khazaka detailed

7/28/2019 Khazaka Detailed

1/12

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 60, NO. 12, DECEMBER 2012 3927

Macromodeling of Distributed Networks FromFrequency-Domain Data Using the

Loewner Matrix ApproachMuhammad Kabir, Student Member, IEEE, and Roni Khazaka, Senior Member, IEEE

AbstractRecently, Loewner matrix (LM)-based methods were

introduced for generating time-domain macromodelsbased on fre-quency-domain measured parameters. These methods were shownto be very efficient and accurate for lumped systems with a largenumber of ports; however, they were not suitable for distributedtransmission-line networks. In this paper, an LM-based approachis proposed for modeling distributed networks. The new method

was shown to be efficient and accurate for large-scale distributednetworks.

Index TermsDistributed networks, frequency-domain data,Hamiltonian matrix, Loewner matrices (LMs), matrix formattangential interpolation, -parameters, time-domain macro-model, vector fitting, vector format tangential interpolation,

-parameters.

I. INTRODUCTION

I N microwave and high-frequency applications, we are oftenfaced with complex multiport linear structures for which itis impossible to derive accurate physics-based analytical models

in the form offirst-order differential equations suitable for cir-

cuit simulation. However, one can usually obtain accurate fre-

quency-domain or -parameter data describing such struc-

tures through the use of measurement or full-wave simulation

tools. In this paper, we propose a new algorithm for the auto-

matic generation of an accurate SPICE-compatible time-domain

macromodel directly from frequency-domain - or -param-

eter data.

Several algorithms were proposed in the last few decades

for macromodeling based on frequency-domain data. One

approach is the global rational approximation macromodeling

[1], which is based on least-squares approximations, but the

application of such methods is restricted to low-order and

narrow-frequency-band systems due to ill-conditioning. Amoment-generation scheme based on time-domain integration

was proposed in [2], but the procedures of this algorithm

is numerically challenging, as pointed out in [3]. A rational

Manuscript received July 06, 2012; revised September 20, 2012; acceptedSeptember 24, 2012. Date of publication November 19, 2012; date of cur-rent version December 13, 2012. This paper is an expanded paper from theIEEE MTT-S International Microwave Symposium, Montral, QC, Canada,June1722, 2012.

Theauthors are with theDepartment of Electricaland Computer Engineering,McGill University, Montreal, QC, Canada H3A2A7.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMTT.2012.2222915

approximation algorithm based on NevanlinnaPick inter-

polation was presented in [4]. This method uses the mirror

image of the original data points which cannot identify the

original system [5]. A convex programming approach for

generating guaranteed passive approximations was proposed

in [6], but the method is limited to low-order systems with a

smaller number of ports due to the CPU-expensive optimiza-

tion process. Another approach for handling frequency-domain

data is convolution-based techniques [7][11]. However, con-volution, in general, can be computationally expensive since

the convolution operator needs to take into account all of the

past history [7]. Recursive convolution can be used to address

this issue if a pole residue representation of the system can be

found [7]. In fact, the method proposed here can be used in

conjunction with recursive convolution. Recently, the Vector

Fitting method [12][15] was developed and refined [16][21]

as an effective method for addressing this issue. However, this

method can have difficulties modeling systems with a large

number of poles and a large number of ports. More recently, a

new approach based on the Loewner matrix (LM) pencil has

been proposed [22][24]. This method was shown to be veryefficient and accurate compared with Vector Fitting [23], par-

ticularly for systems with a large number of ports. However, the

LM approach cannot model distributed networks that are very

common in microwave applications. In [25], a new LM-based

approach was proposed that can handle distributed networks

and is accurate and efficient for systems with a large number of

ports and a large number of poles.

In this paper, we expand on [25] by providing the full de-

tails of the algorithm so that it can be more easily understood

and reproduced. Furthermore, a new more accurate and efficient

way of computing and is presented in addition to a pas-

sivity checking algorithm. Finally, more detailed examples withpassivity checks and comparisons with the most recent imple-

mentation of Vector Fitting [12], [18], [26] are presented. These

show considerable improvement in terms of accuracy, model

size, and CPU cost. In particular,an improvement of two to three

orders of magnitude in accuracy was observed.

II. PROBLEM FORMULATION

Consider the -port linear system shown in Fig. 1. The ob-

jective of the algorithm described in this paper is to construct

a SPICE-compatible time-domain macromodel based on fre-

quency-domain multiport network parameter data, which can

be obtained through measurement or simulation.

0018-9480/$31.00 2012 IEEE


2/12

3928 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 60, NO. 12, DECEMBER 2012

Fig. 1. -port linear network.

