STEADY STATE ANALYSIS OF MULTITONE NONLINEAR CIRCUITS IN WAVELET DOMAIN

Transcription

1 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 1 of 45 STEADY STATE ANALYSIS OF MULTITONE NONLINEAR CIRCUITS IN WAVELET DOMAIN Nick Soveiko and Michel Nakhla Department of Electronics, Carleton University, Ottawa, Ontario, K1S 5B6, Canada Keywords -- Nonlinear circuits, Time-frequency analysis, Harmonic analysis, Wavelet transforms. Abstract -- This paper introduces a new approach to steady state analysis of nonlinear microwave circuits under periodic excitation. The new method is similar to the well known technique of Harmonic Balance, but uses wavelets as basis functions instead of Fourier series. Use of wavelets allows significant increase in sparsity of the equation matrices and consequently decrease in CPU cost and storage requirements, while retaining accuracy and convergence of the traditional approach. The new method scales linearly with the size of the problem and is well suited for simulations of highly nonlinear, multitone and broadband circuits. I. INTRODUCTION Steady state analysis of nonlinear circuits represents one of the most computationally challenging problems in microwave design. Steady state analysis implies that response of the circuit has to be found at times when all the transients have sufficiently died out [1]. This immediately rules out time marching schemes, especially for stiff circuits, unless a good solution for initial conditions can easily be obtained (shooting methods, [2]). Direct frequency-domain methods are not applicable to the nonlinear circuits either for obvious reasons. Existing methods for steady state analysis of nonlinear circuits combine both frequency domain and time domain analysis and are generally known as the Harmonic Balance. The essence of this technique is to replace the original Initial Value Problem with a Boundary Value

2 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 2 of 45 Problem with periodic boundary conditions and to solve the BVP in an appropriate basis that ensures periodicity of the solution. Harmonic Balance-like methods rely on fast and stable ways of solving nonlinear algebraic equations as well as reasonably fast numerical techniques for going back and forth between time and frequency domain. A great deal of research has been performed on finding ways of accelerating Harmonic Balance (e.g. [3]-[6]), however size and density of the Jacobian matrix remains the principal bottleneck for all methods based on Fourier series expansion. Wavelet techniques 1 were introduced to the area of transient analysis of nonlinear circuits by Zhou et al. in [8] and by Steer and Christoffersen in [9]. Possibility of wavelet expansion for steady state analysis was acknowledged in [9], but the matter was not pursued further. Application of the time domain adaptive wavelet collocation method developed in [8] to the steady state analysis of nonlinear circuits was presented in [10] by Li et al. In this paper, we present a new numerical method for steady state analysis of nonlinear circuits, which was developed independently from [10]. The method is somewhat similar in formulation to the traditional Harmonic Balance technique, but uses a wavelet Galerkin scheme for expansion of the circuit equations. The new method uses degenerated wavelet decomposition tree that allows to reduce density of the Jacobian from a matrix with essentially dense blocks to O(N) bandlimited matrix. We also present computational cost analysis for the proposed method that allows to identify application areas where the new method provides significant advantages over Harmonic Balance techniques, namely simulation of highly nonlinear, multitone and broadband circuits. This paper is organized as following. Section II describes generalized matrix formulation for the steady state analysis that is independent of expansion basis. Section III reviews the traditional Harmonic Balance formulation based on the generalized matrix formulation. Section IV introduces wavelet formulation and explains construction of the matrices. Section V performs 1. For a brief introduction to wavelets the reader could be referred to our previous paper [7].

3 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 3 of 45 analysis of the computational complexity for both Harmonic Balance and wavelet formulations and also addresses spectrum truncation issues. Numerical results are presented in Section VI. II. GENERALIZED MATRIX FORMULATION Consider a lumped component 1 nonlinear circuit that is described by nonlinear Ordinary Differential Equations (ODEs) in time domain. Most often these equations are written in the MNA formulation 2 [11]: Cẋ+ Gx + f ( x) + u = 0 (1) Where C and G are N x N x matrices, x is a column vector of unknown circuit variables and u is a vector of independent sources. For steady state analysis we must either assume that the circuit is under periodic excitation, or that the circuit is autonomous and generates periodic output. In both cases solution vector x is periodic with fundamental frequency corresponding to period τ : xt ( + τ) = xt () (2) Equation (1) with boundary conditions (2) can be solved by expanding nonlinear ODEs (1) into a nonlinear algebraic equation for the expansion coefficients of x using approach that is well known as Method of Moments [12]. Suppose that we have an expansion basis m { v k } k = 1, so that we can write a best approximation for x: m x = lim X k v k m k = 1 (3) In order for the solution to satisfy boundary conditions (2), expansion basis must satisfy these boundary conditions as well. In other words, expansion basis must be periodic. Let us assume that [x l ] is a discrete vector containing values of x sampled in time domain at time 1. Formulation for the circuits with distributed parameter components described in frequency domain should be the subject of a separate publication. 2. State space formulation immediately follows from here by assuming C to be identity matrix.

