Efficient Linear Macromodeling via Discrete-Time Time-Domain Vector Fitting

Transcription

1 st International Conference on VLSI Design Efficient Linear Macromodeling via Discrete-Time Time-Domain Vector Fitting Chi-Un Lei and Ngai Wong Department of Electrical and Electronic Engineering The University of Hong Kong, Pokfulam Road, Hong Kong {culei, Abstract We present a discrete-time time-domain vector fitting algorithm, called TD-VFz, for rational function macromodeling of port-to-port responses with discrete time-sampled data The core routine involves a two-step pole refinement process based on a linear least-squares solve and an eigenvalue problem Applications in the macromodeling of practical circuits demonstrate that TD-VFz exhibits fast computation, excellent accuracy, and robustness against noisy data We also utilize an quasi-error bound unique to the discrete-time setting to facilitate the determination of approximant model order Introduction VLSI signal integrity analysis constantly requires efficient modeling and simulation of passive structures such as packages and interconnect networks [] for efficient simulation The ever-rising operating frequencies have, however, posed serious challenges to simulators due to the emergence of high-frequency effects such as interconnect dispersion and mutual couplings [] A full-wave electromagnetic (EM analysis over the global system is impractical A continuous-time frequency-domain system identification technique, called vector fitting (VF [], is recently utilized for package and interconnect macromodeling [] However, frequency-domain macromodeling requires spectral conversion in conventional VF-based signal integrity analysis which involves complicated measurement and relatively long data sequences to be captured Time-domain vector fitting (TD-VF [], similar to other time-domain approximation techniques [5], is developed to directly admit measured or calculated (and usually truncated time-sampled data However, TD-VF does not fully exploit the advantages of time-domain approximation, and requires discretization during approximation The discretization is applicationspecific and introduces extra arithmetics and approximation errors On the other hand, vector fitting has recently been extended to the digital frequency domain, called discrete-time vector fitting (VFz, for digital filter design [] It has been shown that VFz exhibits high computational efficiency and produces highly accurate approximants at low model orders However, the deployment of VFz in the digital time domain has not been explored Since physical measurements or electronic simulations typically result in discrete time-sampled data rather than frequency-domain responses, it is advantageous to perform broadband approximation directly in the time domain In this paper, we reformulate VFz to its discrete-time counterpart, called TD-VFz TD-VFz respects the discrete nature of input/output sequences and works exclusively in the discrete-time domain, and is particularly suitable for generating macromodels based on truncated finite-difference time-domain (FDTD solution or transient response analysis Unlike TD-VF, TD-VFz treats the system as a digital black box and skips the continuous-time interpolation or reconstruction, thereby rendering faster computation and generally comparable or more accurate fitting Though the fitting is performed in the (digital time domain, the response is identified with underlying (digital frequencydomain poles such that global accuracy is achieved further to local matching The algorithm robustness and convergence are discussed An error bound is presented for model order selection Application examples then confirm the remarkable efficiency and accuracy of TD-VFz to generate better macromodel Vector Fitting The original continuous-time frequency-domain VF [] attempts to fit the rational function + d + se ( s a n to the desired response f(s at a set of calculated/sampled data points at frequencies {s k }, k =,,,N s The -9/8 $5 8 IEEE DOI 9/VLSI8 9