A. Frequency-Domain Data

The system can be fully represented in the frequency do-

main by its -parameters

(1)

where is the complex frequency, and

are the vectors of port currents and port voltages, respectively,

is the -parameter matrix, and is the number

of ports. In many practical applications, a closed-form expres-

sion for the -parameters is not available. Instead, measured or

simulated -parameters are available over a certain frequency

range. This frequency-domain data is expressed as

(2)

where is the complex frequency, is the -parameters

at frequency , and , where is the number of

data points.

B. Time-Domain Macromodel

Our goal in this paper is to obtain a SPICE-compatible time-

domain macromodel of the network that matches the fre-

quency-domain data in (2). This macromodel can be expressed

as a linear time-invariant (LTI) system in descriptor system form

with inputs and outputs as

(3)

where and contain the vectors of port volt-

ages and currents, respectively, the matrices ,

, , and define the LTI

descriptor system, and is the order of the system. is gener-

ally singular and the matrix pencil is regular. The polesof the system are the eigenvalues of the pencil .

Note that a closed-form expression of the frequency domain

-parameters of the system in (3) can be expressed as

(4)

Finally it is important to note that both and can be

embedded in the system matrices as shown in Appendix A.

III. LOEWNER MATRIX METHOD

Here, we will present an overview of the LM method [23]

for obtaining a time-domain macromodel as defined in (3) from

frequency-domain data as defined in (2). This method can be

summarized in the following steps.

Fig. 2. Data selection for VFTI.

Fig. 3. Data selection for MFTI.

A. Splitting the Data

The first step of the LM algorithm is to append the frequency-

domain data with the complex conjugates at the negative fre-

quencies, thus resulting in data points or double the originalnumber. The data is then divided into two groups, which we

refer to as the left data set and the right data set as follows:

where , , , and , ,

and are complex frequencies. There are a number of possible

approaches for splitting the data. In this work, we have imple-

mented two that are chosen to result in real matrices that can be

easily expressed in the time domain. The first is associated with

the Vector Format Tangential Interpolation (VFTI) [23] algo-

rithm, and the second is based on the Matrix Format Tangential

Interpolation (MFTI) [24] algorithm.

1) Data Splitting for VFTI [23]: In VFTI, the right data set

contains the first half of the frequency points along with

their complex conjugates, and the left data set contains the

remaining data, as shown in Fig. 2. In other words, for the right

data set, we have

(5)

and, for the left data set, we have

(6)

where and denotes the complex conjugate.Note that the number of frequency samples can be assumed to

be even without loss of generality.

2) Data Splitting for MFTI [24]: In the case of MFTI, the

odd frequency samples along with their complex conjugates are

put in the right data set and the even ones in the left data, set as

shown in Fig. 3. In other words, for the right data set, we have

(7)

and, for the left data set, we have

(8)

where .


3/12

KABIR AND KHAZAKA: MACROMODELING OF DISTRIBUTED NETWORKS FROM FREQUENCY-DOMAIN DATA USING THE LOEWNER MATRIX APPROACH 3929

B. LMs

The next step of the LM algorithm is to construct the LM ,

the shifted LM as well as two other matrices and . This

is done block by block as follows:

(9)

where , , and , rep-

resent the th block entry of and , respectively, and

and are defined as

(10)

where and are the tangential direction

matrices for the right and left data sets, respectively. Then, the

matrices and are constructed as follows:

(11)

Note that the choice of and as well as the number of

columns/rows depends on the type of tangential interpolation

used.

1) Tangential Directions for VFTI [23]: and are vec-

tors in VFTI i.e. . The directions are defined as follows:

(12)

where and is the th column of

the identity matrix of size , if , else

. In other words, , , and vice

versa. For example, the columns of for and is

given as follows:

Note that this choice of tangential interpolation effectively

means that, at each frequency point, only one row/column of

the -parameter matrix is used. The rest of the data is dis-

carded. Furthermore, in VFTI, and defined in (9)

are scalars and the size of and is .

2) Tangential Directions for MFTI [24]: For MFTI, and

are of size , i.e., . The directions are defined as

follows:

(13)

where and is the identity matrix.

Note that the choice of interpolation results in

and , and thus the whole -parameter matrix is

used at each frequency point. In this case, and are

block matrices of size and the size of and is .

C. Real LMs

The LMs as constructed in (9) and (11) are complex. In order

to obtain a real macromodel, real LMs can be computed using

a similarity transformation [23]

(14)

where is a block-diagonal matrix with each block

where is the identity matrix. For VFTI, , so

will simply be replaced by 1. On the other hand, for

MFTI.

D. Time-Domain Macromodel

The third and final step of the LM algorithm is to extract the

time-domain macromodel from the LMs.