4 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 4 of 45 points [t l ], l = 1 N t and that basis { v k } is periodic and has a pair of forward T and inverse T discrete transforms associated with it: X = T[ x l ], [ x l ] = T X (4) The nonlinear term in (1) can be represented in the following form: F( X) = Tf ( T X ) (5) Equation (1) then can now be written in the transform domain as a nonlinear matrix equation: ĈDX + ĜX + F( X) + U = 0 (6) Where Ĉ, D and Ĝ are N t N x N t N x matrices. Ĉ and Ĝ are obtained from C and G respectively by taking their tensor product with a N t N t identity matrix. We denote left side of (6) as Φ( X ) and write it as Φ( X ) = ( ĈD + Ĝ)X + F( X) + U = 0 (7) d Matrix D in (6)-(7) is projection of the derivative operator onto space spanned by { v : dt k } [ D ij ] d = ----v dt i, v j Solution of (7) is usually performed using Newton iterations. Assuming guess for X, the linear matrix equation to be solved at each step becomes X ( 0) (8) as the initial where X () i is the solution of i-th iteration, Φ( X ) is defined by (7) and J( X) is the Jacobian of Φ( X ) : J( X () i )( X ( i + 1) X () i ) = Φ( X () i ) (9) Φ k J( X) = [ J kl ( X )] =, k, l = 1,,( N (10) X t N x ) l Substituting (7) into (10) and applying chain rule, we obtain the following expression for computing the Jacobian [13]: J( X) = ĈD+ Ĝ + T f k T x l (11)

5 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 5 of 45 f k Jacobian is computed as a sum of three matrix components: ĈD, Ĝ and T T. Sparsity x l of the Jacobian becomes equal to the sparsity of the densest of these three components. Matrices C and G result from the MNA formulation and typically have rather sparse structure. Matrix of derivatives D will have sparse structure only if chosen basis allows sparse representation of the derivative operator, i.e. most of the elements in (8) vanish. This naturally happens if { v i } have local support (local support for basis functions means local support for their derivatives and therefore D becomes a bandlimited matrix). Sparsity of the third component in (11) depends primarily on the sparsity of the forward and inverse transform matrices T and T as f k x l for time-invariant systems is just a block matrix consisting of diagonal blocks. For simplicity, we will first consider a scalar case of (1) were both Ĉ and Ĝ matrices in (7) can safely be assumed as being diagonal. III. FOURIER BASIS: HARMONIC BALANCE FORMULATION For many years now, Fourier basis has been the natural choice for solving the steady-state analysis problem. Fourier basis for solution of (1) is usually constructed on an interval that ensures periodicity of the solution, includes 2N f +1 basis functions with base frequencies that are multiples of the fundamental frequency in the circuit [1]: { v i } = { 1, cosωt, sinωt, cos2ωt, sin 2ωt,, cosn f ωt, sinn f ωt} (12) Because complex exponents are natural eigenfunctions of the derivative operator, derivative matrix D in this basis becomes a diagonal matrix in real Schur form with base frequencies on the main diagonal:

6 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 6 of 45 D = ω K K 0 (13) The transform matrix T has dimensions of N t ( 2N f + 1) with N t being the number of time points and N f being the number of frequencies. This matrix has the following structure: 1 cos( ωt 0 ) sin ( ωt 0 ) cos( N f ωt 0 ) sin( N f ωt 0 ) T = 1 cos( ωt 1 ) sin ( ωt 1 ) cos( N f ωt 1 ) sin( N f ωt 1 ) 1 cos ( ωt N t 1 ) sin ( ωt N 1 ) t cos ( N ωt f N 1 ) sin( N ωt f N 1 ) t (14) If N t = 2N f + 1 (15) then T is a square matrix which is nonsingular with a proper choice of time sampling points. If more restrictions are imposed on the time sampling points 1, T can also be made orthogonal: T = T 1 = T T (16) This matrix clearly is dense which would suggest O(N 2 ) operations for computing Fourier coefficients in (4). This cost can be reduced to O(NlogN) by applying Fast Fourier Transform (FFT) algorithm for computing the T and T -1 operators. However, Jacobian in (11) invariably becomes a dense matrix which brings cost of solving (9) up to O N t 3 ( ) at each iteration. In order to reduce the cost of solving (11), one must choose a different basis that provides sparse representation for both D and T matrices. 1. Lengthy discussions of different algorithms for the selection of time sampling grid can be found, for example, in [1] and [2] and are really beyond the scope of this paper.

7 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 7 of 45 IV. WAVELET FORMULATION Let us consider expansion of a scalar form of (1) in a basis of compactly supported wavelets ([14]) in a similar way as traditional Fourier expansion described in Section III. A. Boundary conditions Let expansion basis { v i } be equal to one layer of scaling functions and one layer of wavelets at level J such that 2 J = N f : 2N f { v i } i = 1 = { ψ i, J, ϕ i, J } N f 1 i = 0 (17) where ϕ j, k ( x) and ψ j, k ( x) are translations and dilations of the mother wavelet function and mother scaling function for a given wavelet family [14]: ϕ j, k ( x) = 2 j ϕ 00, ( 2 j x k) (18) ψ j, k ( x) = 2 j ψ 00, ( 2 j x k) (19) that also satisfy refinement equations M ϕ( x) = h j ϕ( 2x m) m = M M ψ( x) = g j ϕ( 2x m) m = M (20) (21) Coefficients [h j ] and [g j ] of the refinement equations are also filter taps for the Quadrature Mirror Filters (QMFs) associated with given wavelet family. Further, let us assume for simplicity that basis (17) is defined on an interval [0; 1] and equation (1) is scaled accordingly such that period of the fundamental frequency in the circuit is also equal to 1. To satisfy the boundary condition of x( 0) = x( 1) we must construct wavelets in such a way that basis becomes periodic on an interval. Such construction can be easily performed when truncated part of the wavelet (or scaling function for that matter) is not discarded, but

