Grivet-Talocia, S.

English

PORTO Publications Open Repository TOrino

213
2006 IEEE Electrical Performance of Electronic Packaging
Delay-Based Macromodels for Long Interconnects
via Time-Frequency Decompositions
S. Grivet-Talocia
Dip. Elettronica, Politecnico di Torino, C. Duca degli Abruzzi 24, 10129 Torino, Italy
Ph. +39 011 5644104, Fax +39 011 5644099 (e-mail grivet@polito. it)
Abstract: We introduce a new macromodeling scheme for electrically long interconnects characterized by tabulated
frequency responses. The transfer function of the interconnect is modeled as a superposition of multiple single-delay
atoms, which are identified via a selective inversion of a Gabor transform of the raw frequency data. Each atom is then
approximated by a delayed rational function, leading to a highly-efficient SPICE-ready macromodel.
1 Introduction and motivations
Macromodeling techniques are now a standard tool for fast system-level simulation of interconnects. The inter-
connect structure of interest is first characterized via full-wave simulation or direct measurement in time or frequency
domain, leading to the matrix transfer function of the structure. The latter is then processed by a suitable fitting algo-
rithm in order to provide a closed-form approximation that can be synthesized into SPICE equivalent for system-level
time-domain analysis.
Several macromodeling techniques are available according to different classes of structures. Interconnects that are
electrically small at the highest frequency of interest can be approximated by lumped blocks characterized by rational
transfer functions. Vector Fitting (VF) [1] is the standard tool for these structures. Long transmission lines can also be
treated in this way since the propagation delay is known explicitly and can be extracted [2]. A complex interconnect
link made of a chain of various blocks (Fig. 1) can be dealt with by first segmenting the structure into its basic parts,
and then by macromodeling each part independently. Unfortunately, this procedure may lead to high-complexity global
network which may take long simulation time for the computation of an eye diagram of the interconnect.
In this work, we explore another possibility based on a global low-complexity macromodel based on delayed ra-
tional approximations. We include multiple delay terms in the macromodel, which results then consistent with the
electromagnetic behavior of the physical structure. Also, we suggest a procedure for the estimation of the macromodel
parameters starting from external port characterization, including direct measurements.
We consider an arbitrary interconnect, as depicted in Fig. 1, made of a chain of cascaded blocks. Each of these
blocks can be a transmission-line structure, a lumped block, or an electrically-long 3D interconnect (e.g., a connector,
which cannot be described as a standard tramsmission line). The topology of the network can be even more complex
than in Fig. 1, including ramifications, stubs, etc. The basic formulation applies to these more complex cases as well.
Let the interconnect structure be described by its scattering matrix S(s) defined at its external accessible ports.
The internal ports are hidden in this description, but any discontinuity which is induced by the junctions of the various
blocks has an effect. In time-domain, and with a suitable pulse excitation, this effect is visible as a reflected/transmitted
distorted and attenuated echo of the excitation pulse, appearing at the external ports after a well-defined propagation
delay. These considerations motivate the functional form which we assume for a global macromodel of the structure.
Denoting as S(s) the scattering matrix of the model, we assume
(k)(,)-sTk Q(k) i ) with Sij(jw) Sijjw) (1)
over a prescribed frequency band, and where N (k) (9 (s) are polynomials of a suitable (small) order. It can be
shown that the above formulation is exact when only ideal transmission lines and lumped blocks are present. If lossy
lines and more complex structures are included, some approximation is obviously involved due to a finite number of
echoes and to the rational approximation of the coefficients Q() (s).