1) Extraction of the Macromodel: A direct relationship be-

tween the LMs and the underlying time-domain macromodel

was shown. In fact, it can be shown that the macromodel can

be obtained by extracting the regular part of the matrix pencil

[22]. The regular part can be extracted, for ex-

ample, by a singular value decomposition (SVD) [22], [23] as

(15)

where , , is a di-

agonal matrix containing the singular values, and are theorthonormal matrices, and denotes the complex conjugate

transpose. Any value of , , will result in

the same SVD, except for the case where is one of the eigen-

values [23]. If a sufficient number of data points is used, the

matrix in general is not full-rank. The regular part

of the system is obtained by taking the first columns of and

to form the following orthonormal bases:

(16)

where and represent the th column of and , respec-

tively, and is the order of the system. Then, the time-domain

macromodel is extracted as follows:

(17)

Note that the matrix is always zero at this stage, and its con-

tribution is embedded inside the other matrices.


4/12


Fig. 4. Normalized singular value plot.

2) Order of the Macromodel and the Impact of : Note that

the order of the system is needed in (16). is determined

from the plot of the normalized singular values

of in (15). A large drop in the plot

indicates that the underlying time-domain macromodel exists.

In that case, there will be a clear separation between the singularvalues corresponding to the singular part and the regular part

[27]. The order is the index of the largest drop in the plot.

For example, the largest drop for the system shown in Fig. 4

occurs at 287, so for that system.

Note that, if is present, another smaller drop in the singular

values occur at , where is the number of ports. In this

case, must be extracted in order to maintain the stability and

passivity of the model [23].

IV. PROPOSED APPROACH FOR MODELING

DISTRIBUTED NETWORKS

The LM method was shown to be very efficient and accurate

for modeling systems with a large number of ports [23]. One of

the key properties of this method is that it is a system identifica-

tion technique that identifies the exact order of the underlying

system and extracts its actual poles. In order to achieve this, the

frequency-domain data must cover most of the bandwidth of

the system. For example, as shown in Fig. 5, the frequency-do-

main data spans the full bandwidth of the underlying lumped

system and the resulting singular value plot clearly identifies

the order of the system. This approach is impossible to apply

to distributed networks which are common in microwave ap-

plications, because these networks have infi

nite bandwidth andan infinite number of poles (an example is shown in Fig. 6).

In this case, it is impossible to completely identify the under-

lying system using a finite-order time-domain representation.

In fact, for distributed systems, any extracted macromodel is a

form of discretized approximation. In this paper, we present a

technique based on the LM method, which generates an accu-

rate time-domain lumped model of a distributed network from

frequency-domain data over a desired bandwidth. The details of

the method are given as follows.

A. LMs

The real LMs are constructed using the standard LM method

in Sections III-AIII-C. Both VFTI and MFTI are possible.

Fig. 5. (a) Frequency-domain data covering the whole bandwidth. (b) Systemidentification based on drops in the singular values.

Fig. 6. Example of a distributed system.

B. Determining the Order of the System

The next step is to determine an appropriate order for

the macromodel. For this, a singular value decomposition is

performed on the LM pencil as described in

Section III-D1 and shown as follows:

(18)

The normalized singular values are then plotted as shown in

Fig. 7. Note that, in this case, the plot does not contain clear

drops identifying the order of the underlying system as was the

case in Fig. 5. This is expected as the underlying system has an

infinite order. Instead our goal here is to select the order that pro-

vides the most accurate nonsingular macromodel. If the number

of frequency points used is sufficient, the normalized singular

values reach the accuracy threshold of the finite precision com-

putation engine, at which point a slope change can be observed,

as shown in Fig. 7. The order of the macromodel is chosen

at the point of this slope change which separates the regular part

from the singular part of the matrices.


5/12


Fig. 7. (a) Frequency-domain data from 0 to 4 GHz for a distributed system.(b) Singular value plot.

C. Extraction of the Regular Part of the LMs

As the order has been determined, the regular part of the

LMs is extracted in the same way as shown in (16) and (17) and

given as

(19)

(20)

resulting in the macromodel

(21)

Note that the matrices and are zero at this stage, and

their contribution is embedded inside the other matrices.

D. Extraction of and

The macromodel extracted in (21) matches the original data

very accurately. However, it generally has unstable poles far

from the origin, as shown in the example in Fig. 8. A similar

problem was observed in the original LM method [23], where

only real unstable poles were observed. This was due to the

embedding of the matrix in the system equations and was

corrected by extracting . In the case of distributed networks,

where both real and complex poles are present, both and

Fig. 8. Poles of the macromodel with and embedded.