8 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 8 of 45 appears on the other boundary of the interval [15]. Periodic basis constructed in such a way naturally enforces periodicity of the solution while retaining all other properties of wavelets. B. Transform matrix Periodic wavelet basis on an interval gives rise to a bandlimited transform matrix T that can be obtained from a non-periodic matrix [16] by introduction of the truncated QMF coefficients into upper right (and, if necessarily, lower left) corners: h 0 g h 2 g 2 h 1 g h 2 g 2 h 0 g 0 0 h M 2 g M 2 h 1 g 1 0 h M 1 g M 1 h M 1 g M 1 h 2 g 2 h 0 g h 1 g 1 0 T = 0 h M 1 g M 1 h 2 g 2 0 (22) h M 1 g M h 0 g h 1 g 1 Lower left corner coefficients will appear if the QMF taps are aligned around the center of the h vector such that h 0 always appears on the main diagonal [17]. For orthogonal wavelets, inverse transform matrix is obtained from (22) via (16). For biorthogonal wavelets, T is constructed by augmenting a nonperiodic transform matrix in a way similar to described above. An example of the sparsity pattern for the periodized transform matrix constructed with orthogonal Daubechies wavelets with 8 filter coefficients can be observed in Fig. 1. Filter

9 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 9 of 45 coefficients are aligned in the following way: h = [ h, 4 h, 3 h, 2 h, 1 h, 0 h, 1 h, 2 h 3 ]. This figure clearly illustrates that for local support wavelets generated by FIR filters, transform matrix remains bandlimited (to a permutation) even in periodic case. With each column containing M nonzero entries, total number of nonzero elements in transform matrix is N NZ = 2MN f or O(N). This is already an improvement over traditional Fourier basis which generates a dense transform matrix. Furthermore, because of the sparsity of the transform matrix and it s banded structure, f k T T x l 2). component of (11) is also a sparse bandlimited matrix with O(N) nonzero entries (Fig. We will proceed with derivation of the D matrix in wavelet basis to determine the overall sparsity pattern of Jacobian. C. Connection coefficients Derivative matrix D in (8) contains 4 types of coefficients produced by discretization of derivative operator in wavelet basis (17) [18]: α l ψ t τ - l = -----, t ψ - N t t τ β l ψ t τ - l = -----, t ϕ - N t t τ γ l ϕ t τ - l = -----, t ψ - N t t τ r l ϕ t τ - l = -----, t ϕ - N t t τ (23) (24) (25) (26) where l = 0,, N t 1. These coefficients obtained by expansion of the derivative operator in a wavelet basis are often called connection coefficients. By substituting (23)-(26) into the refinement equations (20)-(21) one can show ([17],[18]) that α i = 2 g k g k' r 2i+ k k' ( k) ( k' ) (27)

10 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 10 of 45 depend on the number of vanishing moments for a particular wavelet and not on the QMF coefβ i = 2 g kh k' r 2i+ k k' ( k) ( k' ) γ i = 2 h k g k' r 2i+ k k' ( k) ( k' ) (28) (29) I.e., that representation of the derivative operator in a wavelet basis is completely determined by connection coefficients (26) obtained from scaling functions only. For compactly supported (bi)orthogonal wavelets, { r m } is an anti symmetric vector with following properties: r m 0 only for M + 2 m M 2 (30) r 0 = 0 r m = r m mr m = 1 ( m) (31) (32) (33) and, most important: M 2 1 r m = 2 r 2m + -- a 2 2k 1 ( r 2m 2k r 2m + 2k 1 ) k = 1 (34) where a i are autocorrelation coefficients of the low pass QMFs: M i 1 a i = 2 h m h m + i, i = 1,, M 1 (35) m = 0 which can be computed with high precision using the following relationship for a wavelet with L vanishing moments [18]: ( 1) l 1 ( 2L 1)! a 2l 1 = , (36) ( L l)! ( L+ l 1)! ( 2l 1) ( L 1)!4 L 1 l = 1,, L Only odd autocorrelation coefficients have nonzero values. Even coefficients are all equal to zero. We have to note here that not only a i are rational numbers by construction, but they only

11 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 11 of 45 ficients themselves. Therefore, they can be the same for different wavelets with the same number of vanishing moment. Linear algebraic system formed by (30)-(34) is ill conditioned and because of that it s numerical solution is unstable. Fortunately, since the coefficient of this system are rational numbers, it can be solved symbolically. Consequently, basic connection coefficients r m are also rational numbers by construction and can be computed with any required degree of accuracy. Connection coefficients can be precomputed offline according to (30)-(36) and stored for future reference. They have been extensively tabulated in mathematical literature, for example in [17], [18] and [19]. It is interesting to observe that for L=1 (Haar wavelets) connection coefficients r = --, 0, -- are equivalent to a well known finite difference discretization scheme Higher order discretization schemes correspond to wavelets with more vanishing moment. D. Derivative matrix Having obtained connection coefficients for the expansion of derivative operator in basis of scaling functions (26), we can now construct derivative matrix D for (8). We start with constructing matrix R which is projection of the derivative operator onto subspace spanned by scaling functions. Because scaling functions, as well as wavelets, have local support, R is a bandlimited circulant matrix with it s diagonals filled by r m [17]. To extend this construction for periodized wavelets, we need to populate upper left and lower right corners of the matrix as well [19]. Fig. 3 illustrates structure for the derivative matrix R constructed with periodized Daubechies wavelets of third order (6 filter coefficients). From here, matrix D can be easily obtained using (27)-(29). These equations describe convolution of the derivative filter with QMFs, which we can write in matrix form as D = TRT (37) where T and T are forward and inverse transform matrices. Sparsity structure of the matrix D will depend on the sparsity structure of transform matrices, which in it s turn depends on the

