Macromodelling Oscillators Using Krylov-Subspace Methods

Transcription

1 Macromoelling Oscillators Using Krylov-Subspace Methos Xiaolue Lai an Jaijeet Roychowhury Department of Electrical an Computer Engineering University of Minnesota {laixl, Abstract We present an efficient metho for automatically extracting unifie amplitue/phase macromoels of arbitrary oscillators from their SPICE-level circuit escriptions Such comprehensive oscillator macromoels are necessary for accuracy when speeing up simulation of higherlevel circuits/systems, such as PLLs, in which oscillators are embee Stanar MOR techniques for linear time invariant (LTI) an varying (LTV) systems are not applicable to oscillators on account of their funamentally nonlinear phase behavior By employing a cancellation technique to eflate out the phase component, we restore the valiity an efficacy of Krylov-subspace-base LTV MOR techniques for macromoelling oscillator amplitue responses The nonlinear phase response is re-incorporate into the macromoel after the amplitue components have been reuce The resulting unifie macromoels preict oscillator waveforms, in the presence of any kin of input or interference, at far lower computational cost than full SPICE-level simulation, an with far greater accuracy compare to existing macromoels We emonstrate the propose techniques on LC an ring oscillators, obtaining speeups of 0-0 with no appreciable loss of accuracy, even for small circuits I INTRODUCTION Oscillators are important builing blocks in electronic an optical systems For example, they are often use for frequency-translation of information signals in communication systems Voltage-controlle oscillators (VCOs) are key components of phase-locke loops (PLLs), which are wiely use in both igital an analog circuits for clock generation an recovery, frequency synthesis, etc Despite their wiesprea use, the simulation of oscillators an oscillator-base systems still poses significant challenges Traitional circuit simulators, such as SPICE [], are far from ieally suite for simulating oscillators One key problem is that transient simulation accumulates numerical phase errors without limit; furthermore, it is also ifficult to extract phase information from time-omain voltage/current waveforms accurately To improve phase accuracy, many timesteps nee to be taken in each oscillation cycle, with transient simulations of high-q oscillators requiring many thousans of cycles, hence suffering from great inefficiency Since phase responses are of major concern for oscillators, various specialize an approximate techniques (eg, [] [7]) have been evelope for preicting phase information irectly without relying on transient simulation of the full circuit Simulation in the phase omain can result in great speeups However, phase macromoels o not capture amplitue variations at the output of the oscillator, which can be important in many situations For example, in pico-raio systems, raio noes must be ultra-low power, leaing to novel, very simple PLL systems where VCO outputs are in essence irectly fe to analog mixers, with no intervening amplitue stabilization or clipping VCO amplitue variations in such systems change gains of PLL loops ynamically, thereby affecting important phenomena such as jitter, lock/capture behavior, etc Similar esign philosophies are emerging for a variety of low power systems in wireless communication an mobile systems In [8], a technique was presente to macromoel both phase an amplitue in oscillators uner perturbation The phase eviation was calculate via a nonlinear phase equation [9], while the amplitue macromoel was extracte by full Floquet ecomposition [0] of the linearize, time-varying oscillator The metho applies to any kin of oscillator, incluing LC, ring, etc However, it has a rawback: Floquet ecomposition becomes very computationally expensive, an can also suffer from numerical errors, as system sizes increase An oscillator macromoelling technique that scales gracefully with circuit size is of great practical interest, given that on-chip RF oscillators toay can have many thousans of noes In this paper, we present a novel metho to circumvent this issue Instea of performing full Floquet ecomposition on the linearize oscillator equations, we apply the time-varying Paé (TVP) metho [], to reuce the oscillator system to a smaller LPTV system which captures the important amplitue-variation components of the original oscillator accurately