Causal Transient Simulation of Passive Networks with Fast Convolution

Rohan Mandrekarl, Krishna Srinivasan2, Ege Engin3, Madhavan Swaminathan4School of Electrical and Computer Engineering

Georgia Institute of TechnologyAtlanta GA 30332

{rohanl ,krishna2 ,engin3 ,madhavan.swaminathan4 } @ece.gatech.edu

AbstractA methodology for performing causal transient simulation of 0.3 - -_____---__--_

passive networks has been described in [1]. The methodology --- Causality conditionbased signal flow graphs and requires multiple convolu- 05(2nd Transmission)iSDaseclon slgnano w grap lS ana requlre mu hpeconvou-_0.25-3 x Linedelay

tions to be performed at each time step. Since convolution is a - I (8.085 ns)computationally expensive operation, it limits the efficiency of U 0.2the simulation methodology when simulating large sized net-works. This paper describes the use of a fast convolution al- 0.15 Causality violations

0gorithm that improves the simulation efficiency for large sized > 01 -

problems. The paper demonstrates this by simulating a 130-port Causality condition (2.695 ns)S-parameter network using signal flow graphs. 0.05 (1st Transmission)

Time for signal to propagatealong the transmission line

Introduction 0 -

o 0.5 1 1.5At higher frequencies, since the size of passive structures is time (s) x 10-8

comparable to the signal wavelength, distributed effects like de-lay play an important role in the time domain analysis of such Figure 2: Transmission line transient simulation violatingstructures. These distributed effects imply that there are many causalitycausality conditions that need to be satisfied to generate the cor-rect system response in the time domain. As an illustration, Fig- Zure 1 shows the multiple causality conditions due to the finite >velocity of the electromagnetic waves propagating on a trans- s-1

mission line. As seen from the figure the ene-to-end delay of S,f S22 171

the transmission line forms the basis for these causality condi- S12tions. This delay is given by Td I/c where I is the length ofthe line and c is the velocity of propagation of the electromag-netic waves. Macro-modeling techniques for distributed pas-sive structures, that use bandlimited frequency response data do Figure 3: Signal flow graph for the transmission line circuitnot explicitly account for these causality conditions [1]. Hencetransient simulations performed using such macro-models oftenviolate causality. An illustration of this phenomenon is shown technique Using this methodology the circuit in Figure 1 isin Figure 2 where the circuit shown in Figure 1 is simulated represented by the signal flow graph (SFG) shown in Figure 3.using bandlimited frequency response data. Using the delay extraction technique described in [1] this signal

flow graph can be converted into a set of causal SFG equationsCausal transient simulation using signal flow graphs given asA methodology for performing causal transient simulation of V1 (t) = Vs (t) + V3 (t) ® FS (1)

passive networks has been described in [1]. The methodology V2 (t) VI (t)performs causal transient simulation of multi-port passive net- (2)works using signal flow graphs along with a delay extraction V3(t) = V2(t) ® sll(t) + V5(t - Td) 0 sl2min(t) (3)

R V4(t) = V2(t - Td) $ s2lmin(t) + V5(t) s22(t) (4)