Similar representations have already been suggested in the macromodeling literature. The approach in [3] extracts
only one dominant delay term and disregards higher order echoes. However, these can be quite important in the overall
approximation, as we will demonstrate on practical examples. Also, this technique may fail when more the one delay
is present in the raw data. Another approach (see [4] and references therein) and denoted as HPPM (Hybrid Phase-
Pole Model) is a particular case of (1). In this work, only simple structures are considered, the delays are assumed
1-4244-0668-4/06/$20.00 ©2006 IEEE 199
214
Lumped T-line Connector T-line Connector T-line Lumped
Figure 1: Interconnect structures under investigation, including an arbitrary number of transmission-line segments,
lumped discontinuities, and more complex 3D interconnect blocks such as connectors.
to be analytically known, and the estimation of poles/residues of the approximation requires some prior time-domain
simulation of the structure. Here, we rely only on tabulated frequency-domain samples at the external ports, and we
provide a sound numerical procedure for the estimation of all macromodel parameters.
We will denote the elements Q(k) (s)e-Tk in (1) as single-delay atoms. Our proposed method first provides a
decomposition of Sij (s) into single-delay atoms based on an adaptive inversion of the so-called Gabor transform of the
raw data. This is the subject of Sec. 2. Then, each atom is processed by extracting the (unique) delay and computing
the rational approximation via standard techniques. Details are provided in Sec. 3. The numerical results of Sec. 4
illustrate the feasibility of the proposed approach for real interconnect structures. No details on SPICE implementation
will be provided here since it requires standard methods.
2 Time-Frequency decomposition
The mathematical formulation is performed here on a generic scalar transfer function H(w). We define the Gabor
transform [6] ofH as
Q(w,T) = J HQ()WWTQ((9d&: Ww,T(() = W($ w)e Jdr W(4) = ~7-4e 42W2 (2)
-00
The auxiliary function W((), namely a Gaussian with a normalization such that W 2 1, serves as a template
to define the set of "basis" functions WW,T (i). These are defined in (2) as an amplitude-modulated (parameter T
is proportional to the number of oscillations) and translated (parameter w is the center of the translation) version of
W(Q). If W(() were taken to be identically one, the definition in (2) would become (up to a normalization constant)
exactly the inverse Fourier transform of H(Q), which is the system impulse response h(T). Hence, the variable T has
the physical meaning of time or time-delay. The Gaussian window W(y) in (2) plays the role of a sharp bandpass
filter. Therefore, T(w,T) can be regarded as the inverse Fourier transform of H(Q), but retaining only those frequency
components located in a frequency band centered around w. For this reson, g(w, T) belongs to the class of the so-called
time-frequency transforms, since it provides a localization of the various components of H both in frequency w and
time T.
The transform Q(w, T) is highly redundant, since it is obtained by mapping a function of one variable H(s) into a
function of two variables. Nonetheless, an inversion formula is available
H(s) + J J Q(w, T)Ww,T()dWdT (3)
allowing to recover the original function from its coefficients. The theory of Gabor and more general time-frequency
decompositions is well-known and many mathematical properties have been proved (see [6] and references therein).
The main result that we will exploit in this work is that Q(w, T) 2, sometimes called spectrogram, represents the time-
frequency decomposition of the energy of H(S). Also, we will use the fact that the reconstruction formula (3) can be
combined with a time-frequency localization process in order to split H(s) into separate components. More precisely,
K H() I ffQw )wT~dd, K k 2(4
H() Z kH ( k),2()= II(Qk T)Wg,T(()dodT U Qk = X (4)
ki kl~~~~2
In other words, the entire time-frequency plane (w, T) can separated into disjoint subsets Qk. Integration over each of
those regions leads to a partial reconstruction Hk (i). The superposition of all partial components leads exactly to the
original function H(Q,.
200
215
IG(coj)12, constantcoefficients I G(c,o)12, rational coefficients rational coefficients
80 80 80
r 60 60 60&
E EE
E5 40 40 -40 Po
H 20 _ F 20 .._ 20
0 0 0 L
-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 0 0.05 0.1 0.15
Frequency (normalized) Frequency (normalized) E(c)1/2
Figure 2: Time-frequency energy distribution Q(w, T)2 for the synthetic examples of Sec.2. Constant coefficients
(left) and rational coefficients (middle). Right panel is the RMS w-average of the coefficients in middle panel.
The reason for the introduction of the Gabor transform for present application is illustrated by two simple examples.