The transfer function of the LPTV oscillator system is expane into matrix forms using time- or frequencyomain methos (such as FDTD or harmonic balance), an then reuce using Krylov-subspace methos [] [] A major issue face for oscillators, that we solve in this paper, is that the oscillator s LPTV system incorporates both phase an amplitue information, leaing to inaccurate amplitue macromoels if the phase component is not separate out correctly Moreover, because oscillators are funamentally phase-unstable [9] an sustaine small perturbations can change the phase of the oscillator unbounely, aitional issues are face if the phase component is not ealt with specially Instea of using full Floquet ecomposition as in [8], we use a novel alternative technique in this paper, base on canceling out components of the input that excite phase responses, that enables application of Krylov-subspace methos to oscillators The perturbation input to the oscillator is ecompose into two parts with one contributing to phase an the other to amplitue We change the input vector of the LPTV system to remove the input corresponing to phase an apply Krylov-subspace methos to reuce this moifie system The resulting reuce system contains no phase information since the pole corresponing to phase has a very small resiual, thus having been effectively eliminate by the Krylov reuction process Once the reuce amplitue macromoel, without any phase information, has been obtaine, we re-incorporate the phase component by coupling the amplitue macromoel with the nonlinear, scalar phase macromoel presente in [9] The resulting unifie oscillator macromoel is able to preict both phase an amplitue variations accurately at far lower computational cost than that of full SPICElevel simulation Compare to the metho escribe in [8], the key ifference is that the metho in [8] requires full ecomposition of the whole LPTV system, while our metho relies on Krylov-subspace methos, with far lower computational cost Hence, our metho scales well to large oscillator circuits The generate macromoels can be easily encapsulate into other circuit simulation tools (eg, MATLAB/Simulink, Verilog-A, etc) to preict the comprehensive behavior of oscillators in a variety of system-level situations We verify our macromoelling technique on ring an LC oscillators Comparing simulation results between our macromoels an SPICE-level full circuit simulation, we show that our macromoels are able to reprouce the waveforms of oscillators uner various perturbations accurately, while obtaining impressive speeups Even

2 with the relatively small oscillators we have use for testing purposes, we obtain speeups in the range of - orers of magnitue; we expect much greater speeups for larger circuits with complex evice moels, which we are currently in the process of incorporating into our simulation infrastructure The remainer of the paper is organize as follows In Section II, we show that linear perturbation analysis is not vali for oscillators In Section III, we review nonlinear perturbation analysis for oscillators an the nonlinear oscillator macromoel we employ in this work In Section IV, we summarize the time-varying Paé technique for reucing LPTV systems In Section V, we escribe our phase-component eflation technique for macromoelling amplitue variations of oscillators In Section VI, we present simulation results on ring an LC oscillators II LINEAR PERTURBATION ANALYSIS IS NOT SUITABLE FOR OSCILLATORS The traitional metho to analyze perturbe nonlinear system is to linearize the nonlinear equations on its unperturbe orbit In this section, we will show that this metho is not suitable for oscillators A general oscillator uner perturbations can be escribe by ẋ + f (x)=bb(t), () where b(t) is perturbation signal applie to the free running oscillator Since perturbation signal has small amplitue, we can linearize () on its unperturbe steay-state orbit an get a linearize perioic time-varying system ż(t)+g(t)z(t) Bb(t), () where G(t) = f (x) x x s (t) is the linearize system on oscillator s steay-state orbit x s (t) Accoring to Floquet theory [0], the state transition matrix of the homogeneous part of () can be given by Φ(t,τ)=U(t)e D(t τ) V T (τ) () where U(t)=[u (t),u (t),,u n (t)] an V (t)=[v (t),v (t),,v n (t)] are T -perioic nonsingular matrix, satisfying biorthogonality conitions v T i (t)u j(t)=δ ij,and = iag[µ,, µ n ],whereµ i are Floquet exponents The particular solution of () uner perturbation b(t) is given by n t z(t)= u i (t) e µ i(t τ) v T i (τ)bb(τ)τ () i= 0 For an oscillator system, one of the Floquet exponents must be 0 [9] Without loss of generality, we assume µ = 0, an e µ (t τ) term vanishes Thus, we can always choose a perturbation b(t) to satisfy v T (t)bb(t) has nonzero average value, then z(t) will grow unbounely on t even though b(t) have very small amplitue This contraicts the assumption that z(t) is small variation, thus the linear perturbation analysis is inconsistent III PREVIOUS WOEK: NONLINEAR OSCILLATOR MACROMODEL In [9], authors show that the Floquet exponent µ = 0 in () is corresponing to the oscillator s phase eviation, which means the oscillator s phase eviation grows unbounely even though the perturbation signal is very small Due to oscillator s neutral phase stability, linearizing the oscillator on its steay state is meaningless since a small perturbation will change the phase, or the steay-state orbit of the oscillator ramatically An iea to overcome this limitation is to linearize the oscillator circuits over its perturbe time-shifte orbit Base on this iea, in [8], a novel approach is presente to construct the oscillator macromoels The macromoel prouce is a combination of a scalar nonlinear ifferential equation [9] that solves the phase eviation an a reuce linear time-varying system for preicting amplitue variation, which is computationally simpler than the original oscillator system Here, we summarize this metho A Nonlinear Oscillator Phase Macromoel Accoring to [9], the solution of the oscillator uner perturbation can be expresse as x(t)=x s (t + α(t)) + y(t), () where x(t) is the orbit of the oscillator uner perturbation, x s (t) is the unperturbe steay-state orbit of the oscillator, α(t) is the phase eviation, an y(t) is the amplitue variation If we can calculate the phase eviation α(t) an amplitue variation y(t), we can rebuil the waveforms of the oscillator uner perturbations using () The phase eviation α(t) is governe by the nonlinear ifferential equation [9] α(t)=v T (t + α(t)) Bb(t), (6) where v (t) is the perturbation projection vector (PPV) that correspons to oscillator s phase sensitivity to perturbations, an b(t) is the perturbation applie to the oscillator The PPV has perioic waveforms that have the same perio as the oscillator Various methos [9], [] [7], both in time omain an frequency omain, have been presente for calculating the PPV from the oscillator circuit equations When the PPV is available, the phase eviation α(t) can be efficiently calculate by solving this simple one-imension ifferential equation B Amplitue Macromoel In [8], a metho is presente to to construct the amplitue macromoel of the oscillator the oscillator is first linearize on its perturbe time-shifte orbit ȯ(t) f x x s (t+α(t))o(t)+bb(t) = A(x s (t + α(t)))o(t)+bb(t), (7) where x s (t) is oscillator s steay-state orbit, o(t) is small variations ue to perturbation b(t) an α(t) is phase eviation Since A(x s (t + α(t))) is not perioic, we introuce a new variable ˆt = t + α(t) an efine ô(ˆt) =o(t) an ˆb(ˆt) =b(t) After ropping a quaratic term, we can obtain a linear perioic time-varying system ô(ˆt)=a(x s (ˆt))ô(ˆt)+Bˆb(ˆt) (8) Applying Floquet ecomposition to this LPTV system, the solution of this system can be expresse as n ˆt ô(ˆt)= u i (ˆt) exp(µ i (ˆt τ))v T i (τ)bˆb(τ)τ (9) i= 0 where u = 0 is corresponing to oscillator s phase eviation, we nee to rop it The resulting system can be reuce by ropping some less important Floquet exponents The limitation of this metho is that Floquet ecomposition is very slow an numerically unstable when the system size is large IV REDUCED-ORDER MODELING FOR LPTV SYSTEM In [], the time-varying Paé (TVP) metho is presente to reuce the LPTV system A nonlinear system can be expresse as a ifferential-algebraic equation (DAE) q(x(t)) + f (x(t)) = b L (t)+bb(t) t

