## 1. Introduction

[2] Radomes made of lossy dielectric layered materials, including antenna reflectors and feeding sources, are scattering problems whose accurate solution is typically difficult to compute, mainly due to their large size and complex geometries. For these reasons, it is a topic of growing interest which is presently receiving considerable attention by the electromagnetic community [see *Zhao et al.*, 2005; *Sukharevsky et al.*, 2005].

[3] Traditionally, radomes were first analyzed using high frequency methods, such as Ray Propagation in *Gao and Felsen* [1985], or the Geometrical Theory of Diffraction in *Duan et al.* [1991]. Alternatively, rigorous solutions for 2-D problems including canonical reflectors and radomes, combined with complex sources, were employed in *Chang and Chan* [1990], *Svezhentsev et al.* [1995], *Yurchenko et al.* [1999], *Oguzer* [2001], and *Oguzer et al.* [2004]. Although with reduced complexity, the scattering analysis of 2-D structures finds useful applications, for instance in the design of inductive or capacitive waveguide filters widely used in mobile and satellite systems. For these devices, the analysis can be carried out by solving a 2-D scattering problem. Also, cylindrical reflector antennas, which can be modeled as 2-D structures, are used in airborne navigation applications, as recognized in *Nosich* [1999]. These applications show the interest of investigating efficient numerical techniques for the study of 2-D scattering problems.

[4] Recently, solutions for 3-D radomes without reflectors based on volume and surface formulations have also appeared in *Lu* [2003] and *Zhao et al.* [2005], where the analysis has been performed through the Adaptive Integral Method [see *Zhao et al.*, 2005], and the Multilevel Fast Multipole Algorithm proposed in *Li and Li* [2004] and *Lu* [2003]. These 3-D radomes, typically used for fixed earth stations, usually present a spherical geometry. However, several cylindrical scenarios with arbitrary geometries appear in naval radar systems [see *Nosich*, 1999], as well as in telecommunications towers with radiant elements. In these complex real situations, multilayered radome materials are used, and metallic surfaces and reflectors combined with directive complex sources are often found in practice.

[5] In this paper, we propose to use the Equivalence Principle in order to formulate an integral equation problem for the analysis of 2-D cylindrical problems with arbitrary geometries. To set up the formulation we employ the Extinction Theorem to transform the classical volume formulation for dielectric obstacles into a more efficient surface approach. Proceeding in this way, the unknowns of the problem are the equivalent electric and magnetic surface currents defined only over the surfaces of the different homogeneous objects. Therefore, following this approach, a boundary integral equation fully characterizes the problem. That approach was already proposed in *Arvas and Ponnapalli* [1989] for small radomes scattering problems, and a more complete formulation for considering thin dielectric radomes under **TE**^{z} polarization was presented in *Sadigh and Arvas* [1992]. In this paper, a Poggio Miller Chang Harrington Wu Tsai (PMCHWT) field formulation of the type shown in *Kishk and Shafai* [1986] for large dielectric radomes under **TM**^{z} and **TE**^{z} polarization, including also metallic reflectors and complex sources, will be proposed. Furthermore, we have successfully combined such new formulation with the wavelet-like transform. If high accuracy is demanded, then the number of unknowns grows even in the case of surface formulations [see *Peterson et al.*, 1998]. That is the case, for instance, if a simple point-matching Method of Moments (MoM) procedure is used to solve the problem involving electrically large radomes, and/or very complex geometries. Then, CPU cost becomes important, and the wavelet-like transform can be efficiently used to decrease the amount of required memory allocation and CPU computational time.

[6] The use of wavelets to reduce the computational burden associated to MoM solutions is not new, and the idea has been used for instance in *Wang* [1995] to study the scattering from metallic objects. In that work periodic wavelet functions are used as basis functions in the MoM, and the backscattering of several metallic objects are successfully computed. Alternatively, in this paper we propose a surface formulation for the analysis of both metallic and dielectric objects, with subsequent application of the wavelet-like transform. Following this approach, the MoM formulation stays very simple (pulse-point matching), while obtaining big gain in computational cost through the use of the wavelet-like transform.

[7] It will also be shown in this paper that the use of the proposed surface formulation is very convenient for speed acceleration, since it leads to a matrix equation which naturally has a banded submatrix-type structure. This banded structure is very well suited for a subsequent introduction of the wavelet-like transform. An important contribution of this paper is that the wavelet-like transform is applied to each submatrix block, instead of using a global transformation scheme. It has been proved that a considerable gain in computational cost is obtained when the wavelet-like transform is applied following the new introduced subblock scheme. All these gains in CPU time and memory requirements are not possible if other volume based formulations [see *Hsu and Auda*, 1986; *Lu*, 2003; *Sukharevsky et al.*, 2005] are employed. This is because a volume formulation produces full dense matrices, thus loosing some of the matrix sparsity introduced by the proposed surface formulation. Furthermore, the volume formulation presents problems associated to the sorting of the grid elements, which reduce the computation gain related to the application of the wavelet-like transform.

[8] The structure of this paper is the following one. First, the surface formulation employed is briefly described. Then, the wavelet-like transformation is applied in order to obtain very sparse matrices. The theory is validated with general results obtained from the literature, such as canonical reflectors and radomes. Once the novel method is successfully verified, it has been applied to analyze complex shaped reflector-in-radome structures and arrays-in-radome antennas. The results clearly show the validity and usefulness of the new strategy proposed in this paper for the efficient and accurate analysis of this kind of structures.