2 poles a n and residues are either real or in complex conjugate pairs, and d and e are real Starting with a set of N prescribed or approximated poles {α ( n }, n =,,,N, and a scaling function σ(s, the problem is linearized into a separable denominator calculation for the ith iteration, namely, s α (i n + d + se (σf(s ψ γ n s α n (i + σ(s f(s, ( i =,,,N T, where N T denotes the number of iterations when convergence is attained or when the upper limit is reached The unknowns,, d, e and γ n, are solved through an overdetermined linear equation formed by evaluating ( at the N s sampled frequency points The poles are refined iteratively Any unstable pole is flipped about the imaginary axis to the open left half plane for stability Upon convergence, the update in α (i n diminishes and σ(s Recently, it is shown that VF is a reformulation of the Sanathanan-Koerner (SK iteration [], an iterative continuous-time frequency-domain system identification technique [] There exists two problems in VF: The convergence is affected by the choice of initial poles in ( Orthogonal vector fitting (OVF has been proposed to alleviate this degradation [] The algorithm is not robust for noisy data Different techniques have been proposed to alleviate this problem [8,9] Furthermore, a modified VF (MVF [] has recently been proposed for model reduction with estimated time delay (the propagation of the main pulse to significantly reduce the number of poles Discrete-Time Time-Domain Vector Fitting We start by formulating the discrete-time frequencydomain (ie, z-domain VF, called VFz [], and subsequently transform it to the discrete-time time-domain counterpart, called TD-VFz As in VF, VFz uses partial fractions to seek a rational approximation, ˆF (z, to the desired z-domain response F (z, namely, ˆF (z = + d F (z, ( z a n where we assume F (e jω = F (e jω, Ω [ π, π, such that F (z corresponds to a real time-domain sequence Subsequently, and a n are either real or in complex conjugate pairs To ensure stability, the set of poles {a n } in ( must be within the unit circle or a n < As in (, suppose an initial set of poles {α ( n }, α ( n <, are specified, we build z α (i n + d (σf(z ψ γ n z α n (i + σ(z F (z, ( i =,,,N T, where σ(z is matched to unity as z approaches the origin It can be observed that ( constrains (σf(z and σ(zf (z to share the same set of poles, which in turn implies that the original poles of F (z are canceled by the zeros of σ(z Therefore, ( can be re-expressed as N+ Q z β n NQ ( z α n (σf(z F (z (σf(z σ (z N+ Q z e βn NQ ( z α n = σ(z N+ Q N+ Q z β n F (z, (5 z e β n ( Subsequently, solving the zeros of σ(z results in, in the least-squares (LS sense, an approximation to the poles of F (z, ie, {α (i+ (i n } := { β n }, which are then fed back to ( as the new set of known poles for better fitting in the next iteration Next, we apply an input X(z to F (z and let Y (z =F (zx(z be the output The time-domain relationship is then given by the inverse z-transform, NX y[k] dx[k]+ x n[k] γ ny n[k], y n[k] = (α (i n k u[k] y[k], x n[k] = (α (i n k u[k] x[k], ( where denotes convolution and u[k] is the Heaviside unit step sequence NX Computing the Real Poles Continuing from (, suppose N s samples of the input and output sequences, x[k] and y[k], are captured, the following overdetermined linear equation is set up, Av = b, A = x [] x N [] x[] x [] x N [] x[] x [N s ] x N [N s ] x[n s ] y [] y N [] y [] y N [] y [N s ] y N [N s ] v = c c N d γ γ N Λ T, 5, b = y[] y[] y[n s ] Λ T, (8 with N s N + Using the last N elements of the LS solve of v, ie, γ to γ N, σ(z of ( can be reconstructed, whose zeros, denoted by {α (i+ n }, then form the new set of

3 poles in the next TD-VFz iteration Similar to the formulation in VF [], the zeros of σ(z are implicitly obtained as the eigenvalues of Ψ= = α (i α (i N N N 5 h R α (i 5 Λ γ γ N C A γ α (i N γ N where R =+ N i= γ i When only real poles are present, Ψ is a real matrix To ensure stability, it is required that every α (i+ n < Otherwise, its reciprocal is taken, viz α (i+ n := /α (i+ n, such that the pole is flipped back inside the unit circle Here a real α (i+ n is assumed but the flipping of conjugate poles follows exactly by multiplying two conjugate reciprocals Till now, only real poles are considered Special attention must be paid to complex conjugate poles as will be discussed below Modification for Complex Poles The transfer function F (z in ( may, and in fact more often than not, contain complex conjugate poles and residues whose time-domain transforms are also complex conjugate, thus conforming to a real response Apart from accelerating convergence, allowing complex poles in the (real TD-VFz arithmetics is critical, if not necessary, as practical digital systems generally have poles distributed in certain sectors in the complex plane Without loss of generality, we assume α (i = α (i and c = c (γ = γ, so x [k] =x [k] (y [k] =y [k] However, if these complex quantities are directly used in (8, finite-precision arithmetics would almost always result in inexact