3 y(t)= T x(t), (0) where b L (t) is a large signal, which etermines the system s steaystate orbit, an Bb(t) is small perturbations applie to the system B an are input/output vectors For small perturbation analysis, we linearize this system over the orbit generate by the large signal The time-varying Paé (TVP) metho can be applie to reuce this linearize time-varying system Here, we summarize this metho The transfer function of the linearize system can be written as H(t,s)= T ( D [] + sc(t )+G(t )) [B], () t where t D [] is a ifferential operator [] Assume C(t ) an G(t ) to be perioic with angular frequency w 0, an efine W(t,s) to be the operator-inverse in (), we have W(t,s)=( D [] + sc(t )+G(t )) [B] () t ( D [] + sc(t )+G(t ))W(t,s)=B () t Assume W(t,s) is also a perioic function, we can express () in frequency omain where [sc FD + J FD ] W FD = B FD, () J FD =(G FD + ΩC FD ), () C 0 C C C FD = C C 0 C C C C 0 (6) G FD = Ω = jω 0 G 0 G G G C 0 G G G G 0 I I 0I I ii (7) (8) B FD =[,0,0,B T,0,0,] T (9) Then the transfer function in frequency omain can be written as H FD (s)=d T [sc FD + J FD ] B FD, (0) where D = () We can use any Krylov-subspace metho to reuce (0) an obtain a smaller LPTV system H q (s)=lq T [I q q + st q ] R q, () an the reuce system equation is T q x(t)+ x(t)=r q u(t), () where x(t) is a vector of size q, which is much smaller than that of the original system V REDUCING OSCILLATOR SYSTEM USING TVP METHOD Reuce-orer moeling of oscillators is slightly ifferent than that of other systems, such as mixers an converters Oscillator systems are neutral stable, traitional TVP methos may not be able to prouce stable reuce system In this section, we present our metho for moeling amplitue variations of oscillators A Traitional TVP Metho Is Not Suitable For Oscillators TVP methos [] are very useful for reucing LPTV systems, such as mixers an convertors, etc However, the metho is not suitable for oscillators, as oscillator systems are neutral stable We have shown in Section II that the Floquet ecomposition of oscillator system has a Floquet exponent of 0, which contributes to phase response If we expan the oscillator transfer function to frequency omain using the metho escribe in Section IV, the resulting frequency omain transfer function (0) has many poles on the imaginary axis, with interval jπω 0 Figure epicts the pole istribution of an LC oscillator In this figure, the oscillator has poles on ± jnω 0, some poles even have positive real part ue to numerical integration error Performing MOR methos on such a system will prouce unstable reuce system, especially when the expansion point is close to the imaginary axis Moreover, since the phase an amplitue information are mixe together in the oscillator LPTV system, the reuce system is not able to correctly represent the amplitue variation of the oscillator, even though we can obtain a stable reuce system using the TVP metho Fig Imaginary part 0 x Real part x 0 0 Pole istribution of an LC oscillator B Oscillation Pole Cancellation Metho The instability of the reuce system is ue to oscillation poles in (0), or the Floquet exponent µ = 0 in the LPTV system () In [8], Floquet ecomposition is use to eliminate the Floquet exponent µ = 0; however, this metho is expensive an numerically unstable for large systems In this work, we present a novel metho to cancel the effect of µ without performing the Floquet ecomposition

4 Accoring to [8], we can obtain amplitue macromoel of the oscillator by linearizing the oscillator over its perturbe time-shifte orbit x s (ˆt), whereˆt = t +α(t) an α(t) is the phase eviation ue to perturbation The resulting LPTV system is ô(ˆt)=a(x s (ˆt))ô(ˆt)+Bˆb(ˆt) () The solution of above system can be expresse as n ˆt ô(ˆt)= u i (ˆt) exp(µ i (ˆt τ))v T i (τ)bˆb(τ)τ, () i= 0 where µ i are Floquet exponents, µ has a value of 0 [9], u i (t) an v i (t) are T -perioic vectors, satisfying biorthogonality conitions v T i (t)u j(t) =δ ij In[8],theµ term is roppe after the Floquet ecomposition since it contributes to phase eviation To avoi the Floquet ecomposition, we cannot use this metho to eliminate the µ term; instea, we eliminate the input that contributes to phase eviation From (), we know that the input for the µ term is v T (τ)bb(τ) Since u i(t) an v i (t) are biorthogonal, we can eliminate this input using the projection metho We efine a new input vector B = B v (τ)bu (τ) (6) Using this new input vector, the input to the µ term is v T (τ) Bb(τ)=v T (τ)bb(τ) v (τ)v (τ)bu (τ)b(τ) = v T (τ)bb(τ) v (τ)bv (τ)u (τ)b(τ) (7) = v T (τ)bb(τ) vt (τ)bb(τ)=0, an inputs for other µ i s(i > ) are v T i (τ) Bb(τ)=v T i (τ)bb(τ) v (τ)v i (τ)bu (τ)b(τ) = v T i (τ)bb(τ) v i(τ)bv i (τ)u (τ)b(τ) (8) = v T i (τ)bb(τ) 0 = vt i (τ)bb(τ) Hence, the new input vector B only eliminates the effect of the Floquet exponent µ = 0, but preserves inputs of all other Floquet exponents in the system We can replace the original input vector B with B without changing the amplitue response of the LPTV