View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Fukui Repository A Modified Method for Circuit Analysis using Haar Wavelet Transform with Adaptive Resolution —For Circuits with Waveform with Sharp Convex Ranges— Masanori Oishi and Seiichiro Moro Tadashi Matsumoto Department of Electrical and Electronics Engineering, Department of Electrical and Electronics Engineering, University of Fukui Fukui University of Technology 3-9-1 Bunkyo, Fukui, 910-8507 Japan 3-6-1 Gakuen, Fukui, 910-8505 Japan Email: foishi,[email protected] Email: [email protected] Abstract— In this paper, we propose a modified method for the the first-derivative. To overcome this problem, it is considered circuit analysis using wavelet transform with adaptive resolutions. that we can propose a better method if the way to pick out Recently, many approaches to the circuit analysis using the the range using second-derivative value and the previous way wavelet transform have been proposed. Previously, we proposed method which can choose adaptive resolution automatically using are combined. In this paper, we propose a modified method multiresolution analysis. However, the method cannot be applied for the circuit analysis using wavelet transform with adaptive when the waveform has the sharp convex ranges. The proposed resolutions by automatically picking out the ranges which method improves the way to pick out the range and achieves require higher resolution analysis by using both the result of more accurate and efficient calculations. the multiresolution analysis and the slope of derivative value. By the proposed method, it is considered that more accurate I. INTRODUCTION and efficient calculation can be achieved even if the behavior The wavelet transform has been often used in signal process- of the circuit is hardly predicted. In this paper, we confirm the ing because of its orthogonality and multiresolution property. effectiveness of the above method using a simple example. Recently, much attention has been paid to the method for circuit analysis using wavelet transform [7]–[14]. In particular, II. HAAR WAVELET MATRIX Barmada et al. have proposed the Fourier-like approach for Haar functions are defined on interval [0,1) as follows, the circuit analysis using the wavelet transform [11]. In [11], the integral and differential operator matrices are introduced 1 h = p ; (1) to the analysis, and the differential and integral equations are 0 m transformed into the algebraic equations like as using Fourier 8 1 > j k−1 k− or Laplace transform. Moreover, the method can treat time < 2 2 ; ≤ t < 2 ; 1 2j 2j varying or nonlinear circuits. Therefore, this method is useful p × j k− 1 hi = − 2 2 ≤ k (2) > 2 ; j t < j ; for various circuit analyses. m : 2 2 0; otherwise in [0; 1); However, in that method, the use of Daubechies wavelet makes the handling of the operator matrices complicated, espe- i = 0; 1; ··· ; m − 1; m = 2α , cially, in the edges of the interval. Thus, we have proposed the where α is positive integer, and j and k are nonnegative circuit analysis method using Haar wavelet transform [14]. The integers which satisfy i = 2j + k; i:e:; k = 0; 1; ··· ; 2j − 1 Haar wavelet is easy to handle itself, and the operator matrices for j = 0; 1; 2; ··· : using the Haar wavelets are easily derived by introducing the ~y is m × 1-dimensional vector whose elements are the block pulse functions [5], [6]. Moreover, the proposed method discretized expression of y(t) and ~c is m × 1-dimensional can treat the nonlinear time varying circuits. coefficient vector. H is m × m-dimensional Haar wavelet In addition, Haar wavelets have the merit to be able to matrix defined as 2 3 analyze the trajectory near the singular points where the ~ h0 trajectory moves rapidly with high resolution because of 6 ~ 7 6 h1 7 the orthogonality and localization property of the wavelet 6 7 H = 4 . 