; and Fig.18. F show three- dimensional views of the pattern.
• the following design examples refers to the additional constraint of maximum element diameter of 3.4 ⁇ , compatible with antenna requirement of scanning of the beam over the full Earth as seen from a geostationary satellite.
• Fig.19 to Fig.21 F relates to the design choice of the annulus mid-radius accordingly to the Doyle-like optimality condition on the cumulative function (98).
• Example C1 (refer to Fig.l OA and Fig. OB).
• Fig.2 A shows a three-dimensional view of the aperture field of the Array of Maximum Efficiency Annular-Rings of Example D1.
• Fig.21 B azimuthal cuts of the radiation pattern of the Annular Ring Arrays with Maximum Efficiency Radiating Elements of Fig.21A is compared with the desired radiation pattern of Fig.4 (as well as with the radiation pattern of the Array of Maximum Efficiency Annular-Rings of
• Fig.23 to Fig.26F relates to the design choice of the annulus mid-radius accordingly to the Doyle-iike optimaiity condition on the cumulative function (98).
• Example C2 (refer to Fig.10A for the first step).
• Fig.26D shows a colour plot of the radiation pattern of the array of Fig.26A and Fig,26C; and Fig.26E; and Fig.26.F show three- dimensional views of the pattern.
• This front array includes 156 high-efficiency very-compact circular hornswith apertures ranging from 30mm to 50mm.
• Fig.28 shows the front view of the array apertures.
• Fig.lA illustrates the block diagram of a generic array with amplitude tapering known in prior-art (Configuration A).
• Fig.l B shows a typical layout and amplitude tapering (in gray scale) of a generic array with amplitude tapering known in prior-art (Configuration A).
• Fig.l C illustrates the block diagram of a generic Aperiodic Array with Identical Radiating Elements known in prior-art (Configuration B).
• Fig.3B shows an exemplificative desired radiation pattern achievable with the reference linear aperture field of Fig.3A.
• Fig.3C shows a first step of the linear design procedure consisting in determining the elementary ceils by inversion of a "grading function" and a “grading scale”.
• Fig.3D shows a possible second step of the linear design procedure consisting in determining the element centres accordingly to the cumulative curve.
• Fig.3E and Fig.3F 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).
• Fig.3G and Fig.SH 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).
• Fig.5A shows a three-dimensional view of the aperture field of a reference circular aperture (Fig.4).
• Fig.6 shows how the reference circular aperture field of Fig.4 can be radially sampled (at a period of 2 ⁇ ) to derive the excitations of a ring array of equal sized elements (of 2 ⁇ diameter), according to prior art (Configuration A).
• Fig.8C 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).
• Fig.9A shows the basic geometry of a Maximum Efficiency Annular Ring.
• Fig.10A to Fig.14F 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 ).
• Fig.10A 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”.
• Fig.10B shows a possible second step of the circular design procedure consisting in determining the annular rings radial centres accordingly to the cumulative curve.
• Fig.13A shows a three-dimensional view of the aperture field of the Array of Maximum Efficiency Annular-Rings.
• Fig.14D shows a colour plot of the radiation pattern of the array of Fig.14A and Fig.14C.
• Fig. 15 to Fig.18F 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).
• Fig.15 and Fig.16 show, respectively, the aperture fields and the cumuiative functions of the circular array of maximum efficiency annular rings (in black) and reference circular aperture (in gray).
• Fig.17A shows a three-dimensional view of the aperture field of the Array of Maximum Efficiency Annular-Rings.
• Fig.18B azimuthal cuts of the radiation pattern of the Annular Ring Arrays with Maximum Efficiency Radiating Elements of Fig.18A is compared with the desired radiation pattern of Fig.4 (as well as with the radiation pattern of the Array of Maximum Efficiency Annular-Rings of Fig.17A).
• Fig.18D shows a colour plot of the radiation pattern of the array of Fig.18A and Fig.18C.
• Fig. 19 to Fig.2 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: 3.4 ⁇ in the example) together with the design choice of the annular ring radial centre accordingly to the cumulative function optimality condition (Example D1 ).
• Fig.19 and Fig.20 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).
• Fig.21A shows a three-dimensional view of the aperture field of the Array of Maximum Efficiency Annular-Rings.
• Fig.22A shows the aperture field of an exemplificative array according to the second embodiment (Configuration D) obtained substituting the continuous maximum efficiency annular-rings of Fig.21A with a set of maximum efficiency circular radiating elements with element dimensions proportional to the annular ring size.
• Fig.22B azimuthal cuts of the radiation pattern of the Annular Ring Arrays with Maximum Efficiency Radiating Elements of Fig.22A is compared with the desired radiation pattern of Fig.4 (as well as with the radiation pattern of the Array of Maximum Efficiency Annular-Rings of Fig.21A).
• Fig.22C shows the layout of an exemplificative array according to the second embodiment ⁇ Configuration D).
• Fig,22E; and Fig.22F show three-dimensional views of the pattern of the array of Fig.22A and Fig.22C.
• Fig. 23 to Fig.26F 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: 3.4 ⁇ in the example) together with the design choice of annular ring radial centre accordingly to the elementary annu!us mid-radius (Example D2).
• Fig.23 and Fig.24 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).
• Fig.25A shows a three-dimensional view of the aperture field of the Array of Maximum Efficiency Annular-Rings.
• Fig.25B azimuthal cuts of the radiation pattern of the Array of Maximum Efficiency Annular-Rings of Fig.25A is compared with the desired radiation pattern of Fig.4.
• Fig.26A shows the aperture field of an exemplificative array according to the second embodiment (Configuration D) obtained substituting the continuous maximum efficiency annular-rings of Fig.25A with a set of maximum efficiency circular radiating elements with element dimensions proportional to the annular ring size.
• Fig,26B azimuthal cuts of the radiation pattern of the Annular Ring Arrays with Maximum Efficiency Radiating Elements of Fig.26A is compared with the desired radiation pattern of Fig.4 (as well as with the radiation pattern of the Array of Maximum Efficiency Annular-Rings of Fig.25A).
• Fig.26C shows the layout of an exemplificative array according to the second embodiment (Configuration D).
• Fig.26D shows a colour plot of the radiation pattern of the array of Fig.26A and Fig.26C.
• Fig.26E; and Fig.26F show three-dimensional views of the pattern of the array of Fig.26A and Fig.26C.
• Fig.27 summarizes performances of the reported design examples.
• Fig.28 shows the front view of an array, designed accordingly to the described design methods, required to radiate spot beams with a beam diameter of .75° and a side-lobe level lower than -23 dB.
• the array includes 156 high-efficiency very-compact circular hornswith apertures ranging from 30mm to 50mm.
• Fig.29 reports the exemplificative design of three of the eight horns with Maximum Efficiency as required for the array of Fig.28. 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.
• T. T. Taylor Design of Line Source Antennas for Narrow Beamwidth and Low Sideiobes", IEEE Transactions on Antennas and Propagation, Vol. 3, pp. 16-28, January 1955.
• T. T. Taylor “Design of Circular Apertures for Narrow Beamwidth and Low Sidelobe", IRE Trans, on Antennas and Propagation, Vol. AP-8 pp. 17-22, 1960

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