12 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 12 of 45 ordering of basis functions. The traditional way (first introduced in [18]) is to place scaling functions first and then wavelets: 2N f { v i } i = 1 = ϕ 0, J, ϕ, ;ψ, ψ, N f 1, J 0, J N f 1, J (38) This will generate matrix D that has four bandlimited quadrants populated by α, β, γ and r as defined in (23)-(26) [17], which is convenient for generating representation of the derivative operator in so called non-standard form that decouples resolution scales in matrix D. However, as the proposed formulation does not use multiscale resolutions, there is no need to use the nonstandard form either. We can minimize bandwidth of D by reordering basis in such a way, that each scaling function is followed by the overlapping wavelet: 2N f { v i } i = 1 = ϕ 0, J, ψ 0, J, ϕ,, ψ N f 1, J N f 1, J (39) In fact, transform matrices defined by (22) correspond to such bases. Because both T and T in this case are bandlimited matrices, as well as R, resulting matrix D is also a bandlimited matrix with O(N) nonzero entries (Fig. 4). Referring back to the Jacobian in equation (11), we have established that wavelet expansion leads to construction of sparse bandlimited matrices for all components of the Jacobian. Even though derivative matrix in wavelet basis is not diagonal (as in case of Fourier basis), it has only O(N) nonzero entries. Together with sparse, O(N) transform matrix, this results in sparse Jacobian in equation (9). V. ANALYSIS OF COMPUTATIONAL COMPLEXITY We use two computational cost metrics: number of nonzero elements in the Jacobian matrix and net CPU time required for one LU decomposition and Forward/Backward substitution of the Jacobian. The former is independent of all the platform and implementation issues, is a dominating factor for both memory storage and CPU requirements and provides a good measure

13 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 13 of 45 of computational resources required to perform the simulation. The latter is highly dependent on the software and hardware implementation of the simulator, but given pretty much state of the art in both, dominates the CPU cost of a Newton s iteration and provides a real world estimate of the CPU time required for the solution. In this section we will derive analytical estimates for the first metric, while reliable data for the second metric can be obtained only experimentally and will be presented in Section VI. A. Harmonic Balance formulation Let us consider equation (1) in scalar form. Provided square Fourier transform is used, T and T matrices in (4) are square and dense. Jacobian (11) also becomes a dense matrix because of the T( f x)t component. If we denote order of expansion as N t, Jacobian is a 2 dense N t N t matrix that has O( N t ) nonzero elements. Let us generalize this to a vector case. Matrices in (7) and (11) obtain a block structure with each N t N t nonzero block corresponding to one nonzero entry in circuit equation matrices (1). These nonzero blocks have O( N t ) elements for every nonzero entry in matrices C and G and 2 O( N t ) elements for every nonzero entry in [ f k x l ]. Density of Jacobian (11) in this case is dominated by these dense blocks corresponding to nonlinear elements in the circuit. Only the size of these blocks changes with the order of expansion. Overall density of the Jacobian in this 2 case is N NZ O( κ N t ), where κ is constant for a given circuit and therefore 2 N NZ O( N t ) (40) like in scalar case. In it s turn, order of expansion is linearly proportional to the number of frequencies in truncated set: N t = 2N f + 1 (41) with 1 accounting for the DC component. Number of frequencies in truncated set therefore is a critical point for computational cost analysis.

14 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 14 of 45 B. Spectrum truncation issues Let us denote the highest order of intermodulation products retained in simulation as N H. Equation (12) describes frequency set useful only for analysis of circuits excited by a single tone. For multitone analysis, the set should include harmonics of all the tones as well as all relevant intermodulation products: Ω S Ω = ω ω = k s ω s ; k s Z s = 1 (42) This set is infinite. In order to make the problem computationally solvable, we must truncate this set to one that provides an approximate solution. Truncation schemes are the principal source of errors in steady state analysis, primarily due to the aliasing of truncated components [2]. The simplest truncation scheme (we will refer to it as trivial truncation) assumes that all ω s in (42) are commensurate with a single fundamental frequency ω. Trivial truncation then generates an equidistant frequency grid spanning all the frequencies from 0 to N H -th harmonic of the highest frequency in ω s. For example, if tone frequencies are equal to 900 and 910 MHz and N H = 10, the set will span frequencies from 0 to 9100 MHz with step density of the grid as ω = 10 MHz. If we denote ωˆ = ω max{ ω s } (43) then trivial truncation produces a grid that has N f O N H ωˆ (44) frequency components. Together with (40) and (41) this results in the following computational complexity estimation for HB formulation with trivial truncation: N NZ O N H ωˆ (45)

15 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 15 of 45 This effectively renders trivial truncation to be unsuitable for all but the simplest and smallest cases. Another truncation strategy is aimed to retain in Ω H only those frequencies which carry intermodulation components with orders up to N H. This strategy gives rise to box S Ω H = ωω= k s ω s ; k s Z; k s N H s = 1 (46) and diamond S S Ω H = ωω= k s ω s ; k s Z ; k s N H (47) s = 1 s = 1 truncation schemes. In general case, for multitone analysis with S tones and box or diamond truncation, number of frequencies in truncated set is proportional to the volume of a hypercube in S-dimensional space ([1], p.245): However, for a practically interesting case of periodic analysis when all N f S O( N H ) ω s (48) are commensurate with a single fundamental frequency ω, N f grows slower than (48) because with increase in N H frequencies of the new IM products often coincide with already existing in the set. Particularly, if tone frequencies in ω s are evenly spaced (e.g. 900, 910, 920,... MHz), set size grows only as 2 N f O( S N H ) (49) Combining (40), (41) and (49), we conclude that for multitone Harmonic Balance computational cost in terms of the number of nonzero elements in Jacobian is equal to at least 4 N NZ O( N H ) (50) Computational cost in terms of CPU time will actually be slightly higher and also depend on the size and density of the circuit equation (1).

