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On Daubechies Wavelet Based Time Domain Scheme

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On Daubechies Wavelet Based Time Domain Scheme Youri V. Tretiakov and George W. P a n D e p a r t m e n t of Electrical Engineering, A r i z o n a State University Abstmct- Recently, the wavelet-Galerkin time domain (WGTD) approach was preposed by Cheong et al., which signifleantly simplified the multi-resolution time domain technique (MRTD) of Krurnphoh, Katehi et al. In this paper, we provide a rigorous analysis of the WGTD, employing positive Sampling functions and their hiorthogonal dual. This biothogonal system has exact interpolation properties and demonstratea superiority over the FDTD in terms of memory and speed. Numerical examples and compariaons t o the traditional FDTD results are provided. I. INTRODUCTION The multiresolution time domain technique (MRTD) was proposed in [I] using the Battle-Lemarie wavelets in space. In the MRTD, the standard Galerkin procedure converts the two vector curl equations into a system consisting of six updating equations similar to the traditional FDTD scheme of Yee [Z]. For instance, one of the six equations is For i < 0 the coefficients {a,) are given by the symmetry relation a, = -a-1-,. To improve MRTD, the Wavelet-Galerkin time-domain (WGTD) scheme was proposed in 131, employing Daubechies scaling functions of order two (Dz)141 with the support [0,3].In this paper the positive sampling basis and its hiorthogonal dual tesing fuctions are constructed, employing the Daubechies D2 scaling functions. Owing to the exact sampling property of the basis functions and their biorthogonal testing functions, the error estimate can be easily conducted. 11. W G T D SCHEMEBASEDUPON SHIFTEDDAUBECHIEDSz SCALING FUNCTIONS It was remarked by mathematicians 151, that shifted Daubechies Dz scaling func- tions has approximate sampling properties, namely d(k + MI) 6r,o (3) -- cL.-1 Az i=-L, ~1L+1/2~,1,2,......+i+i/.] where MI = J_',"z+(z)dz. Chong et al. immediately recognized these approximate samplingproperties, and constructed the Wavelet-Galerkin time domain (WGTD) algorithm 131. To make use of the shifted interpolation property (3) we recall the following expansion functions (1) where L, denotes the effective support size of the basis function $(z). It has been found numerically [l]that coefficients {a,) for i 2 9 and i 5 -10 are negligible, that is L. 9. The coefficient { a i } represents the weight of the contributions to a node by its neighbors and for i 2 0 is calculated numerically from Plotted in Fig.1 are functions defined in (4) with Az = 1 and m = 0,1,2. Due to finite support of Daubechies scaling functions, the number of nonzero coefficients {a,} is also finite. One can easily verify that ai # 0 for -3 5 i 5 2 and hence L. = 3 in (1). The numerical values of the coefficients {ai} have been tabulated in [3]. From the above discussion it follows that the use of Dauhechies scaling functions is more computationally efficient than that of 0-7803-7070-8/01/$10.00 0 2 0 0 1 IEEE 810 the above-defined sampling function SmL(z) Sm(k) =L , n . (9) The support of the sampling function S,(z) is [m - 1,+CO). Notice that a sampling function {S,(z)} is not orthogonal with respect to its shift Fig. 1. Daubechies shifted scaling functions ( N = 2) for rn = 0 , 1 , 2 . the Batt,le-Lemarie scaling functions. Another advantage is the interpolation propcrty (3). From this property it follows that Therefore the biorthogonal testing functions Q,(z) were introduced in [GI, such that E$(. Qn(x) = - p ) $ ( n - p ) (11) P This will safe time when one has to sam- Due to the finite support of the Daubechies ple field coniponcnts for further numerical scaling functions we can simplify expression computations. (11)to the following compact form 111. T I I EWGTD SCHEMBEASEDON BIORTHOGONAL SAhlPLINC BASE To construct Daubechies based biort,hogonal sampling basis we use the following exprcssion in [6]for a positive sampling function Qn(2)= $ ( ~ - n + 2 ) $ ( 2 ) + $ ( ~ - ~ ~ + 1 ) $ ( 1 ) . (13) + From (13) it follows immediately that Qn(z)is supported on [71 - 2 , n 21. We plotted the sampling function S ( x ) = So(z) and biorthogonal function Q(z) = BO(.) in Fig. 1. wlierc $(z) is t,he Daubcchies scaling func- t,ion, not necessary just Dz. However, i n rest of the paper w e shall rcstrict the Daubcchies scaling functions to only Dz. e, e ~ 1 1pearameter U is equal to -I/&. w e notice here bhat $(l)= $(2) = (see [4]Cor dct,ails) and thus we can rewrite (6)in t,he form I,,. I '.S~, 0 > z I 4 5 Fig. 2. Sampling function S(z)and biorthogonal function Q(z) (7) We define the shifted versions of the sam- pling function S(z) as LL(z) = S(z - m). (8) I t can be shown analytically that the following cxact interpolating property holds for For the WGTD scheme we use the follow- ing hasis for expansion ( 2 s,(z) = S - m) (14) and the biorthogonal testing functions 81 1 After applying the standard Galerkin procedure, The two vector curl Maxwell's equations becomes six updating expressions, analogous to (1) with coefficients { a , } given 1: by ai = Q - ias(l/zx(z)dx~, (16) f u ~ ~ o n t o ~ ~ ~ ~ i ~ n Fig. ~ ~3. Two-dimesianal parallel plate resonator. teo~u~~or~~~ l ~ testing function Q(z), the number of the nonzero coefficients{a,} is small, as in the case of WGTD technique with the shifted Daubechies basis. In fact, we obtained