_______________X_____________ V5(t) = V4(t)$ FL (5)

* ~~~~~~~~~~~wheresll(t) and s22(t) are the respective self impulse re-V(t =0 8 sponses of the transmission line structure, while s12mim (t) and

*,_L_-=>~V(t=O= t<f12S2lmim (t) are the transfer impulse responses after the delay em-1st Rloundtrip . bedded in them has been extracted. Solving these equationsRefet,o=0,t ,2Y t ~ _ 2nd1TransmXission; for the circuit in Figure 1 results in a causal transient response

's shown in Figure 4. The solution to this set of equations at eachtime step requires the evaluation of the convolutions seen on the

Figure 1: Multiple causality conditions on a transmission line right hand side of the individual equations 1 through 5. If the

1-4244-0454-11061$20.OO ©2006 IEEE 61 SPI12006

h(t)0.3 -

- ropagateICausality condition

along the (2nd Transmission)0.25 3 x Line delay

c ~~~I(8.085 ns)CZ 0.2-U,ID ---------

CZ 0.15

o001 1

time (s) X 10-8 work

Figure 4: Transmission line transient simulation enforcing denoted as y(t) =xs(t) 0 h(t) and is given bycausality t

(t) J/ h(t-T)s(T)dT (6)-oo

source and load terminations are resistors as in the case of thecircuit in Figure 1, then 1s and rL arejust one sample in length soss hscnouinitga sipeetda

along~~ ~ ~ ~ ~~~sonethestranvoluhon linea Slmlmnda

and their convolution simplifies down to a single multiplication.However for the above 2-port signal flow graph, the four convo- n- 1lutions given in Equations 3 and 4 still need to be evaluated at y(tn) = E h(tn ti)x(ti)A\t (7)each time step. Generalizing this observation, the simulation of i=-lan N-port S-parameter network using signal flow graphs will where tn is the time at which the convolution is being com-require at least N2 convolutions to be performed to setup the puted, 1 is the length of the impulse response, and At is thesystem matrix for each time step. Since discrete convolution interval between consecutive time samples. The fast convolu-performed using the conventional multiply-and-add algorithm tion method described in [5] decomposes this summation intois a computationally expensive process, it bottlenecks the effi- two parts given asciency of the SFG based simulation approach for systems with s n- 1a large number of ports. y(tn) = E h(tn ti)x(ti)A\t + E3 h(tn ti)x(ti)A\t

Several techniques have been proposed in literature that ad- i=n-li=+dress the problem of improving the computational efficiency (8)of the convolution integral [2][3][4][5]. Amongst these [2] is where n-1 < s < n-1. In [5] the first part of the summa-based on number theoretic transforms and requires prior knowl- tion is computed using Lagrange interpolation while the secondedge of both the sequences involved in the convolution. Hence part is computed using the conventional convolution algorithm.this technique cannot be used in a SFG based framework which For real world systems that are lossy, the variations in h(t) be-proceeds one time-step at a time. [3] uses recursive convolu- come lesser and lesser as tw ch . This is shown in Figure 5tion that requires rational function approximation of the fre- which shows a typical h(t) for a lossy passive network. In Fig-quency response of the network being simulated. Since com- ure 5, a point s cabcnechosen on the time axis such that mostputing the rational function approximation of the multi-port fre- of the energy i e in the darker shaded region. Now ifquency response of a large network often involves solving an h(t) is to be convolved with an input signal s(t) as shown inill-conditioned problem, this technique is unsuitable for simu- Figure 6, the portion of the convolution from the darker shadedlating large sized networks. [4] and [5] describe fast convo- region (which corresponds to the second summation in Equa-lution techniques based on the partitioning of the network im- tion 8) will have a much higher value as compared to that frompulse responses. These techniques are supposed to improve the the lighter shaded region (which corresponds to the first sum-computational efficiency of discrete convolution from 0(N2) mation in Equation 8). This property will remain true even asto 0(Nlog2N) forthfor[4] NansNogsn for[5 resNrisrthewleatsiscomputed for successive values of tn.simulation time. The technique proposedin [] h enud in Also since h(t) itself has minimal variation in the lighter shadedthis work to enhance te efficiency of the SFG based simulation region, the variation in the contribution by the first summationmethodology. in Equation 8 for successive values oftT will also be relatively

smaller. The fast convolution algorithm makes use of this prop-

FastinconvoltionalapprtonachrusinLagrangef itherolati-pontfre erty by computing the first summation in Equation 8 for onlycertaindiscrete points over a block of ime and then using La-

Let y(t) be the convolution result obtained using a time do- grange interpolation for calculating it at each time stepth overmain signal c(t) and an impulse response h(t). Then y(t) is that block of time.

SPI 2006 62

x(t) A0.3-

t

0.2-

~~~~~~~~~~~~~~~~~~~~~~~>0.1Interpolation based 0.05 conventional convolution

ConVentiboal convblution fast convolution

0 0.5 1 1.5

Figure 6: Fast convolution using impulse response partition time(s) x 10-8

Figure 7: Fast convolution using impulse response partitionConsider that y(tn) has been evaluated till ta. Let tb - ta

n- s - = q be the length of a time block over which y(tn)has to be computed. Let the computation of each 0,m requires 0(k) operations where

k = s-n-I + 1, the computation of g(tn) for all q values in

g(tn) = E h(tn- ti)x(ti)At (9) the block requires 0(pq + kp) operations. In comparison withthis a conventional convolution implementation to obtain all qvalues requires O(qk) operations. Since the impulse responses

denote the contribution by the first summation from Equation of lossy passive networks become "smooth" as t -> oc it is8. It is seen the for ta < tn < tb, all values required for the found that the typical value of p required to achieve a desiredcomputation of g(tn) are already known. However g(tn) need accuracy in the evaluation of g(tn) is much much smaller asnot be computed for each time step. Instead we express the compared to k. When this is true, the fast convolution approachsignal h(t - ti) in terms of a Lagrange basis as is much more efficient.