Fig. 9. Pole diagram indicating the separation of the poles.

must be extracted in order to preserve the stability and accuracy

of the macromodel. The algorithm for extracting and

from thesystem matrices is outlined here and can be divided into

two main steps. The first step is to decouple the macromodel in

(21) into two systems such that

(22)

where the system is the desired system and the

system contains the undesired poles that

are the artifact of embedding of and . The second step is

to compute and such that

(23)

which leads us to the final macromodel

(24)

with the closed-form expression defined by

(25)

1) Extraction of the Model With the Desired Poles: First, the

poles of the system in (21) are computed by finding the general-

ized eigenvalues of the matrix pencil . We then iden-

tify the very large poles that are separated by a clear gap from

therest of thepoles. Note that some of these poles may be stable.

An example is provided in Figs. 8 and 9, which show a typical

pole distribution (Fig. 9 is a zoomed-out version of Fig. 8). An

example of the desired system poles is shown in Fig. 10. Once

we have identified the desired and undesired poles, the next step

is to partition the system as shown in (22).


6/12


Fig. 10. System poles with and extracted.

One way to partition is to transform the macromodel to a

block-diagonal one based on the two sets of poles [28]. The

process could be expensive for large-scale systems due to the

generalized Sylvester equation that has to be solved to derive the

block-diagonal structure. Another convenient way is Petrov

Galerkin projection which uses oblique projection of the system

to find a reduced-order model [29], [30]. We employed the pro-jection method using the left and right eigenvectors as projectors

[29] to find the reduced macromodel based on the desired poles.

The right and left eigenvector matrices corresponding to the

system in (21) can be calculated from the following relations:

(26)

where and are the diagonal matrices with the generalized

eigenvalues and and are the corresponding right and left

eigenvector matrices, respectively. The subspaces and

to extract the desired system are then formed by preserving theeigenvectors corresponding to the desired poles as

(27)

where, are the indices of the desired poles. The eigenvectors

are in general complex. The real subspaces are formed by split-

ting the real and the imaginary parts into separate vectors:

where and designate the real and the imaginaryparts,

respectively, and and represent the real subspaces ofand , respectively. QR decomposition is then used to

obtain the orthonormal bases and such that

(28)

The Macromodel corresponding to the desired poles is

formed by an oblique projection using the orthonormal bases

and as projectors

(29)

Fig. 11. Change of error with .

The macomodel is stable but does not include the

contribution of and .

The macomodel based on the undesired poles

is formed following the same pro-

cedure mentioned in (26)(29) as

(30)

where and are the orthonormal bases of the real sub-

spaces spanning the subspaces formed by the eigenvectors cor-

responding to the undesired poles.

2) Compute and : The next step is to compute

and . In order to do that, the error matrices over valuesof equally spaced and spanning the relevant bandwidth are

formed using the model extracted based on the undesired poles

as follows:

(31)

where and and are the real and com-

plex part of the error matrix, respectively. The change of relative

error with is shown in Fig. 11. As can be seen,

data points is more than sufficient in most cases.

is calculated by taking the average of the real part to yield

(32)

For each entry of we have equations which can be

solved by the least-square approach. The following vectors are

formed to apply that approach:

... ...


7/12


Fig. 12. Summary of the proposed algorithm.

and, finally, each entry of is calculated as follows:

(33)

where and .

Now we have a macromodel which is

stable and accurate and includes and explicitly outside.

A summary of the proposed algorithm is provided in Fig. 12.

E. Passivity of the Resulting SystemThe passivity of the resulting system is checked by first veri-

fyingthat is positivedefinite and then using the generalized

Hamiltonian theorem [31]. The matrices and are formed as

follows:

where is a Hamiltonian and is a symplectic matrix. The

system is passive if the matrix pencil has no imaginary

(real part ) eigenvalue. The LM method usually pre-

serves passivity of the underlying macromodel [23]. We were

not able to find a problem for which the model violates the pas-

sivity. However, any passivity violation can be corrected using

the Hamiltonian Matrix perturbation [32], if required.

V. SIMULATION RESULTS

A. Example Circuits

Here, we show a number of numerical examples that demon-

strate the accuracy and efficiency of the proposed method. Ex-

ample 1 is an 18 port transmission line network (Fig. 13) con-

taining nine coupled lines and nine noncoupled lines. Models

for both coupled and noncoupled lines are shown in [33]. The

Fig. 13. Circuit diagram for Example 1.

parameter values for the coupled line are taken from [34]. The

parameter values (per unit length) for noncoupled line are

3.74 , 0 S, 283.7 nH, and 84.6 pF. Thelength of the coupled and non-coupled lines are 0.1 and 0.05 m,

respectively. Example 2 is a 36-port transmission line network

(Fig. 14). This example network is formed by connecting two

networks of Example 1 in parallel using a 500- resistor be-

tween each pair of similar ports. Example 3 is a 72-port network

formed by connecting two networks of Example 2 in parallel

using the same resistor value as Example 2 between the similar

pair of ports. Example 4, shown in Fig. 15, is a 63-port network.