16 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 16 of 45 C. Wavelet formulation Similarly to Harmonic Balance expansion, estimations given in Section IV for wavelet expansion can be generalized to include circuit equations (1), where each nonzero element after expansion becomes an N t N t sparse block, each having O( N t ) nonzero entries (see Fig. 2 and Fig. 4). Total number of nonzero entries in wavelet Jacobian becomes N NZ O( N t ) (51) with N t = 2N f (52) because of the sampling theorem. With wavelets we use trivial truncation that produces an equidistant uniform frequency grid spanning harmonics and IM components up to required N H. This, however, is quite a beneficial trade-off as this scheme produces frequency grid with N f = O N H ωˆ (53) components, where ωˆ is the relative density if the frequency grid (e.g. for base frequencies 99 and 100 MHz expansion is ωˆ = 1%). Combining (51)-(53) we conclude that computational cost of wavelet N NZ = O( N H ) (54) and despite the primitive truncation scheme, with increase in N H and S, wavelet methods very quickly gain significant advantages in computational cost. Comparison of relative computational complexity is shown in Fig. 5. This plot was produced for a scalar case with multitone excitation and closely spaced commensurate tone frequencies (e.g. 1000, 990, 980, 970 MHz,... for ωˆ = 1%). The plot leads to some interesting observations. First, computational complexity of Harmonic Balance with diamond truncation does not depend on the density of frequency grid. This is quite understandable in view of the fact that Fourier series is a frequency localized basis. Good frequency localization is what makes possible

17 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 17 of 45 sophisticated truncation schemes in frequency domain. However, this also causes computational cost to be 4 N NZ O( N H ) and also to depend on the number of tones. Second, computational cost of wavelet formulation with trivial truncation perfectly follows (53) and despite the fact that it depends on the density of frequency grid, it does not depend on the number of tones as long as the newly introduced tones fall into the same grid produced by trivial truncation. This is also quite understandable if we consider the fact that wavelet transform used in our formulation can be represented by a filter bank with one bifurcation. Note, that for (bi)orthogonal wavelets power frequency response of the filter bank satisfies the no distortion condition H ( z)h( z) + G ( z)gz ( ) = 2z 1 (55) which essentially means that the transform covers the whole of the frequency range split into two bands [17]. By comparison, a filter bank associated with Fourier transform has frequency response of a collection of narrowband filters. With trivial truncation, combined frequency response covers the whole frequency range, however, with box or diamond truncation only selected frequencies in the range are covered and introduction of new tones or higher order intermodulation products leads to substantial growth in the number of basis functions (49) and, consequently, in computational cost of the solution. We must emphasize here that trivial truncation is by no means essential to the wavelet expansion. Wavelet transform used in the formulation presented here is constructed specifically to achieve maximum sparsity of the Jacobian matrix and is beneficial primarily to broadband circuits with large number of tones. Different wavelet transforms with frequency domain adaptive schemes must be used for narrowband circuits. Similarly, aperiodic wavelet transforms utilizing boundary adapted wavelets [8], [10] must be used for quasi-periodic and autonomous circuits. VI. NUMERICAL RESULTS All simulations presented in this section were performed in Matlab (R13), running on a SUN Blade-1000 workstation with 900 MHz UltraSPARC-III CPU, 8 MB L2 cache and 5 GB of

18 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 18 of 45 physical RAM. Because Matlab environment uses interpreted programming language [20], CPU time was recorded only for the time required to solve the Jacobian matrix (averaged over several Newton s iterations). Recording total simulation time would include all the overhead associated with the interpreter and possibly other implementation issues and would produce contaminated and therefore misleading results. Matlab 6.x sparse matrix solver relies on the UMFPACK package ([21], [22]). Matlab s left matrix division operator was used to invoke the matrix solver, which in this case performs LU decomposition by Gaussian elimination with partial pivoting. By default the solver performs column approximate minimum degree preordering before performing Gaussian elimination. It was established that explicit utilization of other preordering algorithms is extremely beneficial for the steady state analysis problems. Symmetric approximate minimum degree preordering was used for Jacobians arising from the Fourier series expansion, while for the wavelet expansion it appeared to be possible to use symmetric reverse Cuthill- McKee preordering ([21]). In both examples diamond truncation was used for Harmonic Balance and trivial truncation for wavelet expansion. All the results of wavelet expansion presented in this section were obtained with Daubechies wavelets of second order. Some experiments were also performed with Haar wavelets and higher orders of Daubechies wavelets. Haar expansion produced poor results in both accuracy and convergency, while higher order Daubechies wavelets produced essentially the same accuracy and convergence as the second order, but at a slightly higher computational cost. A simple Ebers-Moll injection model was used for representing BJTs in both examples. A. Case study: cascode LNA A 900 MHz cascode LNA was considered in the first example. The amplifier (Fig. 6) consists of 2 BJTs with DC bias and impedance matching networks. Under these conditions, total size of MNA equations in this example was 25.

