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Keywords:

  • reflector antenna;
  • satellite antennas;
  • physical optics

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Technique
  5. 3. Shape Distortion Determination Algorithm
  6. 4. Application Example: Thermal/Gravitational Errors Determination
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[1] The idealized shapes of satellite reflector antennas are often distorted once they are placed in orbit. The performance of such antennas can be improved by identifying the locations and amount of their surface distortions and then by correcting them using active surface distortion or array feeding. This work presents a method to determine the required discrete surface distortions to correct errors. The algorithm starts by discretizing the entire reflector surface into triangular patches, then by determining the linear relationship between the local distortion and the difference between distorted and undistorted far-fields patterns. A linear system of equations with discrete distortions as unknowns results when this scattered field is sampled at specific observation points. Singular Value Decomposition combined with Tikhonov regularization, is used to solve for the set of distortions, which are translated into a physically realizable continuous surface by projecting onto a Polynomial-Fourier-Series basis. The later scheme is iteratively repeated in order to minimize the residual error. The method has been applied successfully to determine thermal/gravitational distortions on a reflector antenna, with up to 97% accuracy in tenth iteration.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Technique
  5. 3. Shape Distortion Determination Algorithm
  6. 4. Application Example: Thermal/Gravitational Errors Determination
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[2] Satellite reflector antennas are often distorted from their idealized shapes once they are placed in orbit. This distortion may be due to thermal gradients, mechanical stresses, damage from launch, or collisions with foreign objects. Variations from the designed shape lead to antenna performance degradation, compromising the antenna radiation pattern. The shape distortions can often be dynamically corrected with onboard actuators [Lang and Staelin, 1982; Lawson and Yen, 1988; Clarricoats and Zhou, 1991; Pino et al., 2006] or feeding arrays [Acosta, 1986; Cherette et al., 1986; Rahmat-Samii, 1990; Bailey, 1991; Smith and Stutzman, 1992] but it is essential to determine exactly where and how great the distortions are.

[3] Microwave holographic metrology reconstruction [Bennett et al., 1976; Scott and Ryle, 1977; Goodwin et al., 1986; Mayer et al., 1983] uses the far-field amplitude and phase patterns of the antenna to compute the field distribution on the aperture plane; this information combined with ray theory leads to the shape distortion determination [Rahmat-Samii, 1988; Rahmat-Samii and Lemanczyk, 1988].This paper presents a method for finding the shape variations based on the observed far-field pattern. The distortions on the reflector are translated directly to the far-field pattern avoiding the computation of the field distribution at the aperture plane.

[4] The proposed method builds on approaches of the formulation proposed by Westcott [1995] and Martinez-Lorenzo et al. [2005], which approximates the nonlinear association between the position of a patch of reflector and the contribution to the field radiated by this patch it in the far field. By assuming a small positional perturbation, the association is linearized, and thus the perturbed position can be inverted given the observed pattern. Repeated application of this small perturbation inversion quickly leads to the final patch position. As long as the surface deformations are not too great (less than one fourth of a wavelength on average), convergence is likely. Although there have been many approaches to predicting the reflector shape that causes a particular far-field response [Galindo, 1964; Galindo-Israel et al., 1979; von Hoerner, 1978; Pinsard et al., 1997], the current approach makes use of both discretizing the entire reflector surface into triangular patches and linearizing the complex radiated field. By using the physical optics approximatio [Lo and Lee, 1988] for generating the forward model of the far-field pattern, very fast far-field pattern generation is attained, and the inversion can be accomplished quickly.

2. Analysis Technique

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Technique
  5. 3. Shape Distortion Determination Algorithm
  6. 4. Application Example: Thermal/Gravitational Errors Determination
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[5] The electric field, Es, scattered by a perfect electric conductor (PEC), see Figure 1, is computed from the induced current J on the reflector surface Ω with [Balanis, 1989]

  • equation image
  • equation image

where r is an observation point and r′ is a source point on the PEC surface. G, k and Z0 denote the free-space Green's function, the wavenumber, and the free-space wave impedance, respectively. The physical optics (PO) approximation, used for large scattering problems, considers the induced currents J to be proportional to the magnetic incident field to the PEC, Hinc, as:

  • equation image

where equation image is the unitary outward normal vector to the PEC surface.

image

Figure 1. Scattering of single reflector antenna.

