# Orthogonal Methods for Array Synthesis: Theory and the ORAMA Computer Tool They present all the necessary concepts, and set the context for the reader to follow the development of the orthogonal methods in Chapters 4 and 5. It is, in a way, a user's manual for the software. The software offers a menu-driven and user-friendly graphical Windows interface. The package was installed without problems in a few minutes on a laptop running Windows XP. The reviewer concludes that this book could prove to be an indispensable addition for the library of an antenna engineer. The first three chapters are also self contained, and suitable for a compact review of antenna arrays for the industry practitioner, and as an introduction for the newcomer.

The software package that accompanies the book is valuable on its own for its breadth and speed, and could be used as is for several applications of linear antenna arrays. The software is very quick, and it could also serve as the basis of an antenna CAD laboratory course, using the design examples as ready-made problems.

Article :. Date of Publication: Dec. First Page of the Article. During the s, when short waves became popular, the need for antenna arrays was indisputable. It would be useful to go over some of the studies during the last six decades before we come to the discussion of orthogonal methods. Wolff  was a pioneer of the synthesis of uniformly distributed linear arrays with circularly symmetric patterns. His technique was based on the Fourier series expansion. In , Schelkunoff  utilized the relation between the roots of a polynomial on the complex plane and the nulls of the radiation pattern.

Dolph  and Riblet  used the Chebyshev polynomials and offered a control on the side lobe level SLL of the pattern. Woodward and Lawson [5, 6], Van der Maas  and Taylor [8, 9] applied the method of matching the pattern of a continuous source to a certain number of directions or to the ideal Chebyshev factor. Cheng and Ma  used the sampling of a line source and the Z-transforms.

The above-mentioned sample of studies and most of the ones described in the previous chapters assume that the spacing of array elements is uniform. In , Unz [11—13] proposed a synthesis of non-uniformly spaced arrays. He formulated the relation between the element excitations and the complex Fourier expansion of the radiation pattern in a matrix form. Non-uniformly spaced arrays offer more degrees of freedom than uniform arrays of the same number of elements, and, obviously, they are expected to have better performance.

King, Packard and Thomas  and DuHamel and Chadwick  presented non-uniform arrays for broadband applications. Harrington  used a perturbation technique to reduce the size of the side lobes of linear arrays. Also, arrays with uniform excitation and non-uniform spacing have been designed by Pokrovskii  and Brown .

In , Unz  and Lo  applied a formulation with which they transformed a continuous distribution into a nonuniformly spaced array. Since , Sahalos and his colleagues [25—50] have extended and generalized the orthogonal methods for array synthesis. Antenna arrays contain, in general, arbitrarily oriented and non-uniformly spaced elements. Figure 4. If the element positions are pre-assigned and the desired radiation pattern is given, the problem of synthesis is with the determination of the excitations In of the elements.

This is a problem that has been analysed by Kantorovich and Krylov . By solving the Equation , we can derive the excitations In. If the matrix of the system Equation is singular, then the inverse does not exist. In , there is an interesting discussion on the stability of the matrix inversion problems in antennas.

Suppose that the desired array factor AF u is given. The procedure has been discussed in detail by Unz , Uzkov  and Kantorovich and Krylov . The accuracy of the approximation depends on the number and the position of the array elements. Calculation of the excitations Ii of the array elements. In the literature, there are several interesting studies about the well- and ill-conditioned problems in conjunction with the Gram-Schmidt procedure. The Orthogonal Methods 71 4. Linear arrays are of great importance to antenna designers.

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For an array to present maximum directivity, AF u must be an impulse function in the direction in which the maximum of the emission is sought. Results showing maximum broadside directivity versus the element spacing of a uniformly spaced array are presented in Fig. The results are in close agreement with the graphs of .

It can be seen in Fig. The results follow the same behaviour as the one given in . Figures 4. Also, for most angles, directivity improves only by a small value as the size of the array increases. In Chapter 2, an analysis was given of the design possibilities of Chebyshev patterns. Our aim here is to give some useful numerical procedures that are helpful in the orthogonal method. Now, if the desired array factor AF x is a Chebyshev polynomial of the form TL x , then Equation is an equation without any approximation.

