Comparison of Wavelet Packet and Wavelet in Solving Arbitrary Array of Parallel Wires Integral Equations in Electromagnetics

In this paper, wavelets transformation (WT) and wavelet packet transformation (WPT) are used in solving, by the method of moments, a semicircular array of parallel wires electric field integral equation. First, the integral equation is solved by applying the direct method of moments via pointmatching procedure, results in a linear system with a dense matrix. Therefore, wavelet transformation and wavelet packet transformation are used to sparsify the impedance matrix, using two categories of wavelets functions, Biorthogonal (bior2.2) and Orthogonal (db4) wavelets. The far-field scattering patterns and the comparison between wavelets transformation and wavelet packet transformation in term number of zeros in impedance matrix and CPU Time reduction are presented. Numerical results are presented to identify which technique is best suited to solve such scattering electromagnetic problems and compared with published results.


I. INTRODUCTION
he art of computation of electromagnetic (EM) problems has grown exponentially for three decades due to the availability of powerful computer resources. Solving electromagnetic problems requires the application of Maxwell's Equations with the appropriate formulation and boundary conditions. Many powerful numerical analysis techniques have been developed in this area in the last 50 years [1,[4][5]. The use of method of moments (MoM) in EM has become popular since the work of Richmond in 1965 and Harrington [7] in 1967. The method has been successfully applied to a wide variety of EM problems of practical interest such as analysis of electromagnetic wave scattering, radiating, and guiding problems, to mention a few. In conventional MoM, the unknown function is expanded in terms of known basis functions with unknown coefficients. Unfortunately, the conventional bases, trigonometric, Legendre, Bessel, Hermite, and Chebyshev, when applied directly to the integral equations, generally lead to a dense matrix. As a result, the inversion and the final solution of such a system of linear equations are very time consuming [7].
There has been an explosive growth of research on wavelets, particularly in the last decade, resulting in a plethora of books, journals, and research papers dealing with the topic [8]. Recently, to fix the poor frequency localization of high frequency wavelet bases, Coifman, Meyer, and Wickerhauser [1][2][3][4][5][6] have generalized the construction of many other orthonormal bases of timefrequency atoms, corresponding to different tilings of the time-frequency plane. Wavelet packet base is one important example constructed, with time-frequency atoms that split the frequency and the time axis, respectively, in intervals of varying sizes. Wavelet packet bases are designed by dividing the frequency axis in intervals of varying sizes. These bases are particularly well adapted to decomposing signals that have different behavior in different frequency intervals [1][2][3][4][5][6][7][8][9][10][11][12]. The wavelets are a powerful technique for solving integral equations, resulting in sparse impedance matrices [1]. This is due to features of vanishing moments, orthogonality and multiresolution analysis in wavelets. The wavelet approach is used directly on the impedance matrix in order to make it sparse rather than on the operator equations in the form of a set of basis functions to approximate the unknown. The conversion of dense T matrices into a sparse form requires O(N 2 ) operations. The algorithm for solving the resulting sparse system requires only O(Nlog 2N) operations, as shown in [12].
This work is concerned to study of wavelets and wavelet packets for the Solution of Electromagnetic Integral Equations. The paper is divided into three main parts; first the problems of a plane wave by an arbitrary array of parallel wires and scattering are presented. The next section briefly presents the matrix formulation of wavelet transform and we discuss wavelet packets. The numerical examples suggesting sparsity of the transformed matrices and comparisons between wavelet and wavelet packets. The last section contains conclusions and suggestions for future research.

A. BASIC THEORY of SCATTERING by an ARBITRARY ARRAY of PARALLEL WIRES
Consider the problem of computing the scattering of a TM polarized electromagnetic wave from a 2-D an Arbitrary Array of Parallel Wires with the boundary contour. This problem is of more general nature than the scattering wire and antenna wire [2][3]. The far-field scattering characteristics are obtained from the surface current excited by an incident wave. An arbitrary array of N parallel, circular wires of infinite length placed parallel to the z-axis are assumed to exist in unbounded free space is illustrated in Fig. 1 [2][3]. Let a harmonic TM wave be incident on the wires. Assuming a time factor e jωt , the incident wave in phasor form is given by Let a harmonic TM wave be incident on the wires. Assuming a time factor e jωt , the incident wave in phasor form is given by [2] ! " = 0 (1) The incident wave may be, for example, one or more plane waves or a continuous spectrum of plane waves traveling in different directions, If a plane wave has an axis of propagation which makes an angle φ0 with the z axis, its field is given by The angle θi and φi, illustrated in Fig. 2, are the angular coordinates of the axis of propagation in the spherical coordinate system. The incident wave induces current on the surface of wire n. It can readily be shown that the field of a harmonic current In uniformly distributed on a circular cylinder of radius an has a z component given by [2][3] Where −I 5 6 = :;< " J0 is Bessel function of order zero, and H0 is Hankel function of the second kind of order zero. By induction theorem, if In is regarded as the induced current, Eq. (7) may be considered as the scattered field, i.e.
Where the summation is taken over all the N wires. On the surface of each wire (assumed perfectly conducting), Substitution of Eqs. (9) and (7) into Eq. (10) leads to N, And am is the radius of the mth wire. In matrix fro, Eq.(11) can be written as Or Where Once * 6 is calculated from Eq. 14, the scattered field can be obtained as Finally, we may calculate the ''distant scattering pattern,'' defined as The following example, taken from Richmond's work [3], will be used to illustrate the techniques discussed in the latter half of this section.