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[6] The integral equation (1) is numerically computed by dividing the PEC domain Ω into a set of flat triangular subdomains Ωi placed at r′i, for i = 1…NΩ where NΩ is the number of subdomains, and then evaluating the integral on a set of observation points rl, for l = 1…Nobs where Nobs is the number of observation points. When the observation point rl is in the far field region of the PEC surface, the electric field contribution of the subdomain i in that observation point l can be expressed as [Martinez-Lorenzo et al., 2005; Arias et al., 2000]

  • equation image

[7] The complete scattered field from the PEC at the observation point rl, Els, is determined by the summation of the electric field contributions of each subdomain as:

  • equation image

3. Shape Distortion Determination Algorithm

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Technique
  5. 3. Shape Distortion Determination Algorithm
  6. 4. Application Example: Thermal/Gravitational Errors Determination
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[8] Let us consider the problem presented in Figure 2. The ideal undistorted shape, given by position vector r′, produces the scattered field Es. Distortions to this ideal shape produce distorted scattered field Es,Δ which can be observed in the far field. The problem can then be stated as determining the distorted surface, given by a position vector r′Δ, that produces the distorted scattered field Es,Δ. This problem is solved by linearly approximating the field differences between the distorted and undistorted fields. As a consequence, a matrix formulation describing a linear system of equations, with the physical distortions as unknowns, is derived. The matrix is inverted using the Singular Value Decomposition, resulting in the set of discrete distortions of the surface facets. In other to derive a continuous surface for the reconstructed reflector, the unknown points are joined by a Polynomial-Fourier Series (PFS) basis functions in a best fit sense. The error resulting from the linear approximation requires iterative application for accurate results.

image

Figure 2. Overview of the shape distortion determination algorithm.

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3.1. Distorted/Undistorted Fields Linear Relationship

[9] The relationship between the distorted and undistorted fields is determined first by analyzing the perturbations of the initial induced current, Jpo, into equation imagepoΔ, when its corresponding subdomain Ωi is moved from its initial position r′i a quantity τi in the unitary direction equation image. Both amplitude and phase variations occur for the new induced currents, but the amplitude variation can be neglected for small values of τi. The phase variation is given by the projection of the relative displacement, Δr′i = equation imageτi, onto the unitary Poynting vector, equation image, of the wave incident on the domain Ωi (Figure 3). As a result, the new induced currents can be approximated as:

  • equation image
image

Figure 3. Physical optics currents deviation.

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[10] The second step consists of expressing the phase variation introduced in the exponential term in (3) due to the displacement Δr′i (Figure 3). Taking into account both phase terms, an approximation for the distorted scattered field, equation imagel,is,Δ, by the domain Ωi can be written as phase shifted version of the undistorted field, El,is, as:

  • equation image

[11] An approximation for the total distorted scattered fields, equation imagels,Δ, can thus be derived from the undistorted scattered fields by:

  • equation image

[12] The relationship between the distorted and undistorted fields in (6) has an exponential term with an argument proportional to the distortion. For small argument values, the exponential term can be approximated by the first two terms of a series expansion, leading a linear relationship between the linearized distorted fields, equation imagel,is,Δ, and the undistorted ones, with a linear coefficient proportional to the distortion τi as

  • equation image

[13] As a consequence, the linearized approximation for the total distorted scattered fields, equation imagels,Δ, can be written as the contribution of each domain by

  • equation image

3.2. Linear System of Equations

[14] A linear system of equations, with the distortions τi as unknowns, can be derived from (8) and (9) by requiring that the observed distorted fields, Els,Δ, be equal to those predicted by the linear approximation equation imagels,Δ. The system can be written in a matrix form as

  • equation image

where [A] is a Nobs × NΩ matrix containing the value Al,i = equation image · El,isjequation imagel,i in the row l and column i, [x] is a NΩ × 1 column vector containing the unknown value xi = τi in row i, [b] is a Nobs × 1 column vector containing the value bl = equation image · (Els,ΔEls) in the row l, the unitary vector equation image denotes the direction of the electric far-field copolar component of the scattered field as defined by Ludwig [1973] and Knittel [1973]. The scalar product between the copolar vector and the electric field reduces the three dimensional vector problem to a scalar one.