It is noticed that Equations and can be extended for 2-D and 3-D arrays. Next, some interesting results are presented. It is observed that the pattern in Fig. An example is given in Fig. The above given values of x1 , x2 and x3 are not the only ones. One could select a lot of suitable sets that follow certain constraints. The Orthogonal Methods 79 Figure 4. Also, in Table 4. It is obvious that the orthogonal method can give solutions to Chebyshev arrays with different orders and the same number of elements.

We choose x1 , x2 The Orthogonal Methods 81 Figure 4. Finally, we move on to an intermediate array. The orthogonal method can be applied for patterns other than T5 x. Chebyshev polynomial is T7 x instead of T5 x Figure 4. The T5 x polynomial is used For an array with an even number of elements, one can use the orthogonal method for any of the above cases. For a smaller number, we increase the inter-element distance. In Table 2. Dolph-Chebyshev Arrays In the Dolph formulation, it is usually desirable to have broadside arrays. If it is desirable to have a T11 x pattern with the same elements as above and the same SLL, it is possible to apply the orthogonal method.

In fact, the T11 x pattern needs 12 elements. It is observed that the patterns are approximately the same. In Table 4. Table 4. The positions are given in Table 4. As it is observed, the pattern follows the desired constraints. The Orthogonal Methods 91 Table 4. An example is given in Figs.

Three different arrays with the same size of 8. The parameter a is found by Equation The pulses can have the desired amplitudes in certain angular regions. The Orthogonal Methods 95 a b Figure 4. Three different cases will be presented here. The orthogonal method is used in combination with the mean-square-error MSE criterion and the Lagrange technique. It is well known that a useful criterion in array synthesis is the minimization of MSE. Let us assume a desired array factor AF d u that is achieved by a linear array see Fig.

We proceed now to synthesis under constraints. It is desirable to minimize the MSE given in Equation , subject to the constraints that fm. It is desirable to minimize L with respect to Equation It is interesting to give some typical examples. To design the array, we have to null the pattern in the above directions. It is obvious that the pattern follows the desired constraints. Obviously, it is shown that the pattern follows the constraints with a small increase in SLL. It is obvious that nulls follow the constraints but the SLLs increase.

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In the same array, one can have null constraints at the same time by combining the methods given above. It is noticed that n is the excitation amplitude given by the orthogonal method without constraints. It is obvious that the pattern follows the constraint. Again, one can combine all the above-mentioned constraints in one array at the same time. In geometry design, the common characteristic is the adjustment of the spacing among the array elements according to certain optimization criteria. Moffet , Ismail and Dawoud , Ng  and Ng et al.

Moreover, recently, in [38, 44], the orthogonal method for the synthesis of uniformly excited arrays has been given. The basic idea has come from the re-forming of the geometry of an initial array, which is perturbed in such a way so as to approximate the desired pattern.

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## Orthogonal Methods for Array Synthesis: Theory and the ORAMA Computer Tool - PDF Free Download

In the present section, a procedure that combines an iterative technique with the orthogonal method will be given. The initial array can be a uniform or a non-uniform one. It is obvious that, in general, the right non-uniform initial array can give better results than those of a uniform one. We must also say that, in practice, it is preferable to use quantized amplitude excitations. The array can be arranged by certain quantized amplitudes, which come from the corresponding continuous ones.

The number of quantized amplitudes depends on the desired pattern and could be up to the number of the array elements. Most of them use statistical thinning to control the pattern side lobes. Suppose that we have an N -element linear array see Fig. In both factors, the determination of Ii and Ai can be achieved by applying the orthogonal method. If, instead of Ii , we use quantized approximate values, an error to the resultant pattern will occur.

The error depends on the approximation error. It must be pointed out that quantization of the amplitudes offers easier implementation of the arrays. An array with quantized amplitudes can be used as the initial one. It is noticed that the above procedure does not give a successful outcome for any number of quantized amplitudes. The success of the whole issue comes from a suitable selection of the initial array.

It must be pointed out that one can use several algorithms. If at least two of the normalized amplitudes are very close to each other in a margin around 0.

## Download Orthogonal Methods For Array Synthesis Theory And The Orama Computer Tool 2006

Otherwise, their values remain unchanged. The quantized values are the roundup integers of the above amplitudes. If their number is less than L, then we increase the multiplication factor by 1 and the procedure is repeated until at least L quantized values are found. If L is equal to 1, then, no further calculations are needed and all the quantized amplitudes are assigned the value of 1.