The network contains 9 9 noncoupled lines and 6 9 cou-

pled lines. The summary of the example circuits is provided in

Table I. for Example 4 is shown in Fig. 16 to show the

complexity of the problem. The frequency-domain data from0 to 4 GHz was generated using the matrix exponential stamp

[35], which relies on the solution of the telegrapher equations in

the frequency domain.

B. Accuracy and Efficiency Check

We implemented two variations of the proposed method:

MFTI and VFTI. The proposed algorithms were implemented

on an Intel Core i7-2600 CPU (at 3.40 GHz) using MATLAB.

The simulation results are summarized in Tables II and III.

A sufficient number of frequency-domain data was used for

all of the examples to identify the underlying systems. The

number of data for each example was adjusted to keep the

size of the same for both VFTI and MFTI. The proposed

method is compared with the recent implementation of Vector

Fitting. MATLAB source code of VFIT3, an implementation

of Fast Relaxed Vector Fitting (FRVF), was used as the VF

implementation [12], [18], [26]. The accuracy of the model was

measured by the relative error (Appendix B) using 10 000

data points. The Frobenius norm of the errors (Appendix B) for

these 10 000 data points for the models of the example circuits

are provided in Figs. 1720, respectively.

In summary, two possible implementations of the proposed

approach (MFTI and VFTI) were compared with Vector Fit-

ting. In general, MFTI performs better than VFTI in terms of

accuracy and CPU cost. Furthermore, both approaches show


8/12




TABLE ISUMMARY OF THE EXAMPLES CIRCUITS

Fig. 16. for example 4.

considerable improvement over Vector Fitting and scale very

well as the number of ports increases. An improvement in ac-

curacy of two to three orders of magnitude over Vector Fitting

was observed. Note that the examples presented here do not

include noisy data. However, the results in [24] and [36] would

suggest that the data splitting scheme of MFTI is suitable for

noisy data.

C. Time-Domain Simulation

Transient simulations are presented in Figs. 21 and 22 in

order to show the accuracy and stability of the proposed

methods in the time domain. The first simulation in Fig. 21

is based on the model for Example 1, and the input at the near

end was a 1-V, 2-ns pulse with 0.2-ns rise/fall time. The second

simulation in Fig. 22 is based on the model for Example 4, and

the input at the near end is a 1-V, 8-ns pulse with a rise/fall

time of 0.2 ns. The simulations of the proposed model were

done by generating a SPICE netlist of the MFTI model and

simulating it in NGSPICE [37]. In order to verify the accuracy

of the results, a comparison is shown with a time-domain sim-

ulation that we obtained by simple brute-force segmentation

of the transmission lines. As can be seen from the simulation

results, the proposed technique can be used to model systems

with a considerable amount of delay as compared with the

rise/fall time of the signals.

D. Passivity Check

The eigenvalues of the Hamiltonian and the symplectic ma-

trix pencil for all the examples are shown in Figs. 2326,

respectively ( for MFTI and for VFTI). It is evident

from all of the figures that there is no purely imaginary (or very

close to imaginary axis) eigenvalues for any of the examples,

and we also found positive definite for all of the exam-

ples. Thus, the macromodels extracted for all of the examples

are passive according to the passivity theorem mentioned in

Section IV-E.

Furthermore, a brute-force passivity check was performed

on the macromodels for all of the examples. -parameter

matrices were computed using (4) for 20 000 frequency points

from 0 to 4 GHz and 15 000 points from 4 to 20 GHz; the


9/12


TABLE IISIMULATION RESULTS (EXAMPLES 1 AND 2)

TABLE IIISIMULATION RESULTS (EXAMPLES 3 AND 4)

Fig. 17. Frobenius norm of the errors for Example 1.



minimum value of the eigenvalues of are then plotted.

The plots are presented in Figs. 2730, respectively. The


Fig. 21. Time-domain simulation for Example 1.

Fig. 22. Time-domain simulation for Example 4.

minimum eigenvalues are always positive and constant at high

frequency. Thus, all of the macromodels are passive in the


10/12


Fig. 23. Eigenvalues of Hamiltonian matrix pencil for Example 1.




range of frequency of interest as well as out of that band while

we employed two different methods for checking passivity. In

general, the Hamiltonian matrix-based method is sufficient and

recommended.