3.3. Matrix Inversion

[15] In order to obtain the solution vector [x], the matrix [A] is factored using the Singular Value Decomposition method as [Hansen, 1990; Golub and Loan, 1996; Hansen, 1994]:

  • equation image

where [V] = (v1, …, vNΩ) is a NΩ × NΩ matrix containing a set of orthonormal input basis directions for [A], the matrix [U] = (u1, …, uNΩ) is a Nobs × Nobs matrix containing a set of orthonormal output basis vector directions for [A] and, the matrix [Σ] = diag(σ1,…,σmin{NΩ,Nobs}) is a Nobs × NΩ matrix containing the singular values in the diagonal, which can be thought as scalar gain controls by which each corresponding input is multiplied to give a corresponding output, with zeros off the principal diagonal. The symbol '☆' denotes the conjugate transpose of a matrix. The value of [x] for a SVD factorization is given by:

  • equation image

where [Σ+] = diag(1/σ1,…,1/σequation image) is the transpose of [Σ] with every nonzero entry replaced by its reciprocal. By using a SVD factorization, Tikhonov regularization can be applied if the matrix become ill-posed [Hansen, 1994, 1989].

3.4. Surface Continuity

[16] The value of [x] in (12) provides a set of discrete distortions for the centers of every facet. These discrete values must be translated into a physically realizable continuous surface. This goal is achieved by using a set of basis function that best fits the discrete distortion in a least square sense. A suitable set of basis functions are the Polynomial-Fourier-Series (PFS) [Clarricoats and Bergman, 1983; Bergman et al., 1988, 1994]. These have been used for the interpolations involved in the synthesis of collimated-beam as well as shaped-beam reflector antennas for satellite communications. They combine a quadratic polynomial with Fourier-like series expansion. For a centered coordinate frame the PFS may be expressed as:

  • equation image

where fk(u) = 1, sin(u), cos(u), sin(2u), cos(2u), etc., for k = 0,1,2,3,4,…; fm(v) = 1, sin(v), cos(v), sin(2v), cos(2v), etc., for m = 0,1,2,3,4,…; and (u,v) denote conveniently normalized cartesian coordinates (x, y). The coefficients {a1a5} and {C0,0CNu,Nv} are then obtained by a least squares minimization of the residues (zzi) using the NΩ data points (ui, vi, zi). The number of terms (Nu × Nv) in (13) required to yield an acceptable fit is related to the spatial variation of the reflector.

3.5. Iterative Procedure

[17] Note that the error introduced in the series approximation of the exponential term in (6) is of the order of equation imagel,iτi)2. As a consequence, when the error is comparable to the second term of the series expansion, an iterative scheme for the matrix formulation must be adopted. The iterative procedure consists of considering the location of the undistorted domain of the m-th iteration, r′i(m), as the solution of the previous iteration, r′iΔ(m−1), by the following scheme

  • equation image

with initial conditions

  • equation image
  • equation image

It is important to note that for each iteration the distorted reflector surface must be sampled again, as well as the values of El,is and ϕl,i due to their dependence on the position of the domains.

4. Application Example: Thermal/Gravitational Errors Determination

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Technique
  5. 3. Shape Distortion Determination Algorithm
  6. 4. Application Example: Thermal/Gravitational Errors Determination
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

4.1. Antenna Configuration

[18] The methodology of determination of the discrete distortion across the reflector surface is demonstrated using the same geometry that was used in the work of Rahmat-samii [1990]. The characteristic parameters are summarized in Table 1, in accordance with Figure 4.

image

Figure 4. Vertical cross section of the single reflector antenna.