5 ; (3) functions. As circuit analysis methods using this merit, some . ~ methods were proposed to pick out the ranges automatically hm−1 where the trajectory moves rapidly near singular points. [16] ~ However, these methods cannot be applied when the slope of where hi is 1 × m-dimensional Haar wavelet basis vector a curve switches in a short time, because we considered only whose elements are the discretized expression of hi(t). Using 978-1-4244-3896-9/09/$25.00 ©2009 IEEE 299 these vectors and matrix, Haar wavelet transform and inverse Haar wavelet transform are described as follows, respectively, ~c = H~y; (4) ~y = HT ~c (= H−1~c ): (5) III. INTEGRAL AND DIFFERENTIAL OPERATOR MATRICES USING HAAR WAVELET Fig. 1. Van der Pol oscillator circuit. The basic idea of the operator matrix has been firstly introduced by using Walsh function [5]. However, in logical the expression in wavelet domain. See details in [14]. way, the matrices introduced by block pulse function are more fundamental [4], [5]. The block pulse function is the set of m Capacitor: rectangular pulses which have 1=m width and are shifted 1=m each other. − V = C 1(C V + Q I) (11) The integral operator matrix of the block pulse function matrix w 0 0 H T B is defined as the following equation [5], [6]. Cw = Hdiag[C(i0; t0);C(i1; t1); ··· ;C(im−1; tm−1)]H Z i Inductor: B(τ)dτ ≡ QB · B(t); (6) 0 " # I = L−1(L I + Q V ) (12) X1 w 0 0 H 1 1 i T Q × = I × + P ; (7) L = Hdiag[L(i ; t );L(i ; t ); ··· ;L(i − ; t − )]H B(m m) m 2 (m m) (m×m) w 0 0 1 1 m 1 m 1 i=1 Resistor: where B(t) is m × m-dimensional matrix whose elements are the discretized expression of the block pulse functions bi(t), V = RwI (13) i = 0; 1; ··· ; m − 1 "and # T Rw = Hdiag[R(i0; t0);R(i1; t1); ··· ;R(im−1; tm−1)]H i 0 I(m−i)×(m−i) P(m×m) = for i < m, 0(i×i) 0 i V. CIRCUIT ANALYSIS USING ADAPTIVE RESOLUTION P = 0 × for i ≥ m. (m×m) (m m) WITH MULTIRESOLUTION ANALYSIS [16] And the inverse matrix Q−1 is calculated as follows [5]: B(m×m) In this section, we show a previous method for automatically " # picking out the ranges to use the adaptive resolution by mX−1 −1 1 i i analyzing a example as follow. Q = 4m I × + (−1) P : (8) B(m×m) 2 (m m) (m×m) The circuit model is Van der Pol oscillator circuit as shown i=1 in Fig. 1. The characteristic of nonlinear resistor of this circuit As the Haar wavelet matrix H is the set of the orthogonal is; functions, the integral matrix of H is given as follows: 3 2 i(v) = −g1v + g3v = (−g1 + g3v )v; (14) T −1 T T QH = HQBH = HQBH : (9) and branch equations in the wavelet domain are shown as; −1 − IC = C!QH (V V0); −1 Similarly, the differential matrix of H can be written as IL = IL0 + L! QH V; −1 T −1 −1 T −1 T I(V ) = GwV: (15) QH = H(QB) H = H(QB) H : (10) where H~v(0_) = V0;H~iL(0_) = IL0; H~v = V; IV. HAAR WAVELET EXPRESSION OF BRANCH − 2 − 2 ··· − 2 T Gw = Hdiag[ g1+g3v0; g1+g3v1; ; g1+g3vm−1]H . CHARACTERISTICS OF NONLINEAR TIME INVARIANT In this example, we take g1 = 2; g3 = 1;C = 0.05[F], L = CIRCUIT ELEMENTS 0.05[H] , vC (0_) = 0[V], iL(0_) = 1[A]. As the Haar wavelet is defined on interval [0,1), the generic From these characteristics, the nonlinear algebraic equations interval [tmin; tmax) can be rescaled by a new variable τ on are solved numerically with respect to V , and calculation [0,1), where [tmax − tmin)τ + tmin. In this paper, tmin = result for m = 64 without the adaptive resolution is shown in 0 without loss of generality, then capacitance c [F] and Fig. 2. If the waveforms have the singular points, more detailed inductance l [H] are scaled to C = c=tmax and L = l=tmax, analyses are needed by using the smaller intervals. However, respectively. it makes the calculation cost higher, for example, by taking Next, we show the Haar wavelet expression of branch m = 128. Even if waveforms are predicted by experimentally characteristics of nonlinear time varying circuit elements for imagine, we cannot predict the singular points for every circuit 300 Fig. 2. Calculation result for m = 64 without adaptive resolution [16]. Fig. 4. Multiresolution analysis g(5). Fig. 3. Multiresolution analysis for m = 64. Fig. 5. Calculation result with adaptive resolution. in the real world. To overcome this problem, we proposed the resolution wavelets. To avoid the over sampling, we set the method using the results of the multiresolution analysis [16]. sampling width as 1/2α to satisfy 2α−1 < 2p ≤ 2α where p is Figure 3 shows the results of the multiresolution analysis of the number of plots in the picked out ranges. Table I shows the the waveform shown in Fig. 2. f(j) indicates the amounts of relationship between the picked out ranges and the sampling the results for 0; 1; ··· ; j − 1-th resolution wavelets and g(j) width. The result for this method is shown in Fig. 5. The indicates the j-th resolution ones. Note that f(6) = f(5) + total number of plots is 97 and it is less than 128 that are the g(5).