system () can be rewritten as ô(ˆt)=a(x s (ˆt))ô(ˆt)+ Bˆb(ˆt) (9) where õ(t) has size of q, which has size smaller than that of the original system ) Using Oscillator Macromoel: To reprouce oscillator waveforms uner perturbations using our macromoel, we nee to integrate both the phase an amplitue equations The phase equation (6) is solve on time t; however, the amplitue equation (9) is integrate on the shifte time ˆt = t + α(t) Below is the pseuocoe to rebuil the oscillator waveform using our macromoel t=0; α(0)=0; i=0 t=t+ t; i=i+ Calculate α(i) by solving (6) on t ˆt =t+α(i) Calculate amplitue variation o(t)=ô(ˆt) by solving () on ˆt 6 Rebuil oscillator s waveforms using () 7 goto VI NUMERICAL RESULTS In this section, we evaluate the technique presente above using ring an LC oscillators All simulations were performe using MATLAB on an Intel-architecture machine, running Linux We constructe oscillator macromoels using the metho escribe in Section V, simulate oscillator waveforms using the constructe macromoels, an compare the results with SPICE-like simulations of the full oscillator circuits in the same MATLAB environment Experiment results show that our macromoels are able to capture the amplitue variations of oscillators accurately The rebuilt waveforms using our macromoel match the results from full SPICE-level simulation perfectly, with about 0 0 times speeups A -stage Single-Ene Ring Oscillator Our first example is a -stage single-ene ring oscillator, as shown in Figure This oscillator has a system size of 8 an an oscillation frequency of f 0 = GHz V (9) is stable, because we have minimize the resiuals of oscillation poles in (0), even though we o not really eliminate them We can apply the TVP metho escribe in [] to reuce this moifie system an obtain a stable reuce system for amplitue variations 6 C Detaile Proceure Of Oscillator MOR In this subsection, we summarize the proceures for constructing an using the oscillator macromoel ) Constructing Oscillator Macromoel: Following proceure will construct the oscillator macromoel ) Obtain oscillator steay-state x s (t) using time omain or frequency omain methos ) Calculate u (t)=ẋ s (t) ) Calculate the PPV v (t) using numerical methos [9], [] [7] ) Construct new input vector B = B v (t) T Bu (t), where B is a T -perioic vector ) Perform the TVP metho on the new LPTV system an obtain the reuce system ȯ(t)=a(x s (t))o(t)+ Bb(t) (0) T q õ(t)=õ(t)+r q b(t), () Fig Perturbation A -stage single-ene ring oscillator We first calculate the steay-state of the oscillator using the harmonic balance metho, an extract the PPV of the oscillator using the Monoromy metho [9], [] We then linearize the oscillator on its steay-state orbit an reuce the resulting LPTV system using the metho we presente in Section V Since we are intereste in perturbations whose frequency is close to the oscillator s oscillation frequency, we choose the Arnoli expansion point s 0 = π f 0,where f 0 is the oscillator s free-running frequency The original nonlinear system has the size of 8, using our metho, we can reuce the system size to The resulting small LPTV system can be simulate much faster than the original nonlinear system

5 To verify that our macromoel can capture the oscillator amplitue variations correctly, we apply a perioic perturbation current b(t) =0000sin(ω t) to the noe of the oscillator, as shown in Figure We apply ifferent perturbation frequencies ω, rebuil the oscillator s output voltage on noe using our macromoel an compare with SPICE-level full simulation The simulation results are shown in Figure an Figure Our macromoel is able to match the full simulation perfectly, with great speeups The SPICE-level full simulation takes about 6 minutes for a simulation time of 00 cycles; however, it takes only secons to simulate the same number of cycles using our macromoel This gives us approximately 0 times speeup on this small oscillator circuit For larger oscillator circuits with more nonlinear components an complex evice moel, we expect more significant speeups (b) Macromoel Fig Output waveform of the stage ring oscillator uner perturbation b(t)=0000sin(06ω 0 t) (b) Macromoel Rp=0 0p L=n Rb=k Cb=p Re=00 Cp=p Cm=06p Rl=00 C=p C=p noise Fig Output waveform of the stage ring oscillator uner perturbation b(t)=0000sin(0ω 0 t) Fig A GHz Colpitts LC oscillator B GHz Colpitts LC Oscillator Our secon test circuit is a GHz Colpitts LC oscillator The circuit an parameters of