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Table 1. Characteristic Parameters of the Simulated Example
 Parameters
Main Reflector
Reflector diameterD = 1.68 m
Focal lengthF = 1.832 m
Offset heightH = 1.45 m
 
Feed
Frequencyf = 8.45 GHz
Electric field polarizationp = equation image
Offset angleθf = 43.18
Subtended angleθs = 45.11
Taper edge E-planetE = 12 dB
Taper edge H-planetH = 12 dB

[19] For this reflector antenna, thermal/gravitational surface-distortion models were simulated. The mathematical description of this surface distortion is [Rahmat-samii, 1990; Duan and Rahmat-Samii, 1998; Hoferer and Rahmat-Samii, 2002];

  • equation image

where δmax is the maximum deviation between the distorted and undistorted shape in the equation image direction, ρ is the modulus of the vector ρ, ϕ is the angle between the projected vector ρ and the equation image axis (see Figure 4) and N is the spatial-frequency variation of the distorted model. Figures 5a and 6a present the Thermal/Gravitational distortions for N = 2 [Rahmat-Samii, 1990] and N = 4 [Hoferer and Rahmat-Samii, 2002], given by (17) respectively, introduced to the reflector described in Table 1 for a maximum deviation of δmax = λ/4 = 8.9 mm. An important characteristic of thermal/gravitational distortion is that it becomes zero at the center of the reflector, while it is maximum at the border, growing with the third power of the radial component ρ. The spatial-frequency of the reflector surface oscillation with angle increases with N.

image

Figure 5. (a) Analytical thermal/gravitational distortions (in millimeters) for N = 2. Reconstructed thermal/gravitational distortions in the (b) first iteration, (c) fifth iteration, and (d) tenth iteration.

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image

Figure 6. (a) Analytical thermal/gravitational distortions (in millimeters) for N = 4. Reconstructed thermal/gravitational distortions in the (b) first iteration, (c) fifth iteration, and (d) tenth iteration.

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[20] An important issue in solving the problem is to select an adequate number of Fourier-like basis functions, (Nu × Nv), which can accurately describe this thermal/gravitational distortion. For this particular application, a suitable number is Nu = Nv = 5 for N = 2 and Nu = Nv = 7 for N = 4; as a result, the total number of basis functions used to describe the continuous surface is thirty and fifty-four for N = 2 and N = 4, respectively. It is clear that the latter case requires a higher amount of basis functions due to its higher N value. The projection of the real distortion, given by (17), onto the PFS basis functions produces a root mean square RMS error and peak error of (RMS2 = 0.012 mm, Peak2 = 0.042 mm) and (RMS4 = 0.0391 mm, Peak4 = 0.135 mm) for N = 2 and N = 4, respectively. These values provide a bound of the maximum resolution achievable by the inversion method; and the spatial variation of method residual errors are correlated with the errors introduced by the PFS projection.

4.2. Shape Convergence

[21] To ensure the convergence of the method, the number of observation points should be about one third of the number of triangular facets of the discretized reflector, sampled in a regular mesh within a solid of the order of five times the undistorted reflector beam width squared. For this particular case of reconstructing thermal/gravitational distortions, the reflector surface is discretized in NΩ = 5400 triangular facets, and the field is observed in Nobs = 1350 points,with average subdomains size being Hd = 0.788λ. This value imposes the second bound for the maximum resolution achievable by the inversion method, because it is impossible to achieve higher resolution than that of the discretized model. By considering the case of a discretizing paraboloid into flat triangular facets, the peak surface error will be on the order of Hd2/(16f), for a parabolic reflector with focal length f, and for this particular application, it turns out to be in the order of 0.026 mm. In this work equation image is coincident with the reflector paraboloid axis equation image.

[22] Figures 5b–5d present the reconstructed thermal/gravitational discrete distortion, for the first (m = 1) fifth (m = 5) and tenth (m = 10) iteration, respectively, when N = 2. Equivalent results are presented in Figures 6b–6d for the case of N = 4. For the first iteration, the recovered thermal/gravitational distortion has the same shape as the analytical specification with the same angular variation and increasing radial variation. The convergence at the center of the reflector is better than at the border; this is because it is in this area where the analytical distortion reaches its maximum value. There is little change between reconstructions for the fifth and tenth iterations.

[23] The iterative convergence of error, computed as the difference between the analytical and reconstructed distortions, is addressed in Figure 7 for both N = 2 and N = 4. The convergence at the border of the reflector is slower than in the center due to the inherent nature of the thermal/gravitational distortion. Table 2 presents the RMS error and peak error for each iteration. The convergence is likely at the fifth, m = 5, iteration where both the RMS and peak errors reach their minimum value for the method under this particular configuration. The maximum relative peak error for the fith iteration, is 1.48% and 2.71% for N = 2 and N = 4, respectively.

image

Figure 7. Residual error for thermal/gravitational distortions (in millimeters) for first iteration with (a) N = 2 and (b) N = 4, fifth iteration with (c) N = 2 and (d) N = 4, and tenth iteration with (e) N = 2 and (f) N = 4.