cancellation of imaginary parts and lead to erroneous time-domain responses Consequently, (8 should be modified accordingly to restrict all quantities to be real To achieve this, (8 is equivalently cast as Av = b, A = R(x [] I(x [] x[] R(x [] I(x [] x[] R(x [N s ] I(x [N s ] x[n s ] R(y [] I(y [] R(y [] I(y [] R(y [N s ] I(y [N s ] 5, v = R(c I(c d R(γ I(γ ΛT, b = y[] y[] y[n s ] Λ T, ( where R( and I( denote the real and imaginary parts, respectively Modification for other complex conjugate i, (9 poles and residues follows similarly The setup of ( is eased by the fact that ( R(x n [k] = (α (i n k cos(θ n (i k u[k] x[k], (a ( I(x n [k] = (α (i n k sin(θ n (i k u[k] x[k], (b where θ (i n = arg(α (i n R(y n [k] and I(y n [k] are obtained in the same manner by replacing x[k] with y[k] in ( To compute the zeros of σ(z which now contains complex poles, we apply similarity transform to (9 to bring it back to a real matrix Each pair of conjugate poles now manifest as a diagonal block in Ψ For example, when α (i = α (i, Ψ takes the form Ψ= R(α (i I(α(i I(α (i R(α(i 5 5 R h R(α (i γ I(α (i γ i, ( and R is obviously real The entries in the solution of v in ( can be reused in forming the row matrix in (, namely, R(α (i γ =R(α(i R(γ I(α(i I(γ, (a I(α (i γ =R(α(i I(γ+I(α(i R(γ (b Reconstructing the Rational Function Suppose a converged set of poles {α n (NT } are obtained, the final step is to reconstruct the rational function ˆF (z in ( Referring to ( and (, we should now have σ(z and y[k] dx[k]+ NX x n[k], k =,,,N s ( The residues of ˆF (z are computed in the same manner as in (8 or ( by LS fitting, except that the last N columns in A and the last N elements in v are omitted, leaving the unknowns c to c N and d Besides reconstructing the rational function, the parameters can be easily converted into causal parameters and used for recursive convolution for fast simulation [] Algorithmic Convergence and Model Order Selection Similar to the equivalence between VF and SK iteration [], VFz can be regarded as a reformulation of the rational function fitting procedure called Steiglitz-McBride (SM iteration [5] The convergence and error bound properties of VFz, as will be shown below, then carry over to

4 TD-VFz as they are equivalent representations in different domains First, given a transfer function or response F (z, SM iteration replaces the nonlinear LS approximation objective ĜL = Σ Ns k= F (z k P (z k Q(z k with a linearized ĜSM where X N s bg SM = k= fi Q (i (z k fi fiq (i (z k F (z k P (i (z k fi (5 Here P (i and Q (i are respectively the numerator and denominator determined during the ith SM iteration (thus Q (i is assumed predetermined Although ĜSM is not equivalent to ĜL, by using the triangle inequality, if we approximate F (z by an Nth-order system, we get ĜL ĜSM σ N+ Here σ i, which denotes the ith Hankel singular value (HSV of a Hankel-form matrix constructed by the time-domain impulse coefficients h n s of F (z, measures the significance of the ith approximant order [] In general, SM iteration converges to a nearglobal-optimal approximant in the LS sense for noise-free data, with an a priori error bound for an N th-order approximant, Z π deg( Q P =N π π fi F (ejω P (i (e jω / Q (i (e jω dωa σ fi N+ ( Such error bound is important as it provides a certificate for the approximation accuracy and can be used to select the approximant order That is, the order N of ˆF (z should be selected such that σ N+ This also constitutes an analytical and non-heuristic way to determine the model order which, as far as we know, is not available in VF or TD-VF Moreover, unlike point-matching (curve-fitting algorithms, a crucial feature in TD-VFz is that it incorporates the (digital frequency-domain system poles in its (digital time-domain fitting machinery A consequence is that further than local accuracy in the context of the finite captured data set, which forms the objective function of most direct point-matching schemes, TD-VFz achieves global accuracy by simultaneously identifying the intrinsic system poles 5 Noise Robustness Noise robustness and convergence of an algorithm can be attributed to two factors: numerical conditioning of the algorithm itself in finite-precision arithmetics, eg, the solve of the overdetermined equation in (8 or (; sensitivity of the algorithm against noise-corrupted input data Although OVF has been proposed to improve the conditioning of the LS solve in traditional VF [], it is known that the VF convergence will still be hampered even by a small spectral noise, eg, when