If L is equal to 2, then the two quantized amplitudes are the maximum ones and the mean integer value of all the other values. If L is more than 2 then, the next value is found in the same way by calculating the mean integer value of all the other amplitudes. Each one of the amplitudes is approximated by the integer that is closest to its real value from the set of integer values found.

If more than one integer value is equally close to the amplitude, then the larger integer is selected. Three quantized amplitudes are chosen. It is obvious that the solution with the above quantized amplitudes is acceptable. It is understandable that the increased number of quantized amplitudes comes from the type of constraints of the array factor. It is noticed that the same pattern can be realized with only two non-quantized amplitudes. The Orthogonal Methods Figure 4.

The results are shown in Fig. The pattern is given for f0 and 0. Obviously, it is 9. The accuracy of the approximation depends on the desired array factor, the number and the position of the array elements. The pattern of the same array with a Dolph-Chebyshev desired array factor is presented in Fig. It is noticed that the projection of the array at the axis normal to the maximum direction shows nine elements.

The pattern of the same array with a Chebyshev case 1 desired factor is presented in Fig. Here, a T4 x is used with the same mean distance as above. It is obvious that the orthogonal method can give the desired pattern. This is important because it proves that the synthesis problem is not ill conditioned because of the basis of the vector space. For our set of complex functions, it is enough to prove that the Gramian  of the set is different than zero.

This is a real symmetric matrix that can be used to test the linear independence of functions. So, we see that every array is of the same Gramian form. This can be proven. The most usual planar arrays are the rectangular ones. Because of the increase of the variables, the pattern can be controlled and scanned at any point in space. Let us refer to Fig. Let us assume that N isotropic elements are spaced on a circular ring of radius R see Fig.

It is noticed that the present case is different than the case given in Section 4. By adding up the contributions from the dipole and its images, we can derive the total radiation pattern. It is obvious that the array is a circular one. It is noticed that the real pattern is the one in front of the corner. By taking the images into account, we calculate the excitations as above. Another interesting array is a cylindrical one. Parallel circular arrays with their centres on the same axis perform a cylindrical array. This is a special kind of a 3-D array. A linear array in front of a corner with its axis parallel to the edge is a virtual cylindrical array.

Such an array is extremely useful for applications in radio and mobile communications. The excitation can be uniform or not depending on the application. If there are additional constraints on the SLL, then the excitations can be non-uniform on the z-axis. It is obvious that maximum freedom is achieved when no restriction regarding either the location or the orientation of the elements is imposed.

In this section, the design of an array consisting of not only arbitrarily positioned but also arbitrarily oriented dipoles will be presented. These are the directional coordinates that will specify the radiation properties of each dipole. The synthesis follows the steps given above. The only difference is that the determination of the position of the dipoles is also supplemented by their direction. As an example, consider the case of two dipoles normal to each other. Another example involves an antenna array that provides maximum directivity. It is obvious that one can follow a procedure similar to the one given above for arrays with elements other than dipoles.

Here, suppose that we have a wire structure Fig. The currents and the The Orthogonal Methods Figure 4. The resulting patterns for both spacings are shown in Fig. If the mutual coupling is not taken into account, then, the relative input voltages are the same as the currents. In this case, the real patterns uncorrected are different than the desired ones and are also given in Fig.

From Figure 4. By considering the uncorrected patterns, one can see that, for small distances among the elements of the array, the difference between the uncorrected and the desired ones is large. This means that it is almost impossible for the same feed voltages in point source and wire antenna array to produce the same patterns. The array is presented in Fig. The wire-grid model for the antenna structure is used, and the orthogonal method is applied. Thus, the image theory cannot be applied.

Because of the constraints on high gain and low SLL, the array in each sector is designed by the orthogonal method as a Chebyshev one. Each sector contains nine folded dipoles. The geometry of one dipole positioned in front of a mast and its wire-grid model are given in Fig. The geometry of the three sectors with nine folded collinear dipoles, each in front of a common mast, is presented in Fig. The input impedance of the dipoles in each sector and the corresponding input currents are given in Table 4.