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Table 2. RMS and Peak Error for Different Iterations for N = 2 and N = 4a
IterationRMS2Peak2RMS4Peak4
  • a

    Units are in mm.

m = 11.62707.68021.63578.7347
m = 20.49713.23360.59104.9921
m = 30.05680.22550.10630.8846
m = 40.06650.19100.06600.2465
m = 50.04910.10580.06530.2481
m = 60.05710.20570.06460.2373
m = 70.05600.11910.06710.2493
m = 80.04770.10460.06570.2315
m = 90.04710.09200.06220.2349
m = 100.05430.13140.06430.2412

4.3. Field Convergence

[24] The previous subsection shows that adequate convergence is achieved for the distorted shape of the reflector, but it is also interesting to analyze whether the scattered field converges as well. Figures 8 and 9 present the convergence between the far-field pattern and phase of the nominal, distorted and reconstructed reflector in the first (m = 1), fifth (m = 5) and tenth (m = 10) iteration for N = 2. Equivalent results are presented in Figures 10 and 11 for N = 4. After the first iteration, the reconstructed reflector radiates a broader beam, and the phase function is also closer to that radiated by the distorted reflector than that radiated by the nominal reflector. The far-field pattern retains a lobular shape with nulls not been completely removed, as with the distorted reflector pattern. By the fifth iteration, the reconstructed and distorted reflector produce almost a perfect match in both far-field pattern and phase. Table 3 shows that the convergence for the maximum directivity is achieved by the around the fourth/fifth iteration. Although the method has been applied to reconstruct thermal/gravitational distortions on a reflector surface, it is reasonable to consider this method for synthesizing shaped beams, where the distorted field is now replaced by a desired pattern. It is important to mention that both amplitude and phase information of the desired field must be used for the inversion. This approach is quite different from the typical beam shaping procedures, where a nonlinear function of the reflector surface is optimized to produce a desired far-field pattern avoiding phase information. Instead, the approach presented uses a linear function of the discretized reflector and both amplitude and phase could be synthesized. Although this concept promising a physically achievable field specification is not an easily achievable and must be studied in further research works.

image

Figure 8. Comparative of the (a) far-field pattern and (b) phase of the nominal, distorted and reconstructed reflector for the first iteration for N = 2 in the observation plane ϕ = 0.

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image

Figure 9. Comparative between the (a) far-field pattern and (b) phase of the nominal, distorted and reconstructed reflector at fifth iteration for N = 2 in the observation plane ϕ = 0.

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image

Figure 10. Comparative between the (a) far-field pattern and (b) phase of the nominal, distorted and reconstructed reflector at first iteration for N = 4 in the observation plane ϕ = 0.

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image

Figure 11. Comparative between the (a) far-field pattern and (b) phase of the nominal, distorted and reconstructed reflector at fifth iteration for N = 4 in the observation plane ϕ = 0.

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Table 3. Maximum Directivity in dBi
IterationN = 2N = 4
Undistorted42.494342.4943
   m = 141.427041.4362
   m = 240.540740.5583
   m = 340.353340.3772
   m = 440.353040.3677
   m = 540.345340.3676
   m = 640.340440.3664
   m = 740.348540.3674
   m = 840.348040.3666
   m = 940.346340.3670
   m = 1040.344440.3666
Distorted40.350540.3634

5. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Technique
  5. 3. Shape Distortion Determination Algorithm
  6. 4. Application Example: Thermal/Gravitational Errors Determination
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[25] A new method to determine the discrete distortions on reflector antennas has been presented. The method is based on the discretization of the reflector surface into triangular patches. By linearizing the difference between the distorted and undistorted farfiled patterns, a linear system of equations is constructed. The matrix is inverted using a Singular Value Decomposition with Tikhonov regularization. The discrete distortions provided by the matrix inversion are joined into a continuous surface using Polynomial-Fourier-Series basis functions. An iterative scheme must be applied in order to reduce the error introduced from the linearization of the problem. The application of the method for reconstructing thermal/gravitational distortions surfaces yielding accurate results has been presented. The method can be applied to improve the radiation performance of onboard satellite reflector antennas by using active surfaces or phased arrays.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Technique
  5. 3. Shape Distortion Determination Algorithm
  6. 4. Application Example: Thermal/Gravitational Errors Determination
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information