B. The WAVELET TRANSFORM
The idea behind applying wavelets to the Moment Method is to decrease the size of the discretization matrix of the integral equation. This approach is made by applying wavelets to the matrix obtained after discretization by the fast wavelet transform (FWT). Orthogonal wavelets have several properties such as orthonormal bases of L 2 (R), trade-off between orthogonality and continuity, order of regularity, and cancellation property, that are fascinating for electromagnetic field computations. Furthermore, orthogonal wavelets have localization properties in both the spatial and spectral domains [1,[7][8].
In this section, the impedance matrix (IM) obtained by MoM is reduced to sparse one by a wavelet matrix transform and, hence, the computation cost is relatively lowered. By using the MoM, we obtain the matrix equation as Eq.13, where [Z] is a dense impedance matrix. Introducing a wavelet matrix [W], the matrix equation is then transformed as [8]: 6 (20) Here T stands for the transpose of a matrix. One [I] 6 is solved; the desired solution is obtained as: Z 6 has numerous very small elements which can be neglected (threshold) without largely affecting the solution.
Practical considerations require that matrices must satisfy the following design criteria [8].
• The matrix must be (effectively) sparse.
• Matrices must be sparse, so that matrix-matrix multiplications cost only. • The condition number of is not much larger than that of. The last criterium is dictated by the fact that iterative methods converge more slowly for systems with larger condition numbers. The continuous operator and discrete operator are related by where Z F is the approximation of Z projected on Vj and tested in Vj. The nonstandard form method toward the sparsification of an existing impedance matrix is: Since: Where, is dense because both of the expansion and testing functions are scalets. These submatrices represent the interaction between the sources and fields in different subspaces [8]. Submatrix A, B, C, and Z of the matrix Z can be evaluated as follows.
The development of detailed steps relies on the two equations Where h and g are the lowpass and bandpass filter coefficients, respectively.
We may further improve the matrix sparsity in the nonstandard form as follows. Let us consider the lower-left quarter, C j j. Notice the fact that Finally, we are ready to attack the lower-right block C F .

C. WAVELET PACKET
This section is based on the work of Pr. Stéphane Mallat in his book "A Wavelet Tour of Signal Processing the Sparse Way "Ref [1]. Wavelet packets were introduced by Coifman, Meyer, and Wickerhauser [1] by generalizing the link between multiresolution approximations and wavelets. Orthonormal wavelet packet bases are computed with conjugate mirror filters that divide the frequency axis in separate intervals of various sizes.If the signal properties change over time, it is preferable to isolate different time intervals with translated windows. A space Vj of a multiresolution approximation is decomposed in a lowerresolution space Vj+1 plus a detail space Wj+1. This is done by dividing the orthogonal basis nφ F ( − 2 N q *∈P of Vj into two new orthogonal bases [1][2][3][4][5][6][7][8][9][10][11][12]  Theorem of Coifman, Meyer, and Wickerhauser generalizes this result to any space Uj that admits an orthogonal basis of functions translated by n2 j for n∈Z. Let nθ F/B (t − 2 Q/B n)q 5∈R be an orthonormal basis of a space Uj. Let h and g be a pair of conjugate mirror filters. Define The family Is an orthonormal basis of Uj.
Theorem Coifman, Meyer, and Wickerhauser proves that we can set Uj=Wj and divide these detail spaces to derive new bases. This space admits an orthogonal basis of scaling The two wavelet packet orthogonal bases at the children nodes are defined by the splitting relations (43): And Since Where, Any node of the binary tree is labeled by ( j, p), where j-L >=0 is the depth of the node in the tree, and p is the number of nodes that are on its left at the same depth j -L. Such a tree is illustrated in Figure 3. To each node ( j, p) we associate a space W F V , which admits an orthonormal basis by going down the tree [1].

III. NUMERICAL RESULTS
The above theory was implemented has been coded in Matlab language for calculating the induced current, and   [2][3]. We see a good agreement between three methods.
(a)   As seen in Figures 5, the MoM/WPT method gives the matrix sparsity more than the MoM/WT method, for example for Thr = 0.6 the MoM / WPT the rate of sparsity is 97% while for MoM/WT is 94% for the Daubechie wavelet, the same for the Biorthogonal wavelet. We observe also that the number of zeros in IM increases when increasing the threshold; this is due to canceling of the impedance matrix elements.
In order to better explain the benefits of the performed computational model, the following quantities have been also considered [19]:    Table 2 also reveals that the use of MoM/WT and MoM/WPT leads to a CPU Time reduction of 11.05% and 15.10%, respectively, for a threshold of 0.6. This is due to the fact that the CPU time to invert a matrix is proportional to the number of non-zero matrix elements. On the other hand, we can notice that the MoM/WPT(db4) given relative error less than MoM/WT.

IV. CONCLUSIONS
In this paper, the scattering from a circular array of parallel wires has been analyzed using a numerical technique based on the MoM and wavelets. In the first, the wavelet transforms and wavelet packet transforms have been applied to the MoM matrix in order to obtain a sparseness in the system matrix. In the second, a comparison among these techniques, MoM/WT and MoM/WPT, is presented using numerical examples. Several plots for the Scattering pattern presented, regarding various methods. The computed results were compared with published results Ref. [2][3] and very good agreement was observed. In summary, the above-described use of wavelet packet transformations for the solution of electromagnetic integral equations appears to be superior in all aspects of comparison including characterization of the matrix sparsity, and computation time.