In Figs. Examples for such a case can be found in [67, 68]. In such a case, the radiation pattern is found as the product of the element pattern and the array factor. Let us suppose a linear array with parallel-uncoupled elements along the z-axis Fig. The pattern of the antenna array can be expressed as follows. This is a straightforward task, and the orthogonal method for dealing with it has already been presented. It is obvious that the radiation properties of a real antenna array cannot be derived by the uncoupling consideration.

The divergence is evident in signal-processing arrays, which are extremely sensitive to small errors due to the non-linear processing procedures. Under the coupling conditions, the pattern of a linear antenna array can be expressed as follows. This pattern is actually the antenna pattern when only the nth element is excited and the rest are The Orthogonal Methods Figure 4. The coupling between the mth and the nth element can be presented by the equivalent excitation current Cnm. This pattern, as it was mentioned earlier, is the antenna pattern when only the nth element is excited and the rest are short-circuited.

The above formulation can be extended to 2-D and 3-D arrays as well. The results for three different inter-element distances 0. It is obvious that the patterns change as the inter-element distance decreases. It is obvious that the results taken by the present method are closer to the desired ones. The patterns of the elements of the array, on a plane perpendicular to the dipoles, are shown in Figs.

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This array is the same as that given in . The results are presented in Figs. It is obvious that the orthogonal compensation method for small inter-element distance works better than the Fourier method. From the above given examples, it is evident that the orthogonal decomposition method can give acceptable design solutions for the antenna arrays. However, Fig. Several approaches available for the synthesis of non-planar arrays could be applied to conformal ones.

The projection method given in  and the generalized one of Bucci et al. Also, the least mean-square-error LMSE method in a matrix inversion form, as well as in the form of the orthogonal method, offers elegant synthesis solutions [67, 74]. Finally, adaptive and non-deterministic optimization techniques could be used with success [75—77]. Arrays conformed to curved platforms are, in many cases, useful in air- and spaceborne vehicles. They have the characteristic of maintaining the aerodynamic integrity of the air face.

Such arrays are dictated by the geometry of the supporting structure and have to meet the EM Electromagnetic requirements. One of the powerful techniques for the synthesis is the orthogonal method. In the next paragraphs, three different cases of conformal arrays will be presented. One involves microstrip cylindrical arrays, the other involves slot arrays on a perfectly conducting elliptic cone and the third one involves slot arrays on a perfectly conducting paraboloid.

These tools must be able to analyse the whole system, including mutual coupling, and to synthesize the array. The Orthogonal Methods F a b Figure 4. Several applications on aircrafts, missiles, multi-target acquisition, satellites, mobile communications, remote sensing and biomedical systems can be adopted for the above arrays.

Of course, such a choice poses fundamental issues that must be addressed. Microstrip antennas can be easily conformed to objects with a curved surface. Depending on the radius of curvature, different theoretical approaches can be applied. In the case of cylindrical conformal patch arrays, the cavity model and the surface current model can be used.

In the analysis that follows, the two models of the RMSAs are given. Cavity Model In the cavity model, one postulates that magnetic walls enclose the dielectric volume under the microstrip patch. In this way, a cavity is formed. The usual assumption is that because of small height, w, of the cavity, only transverse magnetic TM modes to a The Orthogonal Methods F a F b Figure 4.

The rectangular patch is assumed to consist of two axial and two circumferential slots. The Orthogonal Methods Surface Current Model In the surface current model, the key assumption is that the patch metallization is replaced by a surface electric current density. The size Figure 4. The frequency of operation is 10 GHz, while layer thickness and relative dielectric constant are 1.

The cylinder radius is 0. This is illustrated in Fig. The MSA is the square of the Both the cavity and the surface current models are used to verify the numerical results, and the pattern is presented in Fig. A comparison between the results of both models has shown that there is acceptable agreement.

We start with the assumption that we have an N-element RMSA array that is conformed on a dielectric coated circular cylinder. A complete design procedure optimization scheme for an antenna array is illustrated in Fig. In this, the system parameters are found by using an optimization scheme, which can be a deterministic say a gradient-based one or even a stochastic one.