SNR=dB [8], due to the normalization of the unity basis [the + term in (] This can be alleviated by relaxing that unity basis to a free variable and inserting a relaxed constraint [9] However, extra processing and computation are required in these schemes db Degree Output (a Sampling point (b Normalized radian Original Approx Original Approx Original Approx Error (c Normalized radian Figure Backplane: (a time response for first ns; (b & (c magnitude and phase responses in frequency domain In (TD-VFz, instead of approximating in the whole s- domain wherein the poles can take on largely dynamic magnitudes, the z-domain poles are constrained inside the unity circle which lead to significantly improved numerical conditionings Moreover, the scaling function σ(z in (, which is sensitive to noisy data in VF, is mapped by the ( N inverse z-transform to σ[k] = γ n (α (i n k u[k] + δ[k] in TD-VFz Insensitivity of σ[k] against noisy data (x[k] and y[k] is obviously but the filtering action of the convolution operations in ( These robust features of TD-VFz are verified in the examples that follow Numerical Examples The proposed TD-VFz algorithm is coded in Matlab m- script (text files and run in the Matlab environment on a GB-RAM GHz PC The test example arises from modeling a 5-inch differential transmission channel on a full mesh ATCA backplane [] The time-domain response is obtained by an excitation from 5 MHz to 5GHz The signal is normalized and fitted using TD-VFz with a -pole approximant Time samples are taken at ps intervals for the first 5 points It takes 8 seconds and iterations for TD-VFz to reach convergence Fig (a plots the fitted

5 (a (b 5 (a 5 (b Condition Number Relative Error Sigma (db 5 5 Sigma (db No of iteration No of iteration 5 5 Sampling point 5 Sampling point Figure (a Condition number of overdetermined system matrix and (b relative error in LS solution Figure Magnitude of the scaling function σ [k]: (a noise-free; (b with noisy data (SNR=5dB Table Backplane: comparison between TD- VFz and other algorithms TD-VFz VF MVF TD-VF Number of poles Relative error (db CPU time (sec 8 9 Magnitude Hankel singular value Approximant error time response and shows the excellent accuracy of TD-VFz This is further verified in Fig (b & (c where the fitted results are transformed to the frequency-domain wherein the error is seen to be negligible This demonstrates the power of TD-VFz in achieving multi-domain analyses simultaneously (viz, the time, frequency, and pole-plane domains with fast computation and small models In contrast, obtaining the time-domain response from frequency-domain data as in VF is nontrivial and may require complicated Hilbert transform due to causality consideration [] We also test the same example with VF [], MVF [] (which considers time delay estimation to reduce approximant size, and TD-VF [] The results are shown in Table It is seen that TD-VFz is the fastest, and produces the smallest model with similar accuracy to other algorithms Fig shows the condition number of the system matrix in ( and the relative error in this LS solve in each TD- VFz iteration The fast pole convergence of TD-VFz results in the rapid drop of the condition number after the first iteration to the order of 5, which is much more robust than VF whose condition number remains in the order of 5 in the first few iterations [] Next, we study the robustness of TD-VFz under noisy data We repeat the example but with the output sequence y[k] corrupted by white noise under an SNR of 5dB In this case, TD-VFz converges in runs in 9 CPU seconds with 5 sampling points and a -pole approximant, ending up at a relative error of -db The magnitude of the scaling function σ[k] in the first TD-VFz iteration is shown 5 Model order Figure Hankel singular values and relative error of approximant in Fig, for both the noise-free and noisy cases We note that the ideal σ[k] should be a delta function which corresponds to σ(z =under TD-VFz convergence Apparently, the level of noise thus introduced in σ[k], due to the filtering action discussed in Section 5, is not influential on the delta-function envelope of σ[k] and has not caused too much impact to the TD-VFz convergence Finally, we investigate the use of the HSV in guiding the model order selection Fig shows the HSVs found by the impulse response of the backplane system (cf Section, as well as the relative error of different approximant orders An evidential correlation can be seen between these two parameters, with similar slope and tipping point We can therefore choose the number of poles, ie, the approximant order, to be bigger than or equal to the corner cut-off point Therefore, we choose -pole approximant in the numerical example Such HSV design guideline, as far as we are aware, is unavailable in the continuous-time VF and its variants like MVF, OVF, TD-VF etc The TD-VFz-fitted rational macromodels can then be easily integrated into popular mixed-signal simulation environments like Verilog-A, Synopsys Saber, Matlab etc For

