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MFTI: Matrix-Format Tangential Interpolation for ModelingMulti-Port Systems

Yuanzhe Wang, Chi-Un Lei, Grantham K. H. Pang and Ngai WongDepartment of Electrical and Electronic Engineering

The University of Hong Kong, Pokfulam Road, Hong Kong{yzwang, culei, gpang, nwong}@eee.hku.hk

ABSTRACT

Numerous algorithms to macromodel a linear time-invariant(LTI) system from its frequency-domain sampling data havebeen proposed in recent years [1, 2, 3, 4, 5, 6, 7, 8], amongwhich Loewner matrix-based tangential interpolation provesto be especially suitable for modeling massive-port systems [6,7, 8]. However, the existing Loewner matrix-based methodfollows vector-format tangential interpolation (VFTI), which

fails to explore all the information contained in the frequencysamples. In this paper, a novel matrix-format tangential in-terpolation (MFTI) is proposed, which requires much fewersamples to recover the system and yields better accuracywhen handling under-sampled, noisy and/or ill-conditioneddata. A recursive version of MFTI is proposed to furtherreduce the computational complexity. Numerical examplesthen confirm the superiority of MFTI over VFTI.

Categories and Subject Descriptors

B.7.2 [Integrated circuits]: Design aids-simulation

General Terms

Algorithms

Keywords

matrix-format tangential interpolation (MFTI), Loewner ma-trix, state space, sampling

1. INTRODUCTIONHigh-frequency effects, such as signal delay and crosstalk,

have become dominant factors limiting system performancein IC design. Accurate simulation is required to capture thehigh-frequency behavior of systems, so as to ensure consis-tent design of high-speed electronic systems. To model com-plicated geometry structures, such as packages, boards andRF objects, data-driven macromodeling is usually applied.Linear macromodeling can be classified as a system identi-fication problem [9] wherein circuits or systems are treated

as black boxes, and their responses are measured throughexperiments or calculated by EM simulators. Given thesampled frequency and/or time responses, such as admit-tance or scattering matrices, a model is built which fits thesamples accurately with a satisfactory computational effi-ciency. Several methods have been developed, among whichthe recently proposed Loewner-matrix-based Vector-FormatTangential Interpolation (VFTI) proves to be accurate and

efficient for modeling systems with massive ports [6, 7, 8].VFTI is robust, non-iterative, and does not require pole ini-tialization as in vector fitting. The order of the underlyingsystem is also automatically recognized. However, VFTI in-terpolates only two vectors of a scattering matrix at a time.Hence, it does not explore all the information contained inthe scattering matrices and loses approximation accuracyfor noisy responses. It also suffers numerical problems whenmodeling ill-conditioned samples poorly distributed in thefrequency band of interest.

In this paper, the generalization of VFTI to Matrix-FormatTangential Interpolation (MFTI) is proposed, which resultsin significant improvements over VFTI in practical macro-modeling: (i)by extending the tangential data to matrix for-mat, MFTI interpolates the full sampling matrix instead oftwo vectors. Hence, to achieve the same accuracy, MFTI

requires only 1/ samples compared to VFTI ( being thenumber of ports) to recover the underlying system. This is asignificant improvement considering that frequency-domainsampling usually involves expensive measurement or com-putation. Also, the accuracy is improved when modelingnoisy responses; (ii)by giving different weightings to differentsamples, MFTI is suitable for interpolating ill-conditioneddata (namely, when the sampling frequencies are poorly dis-tributed in the band of interest). It also provides an optionto trade off between computational complexity and fittingaccuracy; (iii)a recursive version of MFTI is developed to au-tomatically select the appropriate set of sampled data for ap-proximation; (iv)a minimal sampling theorem is introducedas theorem 3.5 that guides the number of sampling pointsrequired for an efficient computation.

This paper is organized as follows. In Section 2, the basicsof VFTI are introduced. In Section 3, the formulation ofMFTI is presented. In Section 4, two MFTI algorithms aresummarized. Numerical examples are given in Section 5 andSection 6 draws the conclusion.

2. BACKGROUND

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An LTI system can be expressed as a state-space model:

() = () + (),

() = () + (),(1)

where () , () , () , , , , , . If is singular, (1) iscalled a descriptor system (DS) whose transfer function is

() = ( )1

+ .