Next, the design of an antenna system that consists of probe feed rectangular patches on a dielectric coated circular cylindrical conductor is given. The antenna geometry can be a multi-ring conformal array. In the examples, Chebyshev polynomials are used as the desired patterns because their maximum and the SLL can be controlled. To have a smaller ratio, a larger SLL can be imposed. The radiation pattern and the excitation of the array are given in Figs. Finally, for the vertical pattern, a similar or a simpler procedure can be applied.

The orthogonal method can also be applied for the design of arrays with scanning characteristics. In this case, the maximum of the pattern will be in different directions in different instances. The geometry is shown in Fig. However, because of the radiation characteristics of the RMSA, the grating lobes do not give the acceptable solutions in all cases. So, an approach could be to use more say 69 RMSAs in one ring, from which only 11 are excited. In this case, a multiplexer will control the excitation. The elliptic cone is an interesting geometry in antenna engineering The Orthogonal Methods Figure 4.

The problem of the synthesis of slot arrays conformed on a perfectly conducting elliptic cone shows an increased complexity. The same problem on a circular cone was rarely addressed in the past [87—89]. The present synthesis is based on a rigorous full-wave analysis. The latter are derived from the solution of the wave equation in the spheroconal coordinate system, where the elliptic cone is one of the coordinate surfaces. The subscript 1 or 2 corresponds to Dirichlet or Neumann boundary conditions, respectively, while the superscripts and represent a standing and an outward propagating wave respectively.

In this cone, a typical diagram of the Neumann and Dirichlet eigenvalues is given in Fig. Implementation of the presented theory leads to the analysis of slot arrays positioned on the cone. The centres of both slots reside on the xz plane see Fig. The coupling level in the circumferential case is higher than the one in the radial case.

Both slots exhibit similar behaviour. As the distance of the slot from a b Figure 4. The pattern of a radial slot obeys a cosine function in the same region, even at a small distance from the tip. The above results are qualitatively similar to the ones in  for a circular cone, where the existence of an omnidirectional pattern and a cosine pattern in the lit region is observed.

The nth weight, Wn , in [W ] is the respective complex amplitude-excitation. By employing the Gram-Schmidt procedure, the orthogonal method creates an orthogonal basis. The inclusion of the coupling parameters depends on how Equation is constructed. For the design examples that follow, each of the N slots that comprise the array is fed by its own waveguide.

These factors can be trivially generalized even if modes more than the dominant ones are allowed to exist in the feeding waveguides. Each slot is considered to be connected to an RF chain consisting of a precision attenuator and a precision phase shifter or, alternatively, of a vector modulator. It is assumed that coupling between the slots comes from the space where they radiate. Two different arrays will be presented. One is composed of circumferential Fig. As expected, in the radial case, the HPBW is smaller than that of the circumferential one. For an array with strict requirements low SLL, small HPBW and scanning capability , non-uniform excitation is needed, and this will be provided by the orthogonal method.

The Orthogonal Methods a b c d Figure 4. The radiation pattern of a T20 Chebyshev array Fig. The pattern of a T19 Chebyshev Fig. Slot arrays positioned on a perfectly conducting paraboloid will be presented. Mutual coupling between two slots positioned on the paraboloid is derived and the UTD-OM analysis-synthesis technique is introduced for the design of an array. It is well known that UTD is an asymptotic technique that can be easily implemented with relatively small requirements on computational recourses.

However, UTD is problematic when it is treated in caustics and paraxial regions [78, 79]. In the case of a paraboloid Fig. One of the main steps in the process is the determination of the geodesic paths on the parabolic surface. The key step in computing the geodesics is the determination of the shadow and light separatrix.

In fact, the separatrix is the locus of the shadow boundary. If the positions of the source and the receiver are given, the separatrix contains all possible detachment points Fig. A ray follows a geodesic path on the surface, connecting the source with the detachment point. It is then transmitted to the receiver along a straight line that coincides with the direction of the tangent to the surface at the detachment point. From the rays reaching the separatrix, only one ray in general, and in some cases up to three, will be directed to the receiver.

It is obvious that there must be an angle The Orthogonal Methods Figure 4. In the present case, the concept of equivalent currents is adopted . Correction of caustics is made in order to study radiation from slots positioned on convex surfaces. The expression of this contribution is given in the form of a line integral over the separatrix.