6 Input Output Time [ns] 5 Time [ns] Figure 5 Backplane: random input and the corresponding output example, an output signal response of a high frequency random input signal to the backplane example is shown in Fig 5 using Matlab 5 Conclusions Some remarks are in order: TD-VFz is the fastest macromodeling algorithm among various VF variants studied here In our implementation, TD-VFz is always about 8% faster than TD-VF as there is no need to discretize the continuoustime convolution integrals Perhaps more importantly, the convergence error bound which translates into an analytical model order selection guideline, as far as we are aware, is unique to the discrete-time domain As discussed and exemplified, restriction of poles inside the unit circle renders much better numerical conditioning of (TD-VFz over the conventional (TD-VF which works with poles in the infinite open left half plane Time-domain fitting, due to its inherent convolutional filtering, is much more robust against noisy data than frequency-domain fitting In summary, this paper has generalized the vector fitting (VF algorithm to its discrete-time time-domain counterpart, called TD-VFz, for the fast and robust linear macromodeling of port-to-port responses in time-sampled data The partial fraction basis in digital time domain improves the numerical conditioning, convergence and robustness under noisy data A quasi-error bound unique to the discretetime formulation allows deterministic choice of approximant order Application examples have confirmed that TD- VFz exhibits efficient computation and produces highly accurate approximants, in terms of both time and frequency responses Acknowledgment This work was supported in part by the Hong Kong Research Grants Council under Project HKU E, by the University Research Committee of The University of Hong Kong References [] L P M Celik and A Obadasioglu IC Interconnect Analysis Norwell, MA: Kluwer, [] D Ioan, G Ciuprina, M Radulescu, and E Seebacher Compact modeling and fast simulation of on-chip interconnect lines IEEE Trans Magn, (:5 55, Apr [] B Gustavsen and A Semlyen Rational approximation of frequency domain responses by vector fitting IEEE Trans Power Delivery, (:5, July 999 [] S Grivet-Talocia Package macromodeling via time-domain vector fitting IEEE Microwave Wireless Compon Lett, (:, Nov [5] T Wu, C Kuo, H Chang, and J Hsieh A novel systematic approach for equivalent model extraction of embedded high-speed interconnects in time domain IEEE Trans Electromagn Compat, 5(:9 5, Aug [] N Wong and C-U Lei IIR approximation of FIR filters via discrete-time vector fitting IEEE Trans Signal Processing, accepted [] D Deschrijver, B Haegeman, and T Dhaene Orthonormal vector fitting: A robust macromodeling tool for rational approximation of frequency domain responses IEEE Trans Adv Packag, (: 5, May [8] S Grivet-Talocia and M Bandinu Improving the convergence of vector fitting for equivalent circuit extraction from noisy frequency responses IEEE Trans Electromagn Compat, 8(:, Feb [9] B Gustavsen Improving the pole relocating properties of vector fitting IEEE Trans Power Delivery, (:58 59, July [] R Zeng and J Sinsky Modified rational function modeling technique for high speed circuits In IEEE MTT-S Int Microwave Symp Digest, June [] S Luo and Z Chen Extraction of causal time-domain network parameters from their band-limited frequency-domain counterparts using rational functions IEEE Trans Circuits Syst I, 5(:5, June 5 [] R Mandrekar and M Swaminathan Delay extraction from frequency domain data for causal macro-modeling of passive networks In Proc Int Symp Circuits and Systems, volume, pages 558 5, May 5 [] P Regalia and M Mboup Undermodeled adaptive filtering: an a priori error bound for the Steiglitz-McBride method IEEE Trans Circuits Syst II, (:5, Feb 99 [] C Sanathanan and J Koerner Transfer function synthesis as a ratio of two complex polynomials IEEE Trans Automat Contr, 8(:5 58, Jan 9 [5] K Steiglitza and L McBride A technique for the identification of linear systems IEEE Trans Automat Contr, (:, Oct 95