2.1 Overview of VFTIThe goal of an interpolation problem is to model the

underlying system or circuit from its measured/calculatedinput-output data. Assume that scattering matrices havebeen sampled at different frequencies , = 1, 2, . . . , ,

() =

11() . . . 1()

.

.

.. . .

.

.

.1() . . . ()

. (2)

Our goal is to find a DS as (1) whose transfer functionsatisfies

(2) = (2 )1

+ (). (3)

In VFTI, tangential interpolation data are used instead of

the whole (). Specifically, the right and left tangentialinterpolation data are defined as [6, 7, 8]

{, r,w , r 1

,w = S()r 1

, = 1 ..},

{, l,v , l 1

,v = lS() 1

, = 1..},(4)

whereby + = , , {2}. In (4), the vectors rand l are arbitrarily chosen interpolation directions. Thevectors w and v are called interpolation data. The rightand left interpolation data are used to generate a state spacewhose transfer function satisfies

H()r = w and lH() = v. (5)

Apparently, VFTI has the intrinsic disadvantage that it canonly interpolate two vectors, either row or column, of a sam-ple matrix. Thus (3) is not automatically satisfied by guar-

anteeing tangential constraints (5).

3. MATRIX-FORMAT TANGENTIAL INTER-

POLATION

3.1 Generation of interpolation dataSuppose we have sampled scattering matrices at dif-

ferent frequencies ( is assumed even for simplicity). Thematrix-format right and left interpolation data are in theirgeneral forms

{, , = 2, = () for = 1, 3 . . . 1;

= 1, = 1, = 1 for = 2 , 4 . . . },(6)

{, , = 2+1, = (+1), for = 1, 3 . . . 1;

= 1, = 1, = 1 for = 2 , 4 . . . }.

(7)They can also be expressed in a more compact format (wherein1 = 2, 3 = 4, . . . , 1 = , and the superscript of a ma-trix denotes its dimension) as follows

= diag[1..1 1

, 1.. 1 2

, , 1..1 1

, 1.. 1

],

= [ 11

, 21

, . . . , 11

, 1

],

= [11

, 12 , . . . ,

11

, 1 ], (8)

= diag[1..1 1

, 1.. 1 2

, , 1..1 1

, 1.. 1

],

=

1

1

2

1

.

.

.

1

1

1

, =

1

1

2

1

.

.

.

1

1

1

. (9)

Here we use both the scattering matrices and their conju-gates as tangential interpolation data to guarantee (2)= (2). For systems with identical number of inputsand outputs ( = ), if = and rank() = rank() =(for = 1, 2 . . . ), then all entries in the scattering ma-trices are exploited for interpolation. We can also set tobe different numbers to trade off between speed and accu-racy and/or to give different weightings to ill-conditionedsamples. The goal of interpolation is to find a state-spacerealization whose transfer function [cf. (3)] satisfies the leftand right constraints

() = and () = . (10)

3.2 Block-format (shifted) Loewner matrixSimilar to [6, 7, 8], block-format Loewner matrix and

shifted Loewner matrix are defined as

=

111111

11

1

.

.

.. . .

.

.

.11

1

, (11)

=

11111111

111

1

.

.

.. . .

.

.

.111

1

. (12)

Note that every diagonal block entry in (11) and (12) is a square matrix, thus and are both (1 + 2 + . . . +) (1 + 2 + . . . + ) square matrices. Similar to VFTI,MFTI data fulfill the Sylvester equations,

= ; = . (13)

3.3 State-space realizationThe state-space matrices in (1) can be calculated based

on , , , , and .

Lemma 3.1. If {1 . . . , 1 . . . }, det() =0, then = , = , = , = , = 0 con-stitute a minimal state-space model whose transfer functionsatisfies (10). Furthermore, if = = and , are of

full rank, the transfer function also satisfies (3).

Proof: The proof of the first conclusion is similar to Ap-pendix A-A of [8]. Note that the vector-format tangen-tial interpolation direction and data should be replaced by

matrix-format ones and that there should be replaced by = [0(1+..+1), , 0(+1+..+)].

Then consider the second conclusion. If = = ,(10) becomes ()

= = ()

for =

1, 3 . . . 1. Because the square matrix is chosento be of full rank, we get (2) = () = () for = 1, 3, . . . , 1. Similarly, (2) = () for =2, 4, . . . , . Thus (3) is guaranteed.