Initially, the source produces equivalent magnetic currents on the separatrix through the mechanism of the creeping waves. R is the distance between a point on the separatrix and the receiver. The parameter of the paraboloid is set at unity for the radiation pattern calculations hereafter. Away from the caustic, the results coincide. It is obvious that an equivalent current approach does not show any divergence in the caustic region. This is important for the diffraction problem in curved surfaces.

Shadow region — one ray Lit region — one ray Shadow region — one ray — shadow region of the caustic Shadow region — three rays — lit region of the caustic Shadow region — three rays In the following text, results of the mutual coupling between two slots positioned on a perfectly conducting paraboloid is presented. See Fig. The computed results are presented in Fig. It is assumed that each slot is fed by a rectangular waveguide in the dominant vector mode. The magnetic current modes are used to model the slot radiators.

Following the same procedure as in the previous section, we can apply the orthogonal method to derive the complex amplitudes of the current modes. The slot array under study is given in Fig. The array consists of 21 circumferential slots. The Orthogonal Methods The radiation patterns with different directions of maximum are presented in Figs.

The above method is a useful tool to synthesize arrays conformal to convex surfaces with strict radiation pattern requirements. References  I. E, Vol. Woodward and J. Memo, No. Cheng and M. AP-8, No.

## Telecommunications

Series No. AP-8, pp. King, R. Packard and R. DuHamel and G. Chadwick, Frequency Independent Antennas, eds. AP-9, pp. Bruce and H. AP, p. Sahalos, K. Melidis and S. Sahalos and H. Annals Fac. Thessaloniki, Vol. Melidis and H.

Siakavara and J. Samaras and J. Miaris and J. Tsironas, T. Miaris, H. Margaritis, S. Goudos and J. Kaifas, C. Goudos, G. Miaris, S. Goudos, Chr. Iakovidis, E. The Orthogonal Methods  T. Kaifas, T. Samaras, E. Samaras, K. AP- 53, pp. Kantorovich and V. Derusso, R. Roy and C.

Ismail and M. NG, M. ER, and C. Numazaki, S. Mano, T. Kategi and M. Mailloux and E. Steyskal and J. Bucci, G. Bucci and G. Sureau and K. Olsen and R. Pathak, N. Wang, W. Burnside and R. Pathak and N. Blume and U. Blume, L. Klinkenbusch and U. Blume and V. Villeneuve, M. Behnke and W. Thiele and C. Balzano and T. Sahalos 5. ORAMA has been designed to meet the needs of post-graduate students and professional antenna engineers. ORAMA is based on the orthogonal method OM and has been designed for the element excitation derivation of linear antenna arrays.

This tool has been written through a menu driven form to be userfriendly. Mutual coupling between the elements of the arrays is not taken into account in the present version.

The elements through the phase and amplitude control can vary across the aperture of the array and allow the realization of certain requirements related to pattern characteristics. The antenna array is designed for different types and positions of elements depending on needs. Elements such as dipoles, microstrips and horns are common in most of the applications. In this program, parallel and identical elements are taken into account. It has been designed as a Windows MDI application so that multiple array designs can be performed simultaneously.

The program starts with the screen of Fig. There are four tabs: 1. The user has the following options: 1. Set the design frequency of the array. This frequency is used to convert lengths from wavelengths into metres or inches and vice versa. Set the viewing frequency of the array. This frequency is used to view the radiation pattern of the designed array at a different frequency. Set the number of sources. Set the inter-element distance. This edit box is disabled if elements are at unequal distances. To view or change mean distance in metres or inches the user can select the appropriate unit from the drop-down list box on the right side of the edit box.

Choose the array axis x, y or z. The cases have been selected to provide a great variety of desired factors, and have been parameterized to allow additional degrees of freedom. Figure 5. By clicking one of the patterns see Fig. In the present version, the calculations can be made using the orthogonal method given in paragraph 4. Additional information for element type characteristics is required in most of the above cases. In the case of polar form, the phase can be shown in degrees or radians.

In the dialog box, there is also the possibility of choosing a pre-selected excitation distribution. It is noticed that, besides setting, the pre-selected excitation can be added or multiplied to the one found by the OM of a desired pattern. If a pattern taken by the OM has side lobes higher than the desired ones, an addition or multiplication of the calculated excitation with a pre-selected one could give lower SLL.