[26] This work is supported by CenSSIS, the Gordon Center for Subsurface Sensing and Imaging Systems under the ERC Program of the NSF (award EEC-9986821), and Spanish Government grants FEDER-MEC ESP2005-01894 and TEC2005-03563.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Technique
  5. 3. Shape Distortion Determination Algorithm
  6. 4. Application Example: Thermal/Gravitational Errors Determination
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information
  • Acosta, R. (1986), Compensation of reflector surface distortions using conjugate field matching, paper presented at Antennas and Propagation Society International Symposium (AP-S'86), Inst. of Electr. and Electron. Eng., Philadelphia, Pa.
  • Arias, A. M., J. O. Rubios, I. Cuias, and A. G. Pino (2000), Electromagnetic scattering of reflector antennas by fast physical optics algorithms, Recent Res. Devel. Magnetics, 1), 4363.
  • Bailey, M. C. (1991), Determination of array feed excitation to improve performance of distorted or scanned reflector antennas, paper presented at Antennas and Propagation Society International Symposium (AP-S'91), Inst. of Electr. and Electron. Eng., Ontario.
  • Balanis, C. A. (1989), Advanced Engineering Electromagnetics, 1st ed., pp. 283284, John Wiley, New York.
  • Bennett, J. C., A. P. Anderson, P. A. McInness, and J. T. Whitaker (1976), Microwave holographic metrology of large reflector antennas, IEEE Trans. Antennas Propag., 24(3), 295303.
  • Bergman, J. B., R. C. Brown, P. J. B. Clarricoats, and H. Zhou (1988), Synthesis of shaped-beam reflector antenna radiation patterns, Proc. IEE H, 135(1), 4853.
  • Bergman, J. B., F. J. V. Hasselmann, F. L. Teixeira, and C. G. Rego (1994), A comparison between techniques for global surface interpolation in shaped reflector analysis, IEEE Trans. Antennas Propag., 42(1), 4753.
  • Cherette, A. R., P. T. Lam, S. W. Lee, and R. Acosta (1986), Compensation of distorted offset paraboloid reflector using an array feed, paper presented at Antennas and Propagation Society International Symposium (AP-S'86), Inst. of Electr. and Electron. Eng., Philadelphia, Pa.
  • Clarricoats, P. J. B., and J. B. Bergman (1983), Synthesis of reflector antenna radiation pattern, Proc. URSI Electromagn. Theory Symp. Dig. (URSI'83), 325326.
  • Clarricoats, P. J. B., and H. Zhou (1991), Design and performance of a reconfigurable mesh reflector antenna, IEEE Proc. H, 138, 485492.
  • Duan, D. W., and Y. Rahmat-Samii (1998), A generalized diffraction synthesis technique for high performance reflector antennas, IEEE Trans. Antennas Propag., 46(10), 14491457.
  • Galindo, V. (1964), Design of dual-reflector antennas with arbitrary phase and amplitude distributions, IEEE Trans. Antennas Propag., 12(4), 403408.
  • Galindo-Israel, V., R. Mittra, and A. G. Cha (1979), Aperture amplitude and phase control of offset dual reflectors, IEEE Trans. Antennas Propag., 27(2), 154164.
  • Golub, H. G., and C. F. V. Loan (1996), Matix Computations, 3rd ed, Johns Hopkins Univ. Press, Baltimore, Md.
  • Goodwin, M. P., E. P. Schoessow, and B. H. Grahl (1986), Improvement of the effelsberg 100 meter telescope based on holographic reflector surface measurement, Astron. Astrophys., 167, 390394.
  • Hansen, P. (1989), Perturbation bounds for discrete tickonov regularization, Inverse Problems, 5, L41L44.
  • Hansen, P. (1990), Relations between svd and gsvd of discrete regularization problems in standard and general form, Lin. Alg. Appl., 141, 165176.
  • Hansen, P. (1994), Regularization tools: a matlab package for analysis and solution of discrete ill-posed problems, Numer. Algo., (6), 135.