This lemma indicates that if the number of inputs is iden-tical to the number of outputs (i.e., = ), which is the

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case for a large group of (e.g., MNA) circuits, (3) is satisfiedexactly and (2) = () for all . Note that (3) can-not be guaranteed by VFTI. To guarantee the state-spacematrices to be real, we have the following lemma.

Lemma 3.2. If , , , , and are constructed tosatisfy the conditions in Lemma 3.1, then by using to de-note conjugate transpose, =

, =

, =

, = are guaranteed real and constitute a min-imal DS whose transfer function satisfies (10). Here, =

[1, 3, . . . , 1], = 12

[

], =

1, 3, . . . , 1.3.4 SVD and minimal sampling

Lemma 3.3. Suppose that the interpolation data are gen-erated by sampling an underlying system with state-space ex-pression 0 = 0 + 0 and = 0 + 0. Then for {}

{}, rank( ) size(0) + rank(0).Proof: Note that

=

10(10 0)

1

.

..0(0 0)

1

(0 0)[

(10 0)1

1 . . . (0 0)1

] 0

The Lemma indicates that when the size of or islarger than size(0)+rank(0), the assumption of Lemma 3.1will not hold. Subsequently, to build appropriate state-spacematrices, we need to perform singular value decomposition(SVD) to get rid of the irregular part.

Lemma 3.4. Suppose that for some 0 {}{},

rank(0 ) = rank[ ] = rank[

] . Then

we perform an economic SVD: (0) = (),then = , = , = , = isa realization of the sampling data.

Here 1 + 2 + . . . + is the order of and . Theproof is similar to that in [6] except that = [0

(1+..+1), , 0(+1+..+)] should be used instead of

The SVD approach relies on the assumption made in Lemma3.4. In real implementation, this assumption is generallytrue provided that the sampling is sufficient and thus thesize of () is large enough. The test results [see exam-ple 1] show that the ranks of and satisfy rank() order() and rank() rank( ) order() +rank(0). Here represents the underlying system (0, 0,0, 0, 0) and order() = rank(0) represents the num-ber of poles of . This is partially because that the normof is often much larger than that of. If is chosen tobe 1 or 1,

is close to , thus they have similar

singular value pattern. To recover the system, we shouldperform SVD and use the singular values to determine theregular part to be kept. Thus, the order of and (i.e. )cannot be smaller than order()+rank(0). The followingtheorem follows.

Theorem 3.5. The least number of noise-free samplesto recover the underlying system satisfies order()/ min(, ) (size(0) + rank(0)) / min(, ). Empirically = (order() + rank(0)) / min(, ).

The above theorem implies that the minimum samplesrequired in MFTI are smaller than those in VFTI. The latterneeds at least order() samples.

4. SUMMARY OF ALGORITHMSAlgorithm 1 is proposed by summarizing the results in

Section 3.

Algorithm 1 MFTI of noise-free data

1: Set [1, min(, )] for = 1 . . . , construct orthonormalmatrix-format interpolation direction , ;

2: Construct MFTI data following (6) & (7);3: Construct , from (11) & (12) or solve , from (13);4: Calculate projected ,,, following Lemma 3.2;5: Select an 0 {}

{}, perform SVD on (0 );

6: Obtain state-space representation ( , , , ) of the recoveredsystem following Lemma 3.4.

Real-world data are often noisy, thus the singular valuesin the above algorithm may be disturbed. To achieve a bet-ter accuracy, more samples need to be taken as interpola-tion data in order to minimize random error. But as thecomplexity of the algorithm increases quickly with the num-

ber of samples and orders of

(

) and we cannot estimatehow many samplings are required, algorithm 2 (with Matlab-style matrix notations) is proposed to reduce the complex-ity. When interpolating ill-conditioned samples, can beset as different numbers to give appropriate weightings tothese samples. In each loop, 0 new columns and rows oftangential data are taken into consideration. In Step 4, weonly need to update , , and instead of calculatingthem all from the beginning, which avoids repetitive compu-tation. can be manually set to trade off between speedand accuracy.