19 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 19 of 45 The experiment involves simulation of the cascode LNA circuit under multitone excitation, with two tone input signals of the same power and frequencies of 900 and 910 MHz. Purpose of this experiment is to validate speed and accuracy of the wavelet expansion on computations of the third order in-band intermodulation products at 920 and 930 MHz. Simulation results are shown in Fig. 7 and are in excellent agreement with each other. In each case (HB and wavelets) intermodulation products were computed with N H ranging from 5 to 22 (maximum value for HB given software implementation and available memory). To compare computational complexity of both methods, number of nonzero elements in the Jacobian and average time for one LU decomposition was recorded and is shown as a function of N H in Fig. 8 and Fig. 9 respectively. Detailed data for N H = 12 and 22 can be also found in Tables 1 and 2 respectively. As can be seen from Figures 8 and 9, experimental data for the comparison of computational cost follows the trends predicted by analysis performed in Section V (Fig. 5). Even though trivial truncation results in a large matrix size for wavelet expansion, these matrices are extremely sparse and wavelet method becomes computationally more favourable for N H 10. The trend 4 of O( N H ) derived in (50) is just too powerful and quickly overcomes linear cost of wavelet expansion. CPU time rises even a bit faster than that, which is understandable in view of the computational complexity of the Gaussian elimination for large scale sparse matrices arising from MNA equations ([11]) being O( ( N t N x ) 1 + α ) where 0 < α «1 is a parameter which depends 2 on sparsity ratio of the matrix N NZ ( N t N x ). Both Fourier series and wavelet expansions produce Jacobian with sparsity pattern similar to the original MNA equation except for the fact that with the Fourier series blocks corresponding to nonlinear elements are dense matrices. Because of this, sparsity ratio of HB Jacobian stays essentially the same with increase in N H (2.4% in this example), while sparsity ratio of the wavelet Jacobian decreases as O1 ( N H ) thus compensating for the increased matrix size. In fact, this compensation allows the CPU time metric to stay purpose matrix solver. O( N H ) even when using a general

20 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 20 of 45 An example of the sparsity pattern for Jacobian arising from Fourier series expansion is shown in Fig. 10. Note how the dense blocks corresponding to the nonlinear elements are dominating the nonzero element count. These blocks account for 98.5% of nonzero elements, thus also dominating computational cost in terms of CPU time. Compare this to the sparsity pattern of the Jacobian obtained from wavelet expansion. Jacobian (Fig. 11) has sparse bandlimited blocks in place of dense blocks in the HB Jacobian (Fig. 10). Size of the wavelet Jacobian is much larger, because both were obtained with N H = 12 and wavelet expansion with trivial truncation utilizes larger frequency grid (N f = 2,204 for wavelets versus N f = 313 for Fourier expansion). However, wavelet Jacobian has half the memory storage requirements and it s factorization is 4.6 times faster than that of the Fourier Jacobian (Table 1). By the time N H reaches 22, wavelet expansion requires 10 times less memory and 100 times less CPU time for Jacobian factorization (Table 2). B. Case study: Gilbert cell mixer The second example involves a BJT Gilbert cell mixer circuit that consists of 9 transistors (including 3 as current sources), DC bias and impedance matching networks (Fig. 12). Transformers are assumed to be ideal 1:1 converters. Under these assumptions total size of the MNA equations (1) is equal to 37. The mixer was configured for down conversion with LO input at 1 GHz, RF input at 900 MHz and IF output at 100 MHz. Inputs and output were matched to 50 Ohms active impedance at their respective frequencies. For this experiment input LO power was kept constant at +1 dbm, while performing RF input power sweep. IF output power response for this simulation is shown in Fig. 13. Fig. 14 also illustrates convergence of the Newton iterations from DC solution to operation point P LO = +1 dbm, P RF = -33 dbm with N H = 9. As convergence of the Newton iterations is virtually identical for Harmonic Balance and wavelet formulations, this suggests that spectra of both Jacobians are equivalent, which is important for the convergence of iterative techniques such as Krylov and inexact Newton methods. As can be seen from both plots, both HB and wavelet

21 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 21 of 45 expansion exhibit essentially the same behaviour in terms of both accuracy and speed of convergence, which is not surprising given that both transforms are orthogonal. Comparison of the computational cost of the two methods in terms of the number of nonzero elements in the Jacobian and average time per LU decomposition is shown in Figs. 15 and 16 respectively. Both are in good agreement with the computational cost analysis performed in Section V. Because of the circuit configuration, frequency grid density in this case is 10% ( ω is 100 MHz with 900 and 1000 MHz fundamental frequencies), which means that for N H 9 diamond truncation frequency grid becomes saturated and diamond truncation degenerates into trivial truncation 2 with N f = O( N H ) and N NZ = O( N H ). Detailed computational cost data for N H = 9 and 22 can also be found in Tables 3 and 4. Sparsity patterns for Jacobians arising from Fourier series and wavelet expansion are shown in Figures 17 and 18 respectively. Numerical results for both examples corroborate each other and are in excellent agreement with simulations performed independently in [23]. VII. CONCLUDING REMARKS In this paper we have presented a new approach to the solution of the nonlinear steady state analysis problem that takes advantage of wavelets. Following the traditional Harmonic Balance approach, we converted this problem to a boundary value problem with periodic boundary conditions naturally enforced by periodic basis. Due to the essential local support of wavelets, the proposed approach results in a sparse representation for both the nonlinear and linear (discretization of differential operator) components of the Jacobian matrix. This dramatically reduces computational cost of the analysis, particularly for multitone, highly nonlinear and broadband circuits. Some of preliminary results for the ideas presented in this paper have previously appeared in [24].

22 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 22 of 45 ACKNOWLEDGEMENTS The authors would like to express their gratitude to Dr. Roni Khazaka, Emad Gad and Anestis Dounavis for their invaluable assistance during this research project and to all the reviewers for their comments and suggestion on improving this paper. Simulations were performed with the help of WaveLab, a freeware wavelet toolbox for Matlab.