Fig. 27. Minimum eigenvalues of matrix for Example 1.




VI. CONCLUSION

In this paper, a new LM-based method was proposed for the

modeling of systems based on measured/simulated parameters.

The new approach is suitable for distributed interconnect net-

works which have a very high bandwidth. The new method was


11/12


shown to be accurate and efficient compared with established

techniques such as Vector Fitting, in particular for systems with

a large number of ports. An improvement in accuracy of two to

three orders of magnitude improvement was observed.

APPENDIX A

INCLUSION OF AND INSIDE THE SYSTEM MATRICES

If are the system matrices, and

can be incorporated inside the other matrices

Proof: The -parameters of the original system are given

by

The -parameters of the reduced system are given by

APPENDIX B

ERROR CALCULATION

To evaluate the overall performance of the resulting model,

-norm of the error [23], [38] is used which measures the error

in the magnitude of all of the entries. The same values of are

used to find the measured/simulated -parameter, and the

calculated one [using (4)]. The normalized -norm of the

the error is as follows:

(34)

where is the squared Frobenius norm or the

HilbertSchmidt norm of the matrix.

ACKNOWLEDGMENT

The authors would like to thank Dr. S. Lefteriu for her valu-

able information which helped give a better understanding of

the original method and for providing one of the original codes

to extract . The authors would also like to thank Prof. R Achar

for his valuable advice to improve the paper.

REFERENCES

[1] M. Elzinga, K. Virga, and J. Prince, Improved global rational approx-imation macromodeling algorithm for networks characterized by fre-quency-sampled data, IEEE Trans. Microw. Theory Tech., vol. 48, no.9, pp. 14611468, Sep. 2000.

[2] R. Achar and M. Nakhla, Efficient transient simulation of embeddedsubnetworks characterized by -parameters in the presence of non-linear elements, IEEE Trans. Microw. Theory Tech., vol. 46, no. 12,pp. 23562363, Dec. 1998.

[3] T. Palenius,Efficienttime-domainsimulation of interconnects charac-terized by large RLC circuits or tabulated s parameters, Ph.D. disser-tation, Dept. Electr. and Commun. Eng., Helsinki Univer. of Technol.,Espoo, Finland, Nov. 2004.

[4] C. Coelho, J. Phillips, and L. Silveira, Passive constrained rational ap-proximation al gorithm using nevanlinna-pick interpolation, in Proc.Conf. Design, Automat. Test Europe, Paris, France, Mar. 2002, pp.923930.

[5] A. C. Antoulas,On theconstruction of passive modelsfrom frequencyresponse data, Automatisierungstechnik, vol. 56, pp. 447452, Aug.2008.

[6] C. Coelho, J. Phillips, and L. Silveira, A convex programming ap-proach for generating guaranteed passive approximations to tabulatedfrequency-data, IEEE Trans. Comput.-Aided Design Integr. CircuitsSyst., vol. 23, no. 2, pp. 293301, Feb. 2004.

[7] A. Djordjevic, T. Sarkar, and R. Harrington, Analysis of lossy trans-mission lines with arbitrary nonlinear terminal networks, IEEE Trans.

Microw. Theor y Tech., vol. MTT-34, no. 6, pp. 660666, Jun. 1986.[8] J. Griffith and M. Nakhla, Time-domain analysis of lossy coupled

transmission lines, IEEE Trans. Microw. Theory Tech., vol. 38, no.10, pp. 14801487, Oct. 1990.

[9] S. Lin and E. Kuh, Transient simulation of lossy interconnects basedon the recursive convolution formulation, IEEE Trans. Circuits Syst.

I, Fund am. Th eory A ppl., vol. 39, no. 11, pp. 879892, Nov. 1992.[10] V. Rizzoli, A. Costanzo,F. Mastri, andA. Neri, A generalspicemodel

for arbitrary linear dispersive multiport components described by fre-quency-domain data, in IEEE MTT-S Int. Microw. Symp. Dig., Jun.2003, vol. 1, pp. 912.

[11] T. Brazil, Nonlinear, transient simulation of distributed rf circuitsusing discrete-time convolution, in Proc. IEEE Int. Symp. Circuits,May 2007, pp. 505508.

[12] B. Gustavsen and A. Semlyen, Rational approximation of frequencydomain responses by vectorfitting, IEEE Trans. Power Del., vol. 14,no. 3, pp. 10521061, Jul. 1999.

[13] B. Gustavsen and A. Semlyen, A robust approach for system identifi-cation in the frequency domain, IEEE Trans. Power Del., vol. 19, no.3, pp. 11671173, Jul. 2004.