from (16) that for i 2 3 all coefficients { a i } are exactly zeros. This is due to the specific supports of the functions S(z) and &(z). It was found numerically that for i 5 -4 all coefficients {a,} in (16) are negligiblysmall. For -3 5 i 5 2, we have verified analyticaly that the followingidentity is true E, = 0 and H, = 0. The dimesions are a = 2 m, b = 1 m and the time step At = sec. The electric field values E, were sampled during the time period T, = 216At and the fast Fourier transform (FFT) was performed to obtain the spec- trum of the sampled field E,. Illustrated in Fig. 4 are the numericall results ob- tained with 15 x 7 = 105 Yee cells for both FDTD and WGTD techniques, along with analytical values. It can be seen clearly that WGTD provides better agreement with an- alytical solutions, though it is slower than the FDTD approach. The computational time is 8.93 s for FDTD method and 39.89 s for the WGTD. This means that L. = 3 and { a , ) are exactly the same as in [3]. We conclude that the numerical results will be exactly the same for technique in the previous section and tecnique in this sections, although the sampling function S,(x) obeys the exact interpolation property (9). The advantage of using the biorthogonal sampling system is that the error introduced in the truncation process can be explicitly identified. Hence, the error bound can be estimated easily. In con- To achieve the WGTD accuracy, we decresed the size of the cell in the FDTD. As a result, 40 x 20 = 800 Yee cells for FDTD demonstrated the precision of the WGTD with 105 cells. The results are shown in Fig. 5. The computational time for the FDTD increased to 66.36 sec due to increased number of Yce cells. As can be seen in the figure, both methods give almost the same results for the eigenfrequency,but the WGTD approach here is more efficient in terms of computational time and computer memory. trast, for the shited Daubechies WGTD, the errors are inexplicitly reflected by the numerical values of the Dz scaling func- tions. As mathematicians pointed out that no one knows the exact value of b(&), nor d(l+ L9/4. Iv. NUMERICARLESULTS To validate the newly implemented biorthogonal sampling basis we include three examples. Example 1. Eigenfrequency problem. A 2D parallel plate resonator is depicted in Fig. 3. For the sake of simplicity we analyze only the polarization for which E, = 0, Fig. 4. Magnitude of the electric field component E , in the frequency domain (air-field 2 0 res0"ator). 812 . . . . . . . . .. . I.. . . Fig. 5. Magnitude of the electric field component .~ E. in the freauencv domain fair-field 2D res- onator). Example 2. Partially loaded 2D resonator. Here we consider the nrevious resonator, but filled in part with a dielectric slab, as shown in Fig 3 The additional parameters are h = 0 2 m and = 2 0. We use the same number of Yee cells for the WGTD and FDTD, namely 20 x 10 = 200 From Fig 6 , it can be seen again that the WGTD techniques gives better prediction of the resonance frequencies than the standard Yee’s FDTD approach Fig. 7. Normalized propagation constant @./to frequency (”). tional time was approximately equal t o 150 seconds for WGTD and 204 seconds for FDTD. V. CONCLUSION The Wavelet-Galerkin Time Domain scheme is derived based on biortbogonal sampling system of the Dauhechies D Z scaling functions. The exact interpolation property of the basis functions, the rapid decay and finite support of the testing functions lead to simplified forms of the updating equations. The newly implemented WGTD technique demonstrated better efficiency in terms of computational time and computer memory than the traditional FDTD and also previuosly developed MRTD technique. The new algorithm was tested numerically with a number of examples and showd accurate results. a . . I Y Y Y Y I U Fig 6 Magnitude of the electric field component E , in the frequency domain (partially field with dtelectnc 2D resonator) E x a m p l e 3. Waveguide problem For the last numerical example we model an air-field rectangular waveguide The crosssectional dimensions are a = 2 m, b = 1 m In Fig 7 we plotted the normalized propa- gation constant 8,f k o versus frequency for a few eigenmodes, starting with the dominant mode TE;, To verify our numeri- cal WGTD results we also plotted the dispersion curves from theoretical formulation and from FDTD For WGTD method we used mesh with 20 x 10 = 200 cells To reach a compative precision, the FDTD mesh requires 44 x 22 = 968 cells For each particular value of & the computa- REFERENCES [I] M. Krumphalz and L. P. B.Katehi, “MRTD: New time-domain schemes based on multiresolution analysis,” I E E E %ns. Micmwove Theory Tech.,vol. 44, pp. 555-571, Apr. 1996. 121 K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equation in isotropic media,” IEEE %ns. Antennos Pmpagot., vol. 14, pp. 302-307, May 1996. 131 Y. W. Cheong, Y. M. Less, K . H. Ra, J. G. Kang, and C. C. Shin, “WaveletGalerkin sheme of time-dependent inhomogeneous electromagnetic problems,” IEEE MIcmwave Guided Wove Lett., vol. 9, pp.297299, Aug. 1999. 141 1. Daubechies, Ten Leeturns on Wavelets. Philadelphia, PA: SIAM, 1992. [SI W. Sweldens and R. Piessens, “Wavelet a m - plins techniques,” in Pmc. Statistical Computtng Section, 1993, pp. 20.29. [6] G.G. Walter. Wavelets and other orthogonal systems wrth opplturtsons. CRC Press, Boca Raton, Florida, 1994. 813




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