p Performance analysis of the fast convolution approachh(t - ti) = ilmpn(t)h(cm - ti) (10) Using the fast convolution algorithm, the circuit in Figure 1

m=l1was simulated using causal SFG equations. The comparison of

where 1m is the mlh Lagrange polynomial of degree p - 1 and the simulation result with that obtained using conventional con-cm are Chebyshev nodes given by volution is shown in Figure 7. From the figure it is seen that the

two results virtually overlap each other. But, since the circuit ista + tb tbta (2m -1)7 (I1) a simple 2-port network, the fast convolution does not providecTn + Cos (11)

2 2 2p any appreciable speedup over conventional convolution. To bet-

Using this expansion g(t) can be represented as ter understand the performance improvement achieved by thefast convolution algorithm, a 64-bit microstrip interconnect bus

s p referenced to non-ideal power/ground planes was simulated us-g(t) = S ( E §lm(t)h(cm - ti))x(ti)At (12) ing SFGs. The bus was driven using 64 different random bit

i=n-l m=1 pattern drivers and the transient result was observed at the far

Interchanging the order of the summation in Equation 12 yields end of one of the lines in the form of an eye-diagram. Theentire system was represented by a 130-port S-parameter net-

p S work that was simulated using causal SFGs implementing con-g(t) 5E Tm(t) 5 h(cm - ti)x(ti) At (13) ventional convolution as well as fast convolution. The number

m=1 i=n-l of Lagrange basis functions used for interpolation was varied

Now if we define 0,m given by from 10 to 30 in steps of 5. The eye-diagram obtained from thetransient simulation using conventional convolution is shown inFigure 8. The error introduced in the transient simulation re-

5/m h(Cm - ti)(ti)A\t (14) sult due to the fast convolution algorithm is shown in Figure 9.i=n-l ~~~~~~~~Thefigure plots the maximum error in the transient simulation

we get relative to the eye-opening for various numbers of basis func-P ~~~~~~~~tionsused. It can be seen that even for a moderate number of

g(t) 5Eiim(t)/'m (15) basis functions, the error introduced by the fast convolution al-m=l gorithm is less than 0.05%. The speedup provided by the fast

where &9m is simply the value of g(Cm). Hence the computation convolution algorithm over conventional convolution is shownof each value of g(tn) requires only 0(p) operations. Since in Figure 10. It is clearly seen that for larger problems like

63 SPI 2006

120 _ Conventional convolution- Fast convolution

100 _

0

3 5 ~~~~~~~~80 -.

E-0

0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

=0 ~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~SimulationspanX1-0=5 0 0 5

Figure 11: Time-lines of the simulation progress using normalFigure 8: Eye-diagram observed on the 64-bit microstrip bus convolution and fast convolution

the one considered here, the fast convolution approach providesabout 2.5-2.8X speedup. Since the simulation times for large

For e.g., the conventional convolution based simulation for theabove problem required 121 mn. to complete while the fastconvolution approach using 10 basis functions required a little

0.02over 40 mins to finish. Finally, Figure 11 plots the time-line

a..0Jii12 4

of the two convolution methods for simulating the response ofthe 130-port network for a period of 70 ns. The y-axis plots thetime taken in minutes to perform the convolutions. The plot forthe ast convo ution approac clearly shows its ock-wise na-

OQ01 ture. It is seen that for each block the algorithm requires a setuptime where it needs to compute the coefficients of the Lagrange

.........rbases. However once this is done, the simulation progressesFigure 8: Eye-diagram observed on the 64-bitmicrostrimuch faster as compared to conventional convolution.

Nuberik of basis fuincionsConclusions

This paper demonstrates the use of a fast convolution algo-Fthwrithm in the causal transient simulation of multiport passive net-

works using SFGs. For a 130-port network, the fast convolutionapproach has been shown to give a 2.8X speedup in simulation.

2 __________8__________References[1] Mandrekar, R., Swaminathan, M., "Causality enforcement

in transient simulation of passive networks through delayextraction," Proceedings of SPI, May 2005, pp. 25-28.

27 \lte[2]Agarwal, R., et al. "Number theoretic transforms to imple-ment fast digital convolution," Proceedings of the IEEE,Apr 1975, pp. 50-560.

[3] Oh, K., et al. "An efficient implementation of surface im-sedance boundary con itions for the FDTD method " IEEE

__________________ _ [4]mchifang,r ,eal,"ascoprdtorealentimeaconvolution.agrth o

Number of basis fuflConclusiEEornso i&y u 20,p.3035

Figure 10: SEedup obtainedfor the 130-port network simula- 5 au,S,hta. Efcen iedmi slto ffetion ~ ~~~ ~ ~ ~ ~ ~~~~~~~~lh quntcy uadomaineleents, Prmlao. ofICCAlhpor asv1996tpp

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