[14] D. Deschrijver and T. Dhaene, Passivity-based sample selection andadaptive vectorfitting algorithm for pole-residue modeling of sparsefrequency-domain data, in Proc. IEEE Int. Behavioral ModelingSimul. Conf., Oct. 2004, pp. 6873.

[15] D. Saraswat, R. Achar, and M. Nakhla, A fast algorithm and prac-tical considerations for passive macromodeling of measured/simulateddata, IEEE Trans. Adv. Packag., vol. 27, no. 1, pp. 5770, Feb, 2004.


12/12


[16] D. Deschrijver and T. Dhaene, Stability and passivity enforcement ofparametric macromodels in time and frequency domain, IEEE Trans.Microw. Theo ry Tech., vol. 56, no. 11, pp. 24352441, Nov. 2008.

[17] T. Dhaene and D. Deschrijver, Stable parametric macromodelingusing a recursive implementation of the vector fitting algorithm,

IEEE Microw. Wireless Compon. Lett., vol. 19, no. 2, pp. 5961, Feb.2009.

[18] B. Gustavsen, Improving the pole relocating properties of vectorfit-ting,IEEE Trans. Power Del., vol. 21,no. 3, pp.15871592, Jul. 2006.

[19] B. Nouri, R. Achar, M. Nakhla, and D. Saraswat, z-domain or-thonormal vector fitting for macromodeling high-speed modulescharacterized by tabulated data, in Proc. 12th IEEE Workshop Signal

Propagation on Interconnects, May 2008, pp. 14.[20] P. Triverio, S. Grivet-Talocia, and M. Nakhla, An improved fitting

algorithm for parametric macromodeling from tabulated data,in Proc.12th IEEE Workshop Signal Propagation on Interconnects, May 2008,pp. 14.

[21] A. Chinea and S. Grivet-Talocia, A parallel vectorfitting implementa-tion for fast macromodeling of highly complex interconnects, inProc.

IEEE 19th Conf. Electr. Performance Electron. Packaging Syst., Oct.2010, pp. 101104.

[22] A. J. Mayo and A. C. Antoulas, A framework for the solution of thegeneralized realization problem, Linear Algebra and its Application,vol. 425, no. 23, pp. 634662, Sept. 2007.

[23] S. Lefteriu andA. C. Antoulas, A newapproachto modeling multiport

systems from frequency-domain data, IEEE Trans. Comput.-AidedDesign I ntegr. Circuits S yst., vol. 29, no. 1, pp. 1427, Jan. 2010.

[24] Y. Wang, C. Lei, G. Pang, and N. Wong, MFTI: Matrix-formattangential interpolation for modeling multi-port systems, in Proc.

IEEE/ACM Design Automation Conf., Anaheim, CA, 2010, pp.683686.

[25] M. Kabir and R. Khazaka, Macromodeling of interconnect networksfrom frequency domain data using the Loewner matrix approach, in

IEEE M TT-S In t. Microw. Symp. Dig., Jun. 1722, 2012, pp. 13.[26] D. Deschrijver, M. Mrozowski, T. Dhaene, and D. D. Zutter, Macro-

modeling of multiport systems using a fast implementation of thevectorfitting method, IEEE Microw. Wireless Compon. Lett., vol. 18,no. 6, pp. 383385, Jun. 2008.

[27] G. W. Stewart, Perturbation theory for the singular value decompo-sition, in SVD and Signal Processing, II: Algorithms, Analysis and

Applications, R. J. Vaccaro, Ed. Amsterdam, The Netherlands: Else-vier, 1990, pp. 99109.

[28] P. Benner, Partial stabilization of descriptor systems using spectralprojectors, in Numerical Linear Algebra in Signals, Systems and Con-trol, ser. Lecture Notes in Electrical Engineering, P. V. Dooren, S.P. Bhattacharyya, R. H. Chan, V. Olshevsky, and A. Routray, Eds.Houten, The Netherlands: Springer, 2011, vol. 80, pp. 5576.

[29] P. Krschner, Two-sided eigenvaluealgorithmsfor modalapproxima-tion, Masters thesis, Faculty of Math., Chemnitz Univ. of Technol.,Chemnitz, Jun. 2010.

[30] A. Antoulas, C. Beattie, and S. Gugercin, Interpolatory model reduc-tion of large-scale dynamical systems, in Efficient Modeling and Con-trol of Large-Scale Systems, K. G. J. Mohammadpour, Ed. Berlin,Germany: Springer-Verlag, Feb. 2010.