  • Hoferer, R. A., and Y. Rahmat-Samii (2002), Subreflector shaping for antenna distortion compensation: An efficient Fourier-Jacobi expansion with go/po analysis, IEEE Trans. Antennas Propag., 50(12), 16761687.
  • Knittel, G. H. (1973), Comments on the definition of cross polarization, IEEE Trans. Antennas Propag., 21(6), 917918.
  • Lang, J. H., and D. H. Staelin (1982), Electrostatically figured reflecting membrane antennas for satellites, IEEE Trans. Automatic Control, 27(3), 666670.
  • Lawson, P. R., and J. L. Yen (1988), A piecewise deformable subreflector for compensation of cassegrain main reflector errors, IEEE Trans. Antennas Propag., 36(10), 13431350.
  • Lo, Y. T., and S. W. Lee (1988), Antenna Handbook, Theory Applications and Design, Van Nostrand Reinhold, Hoboken, N. J.
  • Ludwig, A. C. (1973), The definition of cross polarization, IEEE Trans. Antennas Propag., 21(1), 116119.
  • Martinez-Lorenzo, J. A., A. G. Pino, I. Vega, M. Arias, and O. Rubios (2005), Icara: Induced-current analysis of reflector antennas, IEEE Antennas Propag. Mag., 47(2), 92100.
  • Mayer, C. E., J. H. Davis, W. L. Peters, and W. J. Vogel (1983), A holographic surface measurement of the texas 4.9-meter antenna at 86 ghz, IEEE Trans. Instrum. Meas., IM-32, 102109.
  • Pino, A. G., J. A. Martinez-Lorenzo, M. A. Arias, and C. Compostizo (2006), Design and analysis of an adjustable subreflector for the hybrid mechanical-electronic pointing system at the satellite q/v band, paper presented at Antennas and Propagation Society International Symposium (AP-S'06), Inst. of Electr. and Electron. Eng., Albuquerque, N. M.
  • Pinsard, B., D. Renaud, and H. Diez (1997), New surface expansion for fast po synthesis of shaped reflectorantennas, paper presented at International Conference on Antennas and Propatation, Inst. of Electr. Eng., Edinburgh, UK.
  • Rahmat-Samii, Y. (1988), Communicating from space: Applying microwave holographic metrology to space communication antennas, IEEE Potentials, 7(3), 3135.
  • Rahmat-Samii, Y. (1990), Array feeds for reflector surface distortion compensation: Concepts and implementation, IEEE Antennas Propag. Mag., 32(4), 2026.
  • Rahmat-Samii, Y., and J. Lemanczyk (1988), Application of spherical near-field measurements to microwave holographic diagnosis of antennas, IEEE Trans. Antennas Propag., 36(6), 869878.
  • Scott, P. F., and M. Ryle (1977), A rapid method for meassuring the figure of a radio telescope reflector, R. Astron. Soc. Monthly Notices, 178, 539545.
  • Smith, W. T., and W. L. Stutzman (1992), A pattern synthesis technique for array feeds to improve radiation performance of large distorted reflector antennas, IEEE Trans. Antennas Propag., 40(1), 5762.
  • von Hoerner, S. (1978), Minimum-noise maximum gain telescopes and relaxation method for shaped asymmetric surfaces, IEEE Trans. Antennas Propag., 26(3), 464471.
  • Westcott, B. S. (1995), Dual-reflector synthesis based on analytical gradient-iteration procedures, IEE Proc., Microwave Antennas Propag., 142(2), 129135.

Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Analysis Technique
  5. 3. Shape Distortion Determination Algorithm
  6. 4. Application Example: Thermal/Gravitational Errors Determination
  7. 5. Conclusions
  8. Acknowledgments
  9. References
  10. Supporting Information
FilenameFormatSizeDescription
rds5559-sup-0001-t01.txtplain text document0KTab-delimited Table 1.
rds5559-sup-0002-t02.txtplain text document0KTab-delimited Table 2.
rds5559-sup-0003-t03.txtplain text document0KTab-delimited Table 3.

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