Algorithm 2 MFTI of noisy data

1: Set [1, min(, )] for = 1 . . . , construct matrix-formatinterpolation data , , , , , as Algorithm 1;

2: Set II = , iI = {1, 2, . . . , }, = [ 1 : 0 : , 2 : 0 : , . . . , 0 1 : 0 : ];

3: Do {Set ii to be equal to the set of the No. (1), (2), . . .,(0) elements of iI; II = II

ii, iI = iI ii;

4: Use (:, II), (II, :), (:, II), (II, :), (II, II), (II, II) to up-date , , and ;

5: Construct , , , following Algorithm 1. Calculate ();6: Calculate = () + () for iI;

[,] = ();7: If iI = , break;

} while (() > )

5. NUMERICAL EXAMPLES[Example 1] Example 1 is used to illustrate the advan-tages of using MFTI over VFTI in the under-sampled case.The 8 scattering matrices are sampled from an order-150

system with 30 ports. As illustrated in Fig. 1, no obvioussingular value drop can be detected by VFTI, but a sharpdrop can be found in MFTI. Fig. 2 shows the Bode diagram(input 1-output 1) of the original system and the recoveredsystem via both VFTI and MFTI. The MFTI-model fitswell with the original system while the VFTI-model doesnot, which demonstrates that the samples are inadequatefor VFTI while adequate for MFTI. Further experimentsshow that VFTI (using 180 matrix samples) requires about30 times the samples of MFTI (6 matrix samples) to recover

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Table 1: Interpolation of noisy dataAlgorithms Test 1 Test 2

reduced order time(s) relative error reduced order time(s) relative error

VF(10 iterations)n=140 140 3.1734 3.72e-1 140 2.8014 4.89e-1n=280 278 5.2518 7.33e-2 265 3.8492 9.11e-2

VFTI 95 0.4307 1.32e-1 98 0.2563 4.16e-1

MFTI-1 = 2 (weight 1) 190 0.8254 9.60e-3 195 0.7178 3.14e-2 = 3 (weight 2) 252 1.4369 1.70e-3 260 1.3026 4.20e-3

MFTI-2(recursive) 130 0.6002 9.91e-3 147 0.4129 2.51e-2

2 4 6 810

6

104

102

100

102

numbers of singular values

singularvalue

0 50 100 150 200

1030

1020

1010

100

1010

numbers of singular values

singularvalue

L

L

xLL

L

L

xLL

VFTI MFTI

150 180

Figure 1: Singular value pattern of VFTI and MFTI

101

102

103

104

105

102

101

100

101

frequency (Hz)

magnitude

original data

MFTI model

VFTI model

Figure 2: Bode diagrams of original and recoveredsystems using VFTI and MFTI.

the system. Besides, it is noticed that the singular values of

,

and

drop at 150, 180 and 180, respectively,which confirms theorem 3.5.[Example 2] The data in this example are measured froma 14-port power distribution network for INC board [10],with the order of the underlying system unknown. Table 1shows the results. The data in Test 1 come from 100 uni-formly distributed frequency samples, while those in Test2 are 100 poorly distributed samples concentrated in thehigh-frequency band. VF (vector fitting), VFTI, MFTI-1(Algorithm 1) and MFTI-2 (Algorithm 2) are all tested.

The error vector is computed as =(2)()2

()2for = 1, 2 . . . , , whereas is defined as 2/

.

The CPU times are also recorded to compare the speeds ofthese algorithms. For Test 1, = 2 and = 3 demon-strate the dimension of the interpolation matrices. For Test

2, = 2 and = 3 represent two weighting choices (weight1 and weight 2). They both satisfy the condition that for < but more data are utilized for the latter. From Ta-ble 1, we conclude that MFTI is more accurate than VFTI,especially for ill-conditioned samples. Accuracy improvesas increases, which provides option to trade off betweenspeed and accuracy. By employing recursive MFTI-2, theCPU time is only slightly greater than that of VFTI whilethe accuracy is significantly improved. Note that MFTI-2automatically selects appropriate samples and thereby can

yield good accuracy with moderate model size. Besides, it isshown that MFTI is much faster than the popular VF whileat the same time achieves better accuracy.

6. CONCLUSIONMatrix-format tangential interpolation (MFTI), which gen-

eralizes vector-format tangential interpolation (VFTI), hasbeen proposed and validated in this paper. The most sig-nificant advantage of MFTI over VFTI is that MFTI uti-lizes more information contained in the sampled matrices,and thus requires fewer samples to recover the system andyields better accuracy when interpolating under-sampled,

noisy and/or ill-conditioned data. Besides, a minimal sam-pling theorem has been proposed. Numerical examples haveconfirmed the superiority of MFTI over VFTI in practicalmacromodeling.

Acknowledgment

This work was supported in part by the Research GrantsCouncil of Hong Kong under Project HKU 718509E, and inpart by the University Research Committee of The Univer-sity of Hong Kong.

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