[31] Z. Zheng, C. Lei, and N. Wong, GHM: A generalized hamiltonianmethod for passivity test of impedance/admittance descriptor sys-tems, in Proc. IEEE/ACM Computer-Aided Design Conf., San Jose,CA, Nov. 2009, pp. 767773.

[32] W. Yuanzhe, Z. Zheng, K. Cheng-Kok, G. Pang, and W. Ngai,PEDS: Passivity enforcement for descriptor systems via hamil-tonian-symplectic matrix pencil perturbation, in Proc. IEEE/ACMComputer-Aided Design Conf., SanJose, CA, Nov. 2010, pp.800807.

[33] C. Paul, Analysis of Multiconductor Transmission Lines. Hoboken,New Jersey: John Wiley and Sons, Inc., 2008.

[34] A. C. Cangellaris and A. E. Ruehli, Model order reduction techniquesapplied to electromagnetic problems, in Proceedings IEEE Electrical

Performance of Electronic Packaging (EPEPS00), Oct. 2000, pp.239242.

[35] R. Achar and M. Nakhla, Simulation of high-speed interconnects,Proceedings of the IEEE, vol. 89, no. 5, pp. 693728, May 2001.

[36] S. Lefteriu, A. Ionita, and A. Antoulas, Modeling systems based onnoisy frequency and time domain measurements, in Perspectives in

Mathematical System Theory, Control, and Signal Processing, ser.Lec-ture Notes in Control and Information Sciences, J. Willems, S. Hara,Y. Ohta, and H. Fujioka, Eds. Berlin, Germany: Springer, 2010, vol.398, pp. 365378.

[37] Ngspice: A mixed-level/mixed-signal circuit simulator, [Online].Available: http://ngspice.sourceforge.net/

[38] A. C. Antoulas, Approximation of Large-Scale Dynamical Systems.Philadelphia, PA: Soc. Ind. Appl. Math., 2005, ch. 5.1.

Muhammad Kabir (S09) received the B.Sc. degreefrom Bangladesh University of Engineering andTechnology, Dhaka, Bangladesh, in 2005, and theM.Sc. degree from Lakehead University, ThunderBay, ON, Canada, in 2010. He is currently workingtoward the Ph.D. degree in electrical and computerengineering at McGill University, Montral, QC,Canada.

He was a full-time Research Assistant with

Lakehead University, Thunder Bay, ON, Canada,from May, 2010 to July 2010 and was with MotorolaTelecommunication, Bangladesh, as a System Engineer from 2005 to 2008.His research interests include modeling of high-speed interconnect systemsfrom simulated/measured parameters, fast frequency sweep algorithms forhigh-speed modules, parameterization of time-domain macromodels, andextraction of delays from the macromodel.

Mr. Kabir served on the 2012 International Microwave Symposium orga-nizing committee.

Roni Khazaka (S92M03SM07) receivedthe B.S., M.S., and Ph.D. degrees from CarletonUniversity, Ottawa, ON, Canada in 1995, 1998, and2002, respectively, all in electrical engineering.

In 2002, hejoinedthe Department of Electrical andComputer Engineering, Mcgill University, Montral,QC, Canada, where he currently is an Associate Pro-fessor. In 2009, he was a Visiting Research Fellowwith the University of Shizuoka, Japan. He has au-thored and coauthored over 60 journal and confer-ence papers on thesimulation of high-speed intercon-

nects and RF circuits. His current research interests include electronic designautomation, numerical algorithms and techniques, and the analysis and simula-tion of RF ICs, high-speed interconnects, and optical networks.

Prof. Khazaka was the recipient of the IEEE Microwave Theory and Tech-niques Society (IEEE MTT-S) 2002 Microwave Prize, The Natural Sciencesand Engineering Research Council (NSERC) of Canada scholarships (at themasters and doctoral levels), Carleton Universitys Senate Medal and Univer-sity Medal in Engineering, the Nortel Networks scholarship, and the IBM co-operative fellowship. He has served on several IEEE committees and is cur-rently vice chair of the IEEE Montreal section. As a student he was treasurer

of the Carleton University IEEE student branch (19931994) and later a IEEERegion 7 (Canada) student representative on the IEEE Student Activities Com-mittee (1995 to 1998). He was Montreal section treasurer (2005/2006), Montrealsection student activities co-ordinator (2004), and founding chair of the IEEEMontreal Graduate of the Last Decade (GOLD) committee. He is a memberof the technical program committee of the Signal Propagation on InterconnectsWorkshop since 2006 and the technical program review committee of the In-ternational Microwave Symposium since 2012. He served on the organizingcommittee and numerous conferences such as MWCAS, NEWCAS, ISSSE,CCECE, and the 2012 International Microwave Symposium.

khazaka detailed

Documents