this oscillator are shown in Figure The oscillator has a system size of 6 an a free-running frequency of f 0 = GHz We apply similar metho as we escribe in the ring oscillator case to calculate the PPV an construct the amplitue macromoel We evaluate our macromoel with a current perturbation injecte into the noe of the oscillator, as shown in Figure The perturbation current has an amplitue of 0mA We simulate the oscillator s output voltage on the noe uner ifferent perturbation frequencies using our macromoel an compare with SPICE-level full simulation The simulation results are shown in Figure 6 an Figure 7 Our macromoel is able to match the full simulation with acceptable accuracy In this case, we obtain about 0 times speeup This speeup is not as goo as the ring oscillator case, because this LC oscillator has only one nonlinear component, hence, its full SPICElevel simulation is very fast VII CONCLUSIONS We have presente a novel technique to extract simple amplitue macromoels from SPICE-level circuit escriptions of any oscillator circuit Combining it with the oscillator phase macromoel presente in [9], we can rebuil the output waveforms of any perturbe oscillator We have teste our macromoel using LC an ring oscillators an provie etaile comparisons against full SPICE-level circuit simulation Numerical results show our macromoels are able to preict oscillator amplitue an phase variations well in the presence

6 (b) Macromoel Fig 6 Output waveform of the Colpitts LC oscillator uner perturbation b(t)=0000sin(0ω 0 t) (b) Macromoel Fig 7 Output waveform of the Colpitts LC oscillator uner perturbation b(t)=0000sin(0ω 0 t) of perturbations, with great speeups over SPICE-level simulation Currently, we are working on valiating our technique on larger circuits, for which we expect speeups of - orers of magnitue REFERENCES [] L Nagel SPICE: A Computer Program to Simulate Semiconuctor Circuits Electron Res Lab, Univ Calif, Berkeley, 97 [] M Garner Phase-Lock Techniques Wiley, New York, 966 [] A Hajimiri an TH Lee A general theory of phase noise in electrical oscillators IEEE Journal of Soli-State Circuits, (), February 998 [] A Demir, E Liu, AL Sangiovanni-Vincentelli, an I Vassiliou Behavioral simulation techniques for phase/elay-locke systems In Proceeings of the Custom Integrate Circuits Conference 99, pages 6, May 99 [] A Costantini, C Florian, an G Vannini Vco behavioral moeling base on the nonlinear integral approach IEEE International Symposium on Circuits an Systems, :7 0, May 00 [6] L Wu, HW Jin, an WC Black Nonlinear behavioral moeling an simulation of phase-locke an elay-locke systems In Proceeings of IEEE CICC, 000, pages 7 0, May 000 [7] M F Mar An event-riven pll behavioral moel with applications to esign riven noise moeling In Proc Behav Moel an Simul(BMAS), 999 [8] X Lai an J Roychowhury Automate oscillator macromoelling techniques for capturing amplitue variations an injection locking In IEEE/ACM International Conference on Computer Aie Design, November 00 [9] A Demir, A Mehrotra, an J Roychowhury Phase noise in oscillators: a unifying theory an numerical methos for characterization IEEE Trans on Circuits an Systems-I:Funamental Theory an Applications, 7():6 67, May 000 [0] R Grimshaw Nonlinear Orinary Differential Equations Blackwell Scientific, New York, 990 [] J Roychowhury Reuce-orer moelling of time-varying systems IEEE Trans Ckts Syst II: Sig Proc, 6(0), November 999 [] P Felmann an RW Freun Efficient linear circuit analysis by pae approximation via the lanczos process IEEE Trans on Computer-Aie Design, :69 69, May 99 [] A Oabasioglu, M Celik, an LT Pileggi PRIMA: passive reuceorer interconnect macromoelling algorithm In Proceeings of IEEE ICCAD 997, pages 8 6, November 997 [] RW Freun Reuce-orer moeling techniques base on Krylov subspaces an their use in circuit simulation Technical Report TM, Bell Laboratories, 998 [] A Demir Phase noise in oscillators: Daes an colore noise sources In IEEE/ACM International Conference on Computer-Aie Design, November 998 [6] A Demir, D Long, an J Roychowhury Computing phase noise eigenfunctions irectly from steay-state jacobian matrices In IEEE/ACM International Conference on Computer Aie Design, pages 8 88, November 000 [7] A Demir an J Roychowhury A reliable an efficient proceure for oscillator ppv computation, with phase noise macromoelling applications IEEE Trans on Computer-Aie Design of Integrate Circuits an Systems, ():88 97, February 00