23 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 23 of 45 REFERENCES [1] Paulo J.C. Rodrigues, Computer-aided analysis of nonlinear microwave circuits, Norwood: Artech House, [2] Kenneth S. Kundert, Jacob K. White and Alberto Sangiovanni-Vincentelli, Steady-state methods for simulating analog and microwave circuits, Boston: Kluwer Academic Publishers, [3] V. Rizzoli, F. Mastri, F. Sgallari, and G. Spaletta, Harmonic-balance simulation of strongly nonlinear very large-size microwave circuits by inexact newton methods, IEEE MTT-S Int. Microwave Symp. Dig., pp , [4] P. Feldmann, B. Melville, and D. Long, Efficient frequency domain analysis of large nonlinear analog circuits, Proc. IEEE Custom Integrated Circuits Conf., pp , [5] D. Long, R. Melville, K. Ashby, and B. Horton, Full-chip harmonic balance, Proc. IEEE Custom Integrated Circuits Conf., pp , [6] E. Gad, R. Khazaka, M. Nakhla and R. Griffith, A circuit reduction technique for finding the steady-state solution of nonlinear circuits, IEEE Trans. Microwave Theory Tech., vol. 48, pp , Dec [7] N. Soveiko and M. Nakhla, Efficient Capacitance Extraction Computations In Wavelet Domain, IEEE Trans. Circuits Syst. -- I: Fundamental Theory and Applications, vol. 47, pp , May [8] D. Zhou, W. Cai and W. Zhang, An adaptive wavelet method for nonlinear circuit simulation, IEEE Trans. on Circuits and Systems -- I: Fundamental Theory and Appl., Vol. 46, pp , Aug [9] M. Steer and C. Christoffersen, Generalized circuit formulation for the transient simulation of circuits using wavelet, convolution and time-marching techniques, Proc. of the 15th European Conference on Circuit Theory and Design, Aug. 2001, pp

24 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 24 of 45 [10] X. Li, B. Hu, X. Ling and X. Zeng, A wavelet-balance approach for steady-state analysis of nonlinear circuits, IEEE Trans. on Circuits and Systems -- I: Fundamental Theory and Appl., Vol. 49, pp , May [11] Jiri Vlach and Kishore Singhal, Computer methods for circuit analysis and design, New York: Van Nostrand Reinhold, [12] Donald G. Dudley, Mathematical foundations for Electromagnetic Theory, New York: IEEE Press, [13] Kenneth S. Kundert, Gregory B. Sorkin and Alberto Sangiovanni-Vincentelli, Applying Harmonic Balance to Almost-Periodic Circuits, IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-36, pp , February [14] Ingrid Daubechies, Ten Lectures on Wavelets, Philadelphia: SIAM, [15] Neil H. Getz, A Perfectly Invertible, Fast, and Complete Wavelet Transform for Finite Length Sequences: The Discrete Periodic Wavelet Transform, Mathematical Imaging: Wavelet Applications in Signal and Image Processing, Proceedings of the SPIE - The International Society for Optical Engineering, vol.2034:332-48, July 1993 San Diego. [16] Yves Nievergelt, Wavelets Made Easy, Boston: Birkhauser, [17] Stefan Goedecker, Wavelets and Their Applications for the Solution of Partial Differential Equations in Physics, Lausanne: Presses polytechniques et universitaires romandes, [18] G. Beylkin, On the representation of operators in bases of compactly supported wavelets, SIAM J. on Numerical Analysis, Vol. 6, No. 6, pp , June [19] Juan Mario Restrepo, Gary K. Leafy, Inner Product Computations Using Periodized Daubechies Wavelets, International Journal of Numerical Methods in Engineering, 40, pp , [20] Matlab User Manual, MathWorks Corporation, helpdesk/help/techdoc/matlab.shtml

25 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 25 of 45 [21] John R. Gilbert, Cleve Moler and Robert Schrieber, Sparse Matrices in Matlab: Design and Implementation, [22] T.A. Davis, UMFPACK Version 4.0 User Guide ( sparse/umfpack/v4.0/userguide.pdf), Dept. of Computer and Information Science and Engineering, Univ. of Florida, Gainesville, FL, [23] Roni Khazaka, Projection Based Techniques For The Simulation Of RF Circuits And High Speed Interconnects, Ph.D. Thesis, Ottawa: Carleton University, [24] N. Soveiko and M. Nakhla, Wavelet Harmonic Balance, IEEE Microwave and Wireless Components Lett., Vol. 13, No. 6, pp , July 2003.

26 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 26 of nz = 512 Fig. 1. Sparsity pattern for the periodized transform matrix.

27 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 27 of nz = 384 Fig. 2. Sparsity pattern for the T( f x)t component of Jacobian expanded in a basis of periodic orthogonal Daubechies wavelets of order 2.

28 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 28 of r 1 r2 r 3 r r 4 r r 2 r nz = r 0 Fig. 3. Derivative matrix R for periodized wavelets.

29 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 29 of nz = 1152 Fig. 4. Sparsity pattern for the 64x64 derivative matrix D constructed in basis of order 3 periodized Daubechies wavelets.

30 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 30 of 45 Comparison of computational complexity Fourier: diamond truncation Wavelets: trivial truncation 10 9 Asymptotic slopes tones 5 tones 4 tones 3 tones 2 tones Relative computational complexity f=0.1% f=1% f=10% O(N 4 ) O(N) Max order of IM products Fig. 5. Comparison of computational complexity in terms of the number of nonzero elements in the Jacobian.

