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

and Applied Analysis 3 Daube is composed of QMFs coefficients k and k as follows:


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][2][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.

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 () = ∑   V  ().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]  ẋ +  +  () +  = 0, Abstract and Applied Analysis 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 {V  } be the basis; then the unknown function () can be expanded () = ∑   V  ().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 [  ],  = 1, . . .,   .Then (1) can be written in the transform domain as a nonlinear algebraic equation system: where Ĉ, , and Ĝ are     ×     matrices, especially, the matrix  is a representation matrix of the derivative operator / in expansion basis and, finally,  and  −1 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 Φ() Hence, the sparsity of this Jacobian matrix () affects the computational cost of iterative method.Because these matrices Ĉ and Ĝ 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 ,  −1 , and the representation matrix  of the differential operator /.
Given the base {V  }  =1 , the matrices , , and  −1 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.

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 {ℎ  } −1 =0 and {  } −1 =0 are the quadrature mirror filters (QMFs) of length   .The quadrature mirror filters {ℎ  } 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  = 2 for the wavelets constructed in [9].We observe that once the filter {ℎ  } has been chosen, the functions  and  can be confirmed.Moreover, due to the recursive definition of the wavelet bases, via the twoscale equation, all of the manipulations are performed with the quadrature mirror filters {ℎ  } 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 [0, ] are required.Hence, periodic Daubechies wavelets on the interval [0, ] are constructed by periodization.Here, we describe the discrete wavelet transform matrix by the periodic Daubechies wavelets.The discrete wavelet transform with the period  = 2  can be considered as a linear transformation taking the vector f  ∈   determined by its sampling data into the vector where c  stands for the scaling coefficients of the function () and d  for the wavelet coefficients.This linear transform can be represented by the  = 2  dimensional matrix  Daube such that If the level of the DWT is  ≤ , then the DWT of the sequence has exactly 2  coefficients.The transform matrix  Daube is composed of QMFs coefficients {ℎ  } and {  } as follows: where  is the length of the filters.
The periodized Daubechies wavelet HB formulation has been formulated in [4], so we have Daube , where  is a band limited circulant matrix with its diagonals filled by   in [10], where with the following properties: And the matrix  −1 Daube is the inverse matrix of the forward transform matrix  Daube which satisfies  −1 Daube =   Daube due to the orthogonality of the matrix  Daube .

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  cubic and the differential operator representation matrix  cubic 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 [0, ],  > 4. For convenience, we still denote by V  () the periodic function.Let us assume that expansion bases are where   = 2  ,   = 2   − 1. 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 as follows: Substituting the expressions into (1), we obtain nonlinear discrete algebraic systems.Denote by  cubic the cubic spline wavelet transform matrix.We introduce an inverse wavelet transform (IWT)  −1 cubic which maps its wavelet coefficients x to discrete sample values f  with length   ; that is  −1 cubic x = f  .The inverse transform matrix  −1 cubic is where  denotes a tridiagonal matrix with dimension  + 2 and   is a tridiagonal matrix with dimension 2  .
We obtain the derivative matrix  cubic in [11] as follows: where and these constants in these matrices  1 and  2 can be referenced from the formulae (2.20a)-(2.20d) in [11].
For the whole nonlinear equation system where  1 ,  2 , and  −1 cubic are tridiagonal matrices, the triangular decomposition of the tridiagonal matrix can be used to decompose the Jacobian iterative matrix.

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.
For the cubic spline interpolation wavelets, the transform matrix  −1 cubic has the following property:

Comparison of the Derivative Matrices 𝐷 𝐷𝑎𝑢𝑏𝑒 and 𝐷 𝑐𝑢𝑏𝑖𝑐 .
The sparsity of the derivative matrix  is an important property of Jacobian matrix ().According to the approach in [4], matrix  Daube is composed of  Daube  −1 Daube , where  −1 Daube is the transpose of the matrix  Daube .Since both  Daube and  −1  Daube in this case are band-limited matrices, as well as , the resulting matrix  Daube is also a band-limited matrix.Especially, the nonzero element numbers of matrix  are ( − 5) ×  + 2 ∑ −2 =1  ∼ (), so we have By the formulation in Section 2.3, the number of nonzero element of matrix  cubic =  −1 1  2 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.

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 Ĉ, Ĝ, and [/  ] are diagonal.Figure 1 is the sparsity of  Daube using the periodized D4 Daubechies wavelets.
In Figure 2 we plot the sparsity of the derivative matrix of periodic Daubechies wavelets and the matrix  2 or  1 of the derivative matrix  cubic .

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

3. 1 .
The Comparison of the Wavelet Transform Matrixes  Daube and   .Now we analyze the sparsity of the matrixes  Daube and  cubic .The computation cost [  /  ] −1 of Jacobian matrix () results from the number of nonzero elements of the wavelet transform matrix .We analyze the nonzero elements (NZ) of matrices  Daube and  cubic .Let the maximum level of wavelet decomposition be ,  = 2  ,  ≤ .From [12], we have the nonzero numbers NZ  Daube of the matrix  Daube are

Figure 1 :Figure 2 :
Figure 1: (a): Sparsity pattern of the periodized transform matrix  Daube by the periodized D4 wavelets; (b): sparsity pattern of the inverse transform matrix  −1 cubic of the cubic spline wavelet.