Abstract and Applied Analysis

Volume 2014 (2014), Article ID 634974, 6 pages

http://dx.doi.org/10.1155/2014/634974

## Comparison Analysis Based on the Cubic Spline Wavelet and Daubechies Wavelet of Harmonic Balance Method

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an 710049, China

Received 25 January 2014; Revised 25 March 2014; Accepted 5 April 2014; Published 22 April 2014

Academic Editor: Eugene B. Postnikov

Copyright © 2014 Jing Gao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper develops a theoretical analysis of harmonic balance method, based on the cubic spline wavelet and Daubechies wavelet, for steady state analysis of nonlinear circuits under periodic excitation. The properties of the resulting Jacobian matrix for harmonic balance are analyzed. Numerical experiments illustrate the theoretical analysis.

#### 1. Introduction

The rapid growth in integrated circuits has placed new demands on the simulation tools. Many quantities properties of circuits are of interest to circuits designer. Especially, the steady-state analysis of nonlinear circuits represents one of the most computationally challenging problems in microwave design.

Harmonic balance (HB) [1–3] is very favorable for periodic or quasiperiodic steady-state analysis of mildly nonlinear circuits using Fourier series expansion. However, the density of the resulting Jacobian matrix seriously affects the efficiency of HB based on Fourier series expansion. More effective simulators are required to study steady-state analysis. Soveiko and Nakhla in [4] have provided the elaborate formulation for HB approach applying Daubechies wavelet series instead of Fourier series and obtain the sparser Jacobian matrix to reduce the whole computational cost. And Steer and Christoffersen in [5] have given the possibility of wavelet expansion for steady-state analysis. One advantage of wavelet bases is a sparse representation matrix of operators or functions which is favorable for solving the nonlinear system by Newton iterative method. But the main disadvantage is the waste of much time in storing Jacobian matrix due to the complex computation of Daubechies wavelet functions. And few studies have been reported on the efficient wavelet matrix transform which is very important in the wavelet HB approach. The cubic spline wavelet in [6, 7] has the explicit form and sparse transform matrix and derivative matrix. In this paper, we provide the theoretical analysis for HB method by using the cubic spline wavelet and Daubechies wavelets.

The remainder of this paper is organized as follows. In Section 2, we develop the HB method based on the cubic spline wavelet and Daubechies wavelets in [4] for nonlinear circuits simulations, respectively. Section 3 provides the theoretical comparison analysis in the sparsity and computation of Jacobian matrix obtained by the transform. And it is shown that the cubic spline wavelet HB method has sparser Jacobian matrix. Numerical experiments are provided in Section 4. It is concluded in Section 5.

#### 2. HB Formulation Based on the Cubic Spline Wavelets and Daubechies Wavelets

##### 2.1. Generalized HB Formulation

The harmonic balance (HB) method is a powerful technique for the analysis of high-frequency nonlinear circuits such as mixers, power amplifiers, and oscillators. The basic idea of HB is to expand the unknown state variable in electrical circuit equations by some series . Then the problem is transformed into the frequency domain focusing on the coefficients .

Let us consider the general approach of HB which assumes obtaining the solution of the nonlinear modified nodal analysis (MNA) equation in [8] which satisfies the following periodical boundary condition: where and are matrices, is a dimensional column vector of unknown circuit variables, and is a dimensional column vector of independent sources. Let be the basis; then the unknown function can be expanded . To solve (1) with periodic boundary condition (2), assume that the expansion basis is periodic with period and is a discrete vector containing values of sampled in the time domain at time points , . Then (1) can be written in the transform domain as a nonlinear algebraic equation system: where , and are matrices, especially, the matrix is a representation matrix of the derivative operator in expansion basis and, finally, and are the matrices associated with the forward and inverse transform arising from the chosen expansion basis. The nonlinear matrix system (3) can be solved by Newton iterative method where is the solution of the th iteration and is the Jacobian matrix of

Hence, the sparsity of this Jacobian matrix affects the computational cost of iterative method. Because these matrices and have a rather sparse structure due to the MNA formulation and for time-invariant systems is just a block matrix consisting of diagonal blocks, the sparsity of the Jacobian matrix is determined by three matrices , , and the representation matrix of the differential operator .

Given the base , the matrices , , and are constructed before those iterative methods are used. So the sparsity of the Jacobian matrix based on these different basis functions indicates how to solve the nonlinear algebraic system. Next, we give the formulation for two kinds of wavelet bases.

##### 2.2. Description of the Periodic Daubechies Wavelets

Two functions and are the wavelet function and its corresponding scaling function described by Daubechies [9]. They are defined in the frame of the wavelet theory and can be constructed with finite spatial support under the following conditions: where the coefficients and are the quadrature mirror filters (QMFs) of length . The quadrature mirror filters and are defined

The function has vanishing moments; that is, The number of the filter coefficients is related to the number of vanishing moments , and for the wavelets constructed in [9].