31 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 31 of 45 V cc =3V + 1kΩ 600Ω 100pF 29.3nH 100pF 1kΩ 0.69pF 50Ω 50Ω 17.2nH 100pF V RF 6.9pF 5 µα 0.1nH Q=10 Fig. 6. Cascode LNA circuit.

32 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 32 of Harmonic Balance Wavelets Two tone input, MHz MHz Output power, dbm MHz MHz MHz input power, dbm Fig. 7. Two tone input simulation results for the cascode LNA in Fig. 6.

33 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 33 of 45 Comparison of memory requirements 10 8 Fourier series Wavelets Asymptotic slopes 10 7 Number of nonzero elements 10 6 O(N) 10 5 O(N 4 ) Maximum order of IM components Fig. 8. Number of nonzero elements in Jacobian for the cascode LNA in Fig. 6.

34 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 34 of 45 CPU time comparison 10 4 Fourier series Wavelets Asymptotic slopes 10 3 CPU time, seconds O(N) O(N 4 ) Maximum order of IM components Fig. 9. Average CPU time per LU decomposition of Jacobian for the cascode LNA in Fig. 6.

35 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 35 of 45 Fig. 10. Sparsity pattern for the Jacobian arising from Fourier series expansion for cascode LNA in Fig. 6 with N H = 12. Dense blocks account for 98.5% of nonzero elements.

36 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 36 of 45 Fig. 11. Sparsity pattern for the Jacobian arising from wavelet expansion for cascode LNA in Fig. 6 with N H = 12.

37 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 37 of V 580Ω 580Ω 375nH 6.17pF 50Ω 50Ω 13.8pF 1.7V 100Ω 200Ω V RF 5.8pF + 200Ω 50Ω 22.1nH 1.2V 100Ω 200Ω 100Ω V LO.82pF + 200Ω 1200Ω + 3V 36Ω 36Ω 36Ω Fig. 12. Gilbert cell mixer circuit

38 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 38 of 45 0 Harmonic Balance Wavelets 5 IF output power at 100 MHz, dbm RF input power, dbm Fig. 13. IF output power at 100 MHz with respect to RF input power for Gilbert cell mixer circuit in Fig. 12.

39 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 39 of Convergence of Newton s iterations Wavelets Harmonic Balance 10 4 Eucledian norm of the residue number of Newton s iterations Fig. 14. Convergence of the two tone simulation of the Gilbert cell mixer circuit in Fig. 12.

40 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 40 of 45 Comparison of memory requirements 10 8 Fourier series Wavelets Asymptotic slopes 10 7 O(N 4 ) Number of nonzero elements O(N 2 ) O(N) Maximum order of IM components Fig. 15. Number of nonzero elements in the Jacobian for Gilbert cell mixer in Fig. 12.

41 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 41 of 45 CPU time comparison 10 4 Fourier series Wavelets Asymptotic slopes 10 3 CPU time, seconds O(N 4 ) O(N) Maximum order of IM components Fig. 16. Average CPU time per LU decomposition of Jacobian for Gilbert cell mixer in Fig. 12.

42 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 42 of 45 Fig. 17. Sparsity pattern for the Jacobian arising from Fourier series expansion for Gilbert cell mixer in Fig. 12 with N H =9. Dense blocks account for 98.2% of nonzero elements.

43 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 43 of 45 Fig. 18. Sparsity pattern for the Jacobian arising from the wavelet expansion for Gilbert cell mixer in Fig. 12 with N H =9.

44 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 44 of 45 TABLE 1. COMPUTATIONAL COST COMPARISON FOR CASCODE LNA (FIG. 6) AT N H = 12. Harmonic Balance Wavelets Frequency grid size 157 1,102 Time grid size 313 2,204 Jacobian size Number of nonzero elements ,489, ,174 Sparsity ratio 2.4% 0.025% Memory storage size, MBytes Average CPU time for preordering, seconds Average time per LU/FBS, seconds Number of Newton iterations 5 5 TABLE 2. COMPUTATIONAL COST COMPARISON FOR CASCODE LNA (FIG. 6) AT N H = 22. Harmonic Balance Wavelets Frequency grid size 507 2,012 Time grid size 1,013 4,024 Jacobian size Number of nonzero elements ,462, ,400,370 Sparsity ratio 2.4% 0.014% Memory storage size, MBytes Average CPU time for preordering, seconds Average time per LU/FBS, seconds 2, Number of Newton iterations 5 5

45 2169-IF-TH-21: Steady State Analysis Of Multitone Nonlinear Circuits In Wavelet Domain Page 45 of 45 TABLE 3. COMPUTATIONAL COST COMPARISON FOR GILBERT CELL MIXER (FIG. 12) AT N H =9. Harmonic Balance Wavelets Frequency grid size Time grid size Jacobian size Number of nonzero elements ,134, ,016 Sparsity ratio 1.2% 0.1% Memory storage size, MBytes Average CPU time for preordering, seconds Average time per LU/FBS, seconds Number of Newton iterations 8 8 TABLE 4. COMPUTATIONAL COST COMPARISON FOR GILBERT CELL MIXER (FIG. 12) AT N H = 22. Harmonic Balance Wavelets Frequency grid size Time grid size Jacobian size Number of nonzero elements ,539, ,475 Sparsity ratio 1.2% 0.042% Memory storage size, MBytes Average CPU time for preordering, seconds Average time per LU/FBS, seconds 2, Number of Newton iterations 8 8