If it is desirable to lower SLL, then we can add a triangular pre-selected excitation. The procedure can be continued until the desired SLL is achieved. In Table 5. It is noticed that instead of the triangular distribution one could choose another one on the menu and compare the results. If it is desirable to rotate the main lobe, a phase difference can be added through the dialog box. So, with the excitation of Table 5. The user has the option of customizing the plot view. Table 5. The option of plotting the total factor, the element or the array factor is also possible.

The graph can be viewed in either polar or rectangular form. The planes shown Fig. A 3-D graph option Fig. The user can save, open or close an array design. The user should pay attention to array and element size. Although it is physically unrealizable, the isotropic source is frequently used in the array design process because it helps in the exploitation of array factor characteristics.

An isotropic source is a point source that can be simulated by a delta function excitation . The dipoles can be parallel to the x, y or z-axis. Therefore, only the results will be presented. For more information, the reader is referred to [1, 2]. The length of the dipoles is derived in wavelengths, metres or inches. The dipoles can be on the xy, yz or xz plane. Next, the formulation for the rectangular and the sectoral horns with smooth or corrugated walls will be given.

It is assumed that the aperture is on the xy plane. It is assumed that the TE10 mode exists in the rectangular waveguide. Three separate cases are the most interesting ones. Finally, the third one derives the horn dimensions under constraints on the ratio of the HPBWs on the two main planes and the directivity. By taking into account the geometry given in Fig. There are however, cases for which HPBWs on the two main planes or their ratio are just as important as directivity maximization.

SET 1. Then, from s is found. Finally, the horn length R is calculated from The results do not give the exact outcome. So, in order to improve the design an iterative procedure between Equations and can be followed. SET 2. Finally, the axial length R is calculated from It is noticed that as in the previous case, the calculated results are approximate. One of the techniques gives the best solution. The choice of the optimum parameter s or t and the use of the appropriate set of equations depend on whether k is bigger or smaller than kopt. The rest of the methodology described in the previous section can be followed.

The mode on the horn aperture is hybrid. A converter section is used to convert the waveguide mode into the hybrid one. In this case, I1x or I2y are simpler. If the design frequency is outside the one covered by the available standard waveguides, ORAMA asks the user to enter the dimensions of a custom waveguide. By pushing the above button, a dialog box appears see Fig. Going back to the main menu of ORAMA, one can calculate the pattern characteristics of a single element.

We choose the pyramidal horn as element type and we give the following design examples: 1. By plotting the pattern of the designed horn, it is found that the resulting gain is The resulting patterns of the horn are given in Fig. If we choose the same dimensions for a corrugated horn, the resulting gain becomes equal to We design the horn again for Pyramidal horn with desired HPBW ratio equal to 1.

After iteration, for a gain of We design the smooth-walled horn again for a gain equal to The geometry of a rectangular patch is presented in Fig. The radiation of the patch comes from the fringing effect that occurs at the open transmission line ends see Fig. Each slot has a uniform excitation with a length equal to the patch width W. It also has a width equal to the thickness h of the substrate. The radiation pattern is found by using the standard aperture theory . If a rectangular patch is selected see Fig. The dielectric constant of the substrate is not requested, because it is ignored in the calculations.

If a circular patch is selected see Fig. In this paragraph, some supplementary design cases will be given. First, the pattern of a radar is presented. The success of the OM is obvious. Suppose, it is desirable to illuminate an area with a base station antenna where the received signal at the mobile units is the same.

It is known that a cosecant-square—shaped beam power pattern is necessary. In the same table, the excitation of a similar case given in  is presented. This value is exactly the same as the corresponding one in Table Orthogonal Methods for Array Synthesis: Theory and the ORAMA Computer Tool Orthogonal Methods for Array Synthesis: Theory and the ORAMA Computer Tool Orthogonal Methods for Array Synthesis: Theory and the ORAMA Computer Tool Orthogonal Methods for Array Synthesis: Theory and the ORAMA Computer Tool Orthogonal Methods for Array Synthesis: Theory and the ORAMA Computer Tool Orthogonal Methods for Array Synthesis: Theory and the ORAMA Computer Tool Orthogonal Methods for Array Synthesis: Theory and the ORAMA Computer Tool

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