We observe that once the filter has been chosen, the functions and can be confirmed. Moreover, due to the recursive definition of the wavelet bases, via the two-scale equation, all of the manipulations are performed with the quadrature mirror filters and . Especially, the wavelet transform matrix and the derivative matrix for the differential operator can be obtained by the filters.

In HB method the wavelets on the interval are required. Hence, periodic Daubechies wavelets on the interval are constructed by periodization. Here, we describe the discrete wavelet transform matrix by the periodic Daubechies wavelets. The discrete wavelet transform with the period can be considered as a linear transformation taking the vector determined by its sampling data into the vector where stands for the scaling coefficients of the function and for the wavelet coefficients.

This linear transform can be represented by the dimensional matrix such that If the level of the DWT is , then the DWT of the sequence has exactly coefficients. The transform matrix is composed of QMFs coefficients and as follows: where is the length of the filters.

The periodized Daubechies wavelet HB formulation has been formulated in [4], so we have , where is a band limited circulant matrix with its diagonals filled by in [10], where with the following properties: in which are autocorrelation coefficients of the QMFs And the matrix is the inverse matrix of the forward transform matrix which satisfies due to the orthogonality of the matrix .

##### 2.3. The Cubic Spline Wavelet Basis

Consider the cubic spline wavelets as the expansion base in HB technique. The cubic spline wavelets are constructed in [7], which are semiorthogonal wavelets. The high approximation rate and the interpolation property can be inherited from spline functions. Therefore, the cubic spline wavelet transform matrix and the differential operator representation matrix have the following properties which are suitable for HB method.

Due to the periodic condition , we must use the periodization functions of the cubic spline wavelets on the interval . For convenience, we still denote by the periodic function. Let us assume that expansion bases are where . Correspondingly, the unknown state variable is approximated by the bases of these spaces where Based on the interpolation property of the cubic spline wavelets, we have Denote the expansion coefficients by a dimensional vector , that will be determined by satisfying the collocation conditions, . Interpolate at the collocation points as follows: Substituting the expressions into (1), we obtain nonlinear discrete algebraic systems.

Denote by the cubic spline wavelet transform matrix. We introduce an inverse wavelet transform (IWT) which maps its wavelet coefficients to discrete sample values with length ; that is . The inverse transform matrix is where denotes a tridiagonal matrix with dimension and is a tridiagonal matrix with dimension .

We obtain the derivative matrix in [11] as follows: where and these constants in these matrices and can be referenced from the formulae (2.20a)–(2.20d) in [11].

For the whole nonlinear equation system where , , and are tridiagonal matrices, the triangular decomposition of the tridiagonal matrix can be used to decompose the Jacobian iterative matrix.

#### 3. Comparison Analysis

Using HB method to solve nonlinear ODEs, the Newton iterative form is obtained. Here we want to analyze the sparsity of derivative matrix and wavelet transform matrix of the Jacobian matrix based on two wavelets.

##### 3.1. The Comparison of the Wavelet Transform Matrixes and

Now we analyze the sparsity of the matrixes and . The computation cost of Jacobian matrix results from the number of nonzero elements of the wavelet transform matrix . We analyze the nonzero elements (NZ) of matrices and . Let the maximum level of wavelet decomposition be , . From [12], we have the nonzero numbers of the matrix are For the cubic spline interpolation wavelets, the transform matrix has the following property:

##### 3.2. Comparison of the Derivative Matrices and

The sparsity of the derivative matrix is an important property of Jacobian matrix . According to the approach in [4], matrix is composed of , where is the transpose of the matrix . Since both and in this case are band-limited matrices, as well as , the resulting matrix is also a band-limited matrix. Especially, the nonzero element numbers of matrix are , so we have

By the formulation in Section 2.3, the number of nonzero element of matrix is . It follows that cubic spline wavelets yield a sparser derivative matrix than that of Daubechies wavelets. Thus, the Jacobian matrix of the cubic spline wavelets is much sparser than the periodic Daubechies wavelet.

#### 4. Numerical Experiments

In this section, we will give the sparsity figures of the transform matrix and the derivative matrix based on two kinds of wavelets. For simplicity, assume the matrices , , and are diagonal. Figure 1 is the sparsity of using the periodized D4 Daubechies wavelets.

In Figure 2 we plot the sparsity of the derivative matrix of periodic Daubechies wavelets and the matrix or of the derivative matrix .

#### 5. Conclusions

In this paper, we formulate the comparison analysis of harmonic balance method based on the cubic spline wavelets and periodic Daubechies wavelets. It is shown that the cubic spline wavelet HB method has the special structure for Jacobian matrix compared to the Daubechies wavelet HB method to solve steady-state analysis of nonlinear circuits.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The work was supported by the Natural Science Foundation of China (NSFC) (Grant nos. 11201370 and 11171270) and the Fundamental Research Funds for the Central Universities.

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