First, we consider a synthetic response of the form
K
S(s) S Ake-sTk, with K = 4, Ak= (0.5)k-1, Tk= 10 + 20k. (5)
k=1
over a normalized unitary frequency band. This example represents the transmission response of a mismatched ideal
transmission line with resistive terminations. The (normalized) time-frequency energy distribution 9 (w, T) 2is de-
picted in the left panel of Fig. 2. The delays are well resolved by the transform, which also shows that the frequency
behavior of the single-delay atoms is constant, as expected. Conversely, the amplitude of the single-delay atoms is
decreasing for increasing delay, coherently with (5).
The second example is similar, but we replace the constant coefficients in (5) with rational frequency-dependent
coefficients Ak (s) constructed as fourth order rational functions with randomly-selected poles and residues. The cor-
responding energy distribution Q9 (w, T) 2 is depicted in the middle panel of Fig. 2. It should be noted that although the
coefficients are now frequency-dependent, the localization of the time delays is still excellent. This fact will enable the
separation of the single-delay atoms, as described in Sec. 3.
A final remark on discretization issues. The transform in (2)-(3) is continuous and defined on an infinite domain.
Therefore, some discretization and truncation must be performed in order to apply this tool to finite-bandwidth sampled
responses. The natural choice is to constrain w = moo and T= mTu. This discretization leads to technical complica-
tions, since a pair of dual window functions {W(), W(() } must be used for the transform and its inverse, respectively.
The actual shape of the dual window depends on the resolution of the time-frequency discretization lattice. The theory
offrames provides all mathematical details [6]. Of course, all the results in this work have been produced with the
discrete Gabor transform, although the theoretical formulation adopts the continuous transform for simplicity.
3 Separation and approximation of single-delay atoms
We now describe how the single-delay atoms can be separated based on the Gabor coefficients. We average the
time-frequency energy distribution via
8(T)=(w,T) dw. (6)
This quantity presents well-pronounced maxima corresponding to the delays Tk, as clearly seen from the right panel
of Fig. 2. The minima between each pair of maxima, which we denote as Tk, can be used to separate the single-delay
atoms. Therefore, we define
Qk {(w,T) : T C (Tk,Tk+1), Vw} (7)
as horizontal strips in the (w, T) plane. The single-delay atoms Hk (() are then obtained by applying Eq. (4).
Each local maximum Tk of S(T) provides a good estimate for the delay associated to the corresponding single-delay
atom. However, we refine this estimate by applying a procedure similar to [5]. Assuming that the true delay is Tk, a
tentative delay-free atom is defined as Hk( )e±J+Tk. Then, VF is applied to compute its rational approximation, and
the VF approximation error k (Tk) is computed. The optimum delay Tk is determined by minimizing the error k (Tk)
over a suitable (small) interval centered at Tk.
201
216
Magnitude 1 Angle 0 S1 IG(w ,)121 lll4 ll 4X 10
0.8 Data 2 33
0.6 Model 2
0
0.4 - Dataw 1
0.2 -2 Model
_
0
-4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
0.5 1 1.5 2 0 0.5 1.5 2 -2 -1 0
Frequency, MHz x 104 Frequency, MHz 104 Frequency, MHz x 104
S1 3' Magnitude S1 3' Angle 103 S1 3' G(o),1
\l\\\4 4
Dat
0.8 Data 2 Moe 3
0.6 ~~~~~~~~Model l~ lm
0
-C0.4 i
0.2 2 0-
0 -410 0.5 1 1.5 2 0 0.5 1.5 -2 -1 0
Frequency,MHz x 104 Frequency, MHz 104 Frequency, MHz x 104
Figure 3: Model validation for Sl,1 (top row) and Sl,3 (bottom row). Right panels depict the Gabor transform energies.
The above procedure results in a delayed rational approximation of each single-delay atom, as formulated in Eq. (1).
The algorithm is fully automatic and is applicable to "black-box" frequency responses, such as direct measurements.
The only approximation error which is involved in this global macromodeling scheme is due to the rational fit, since
the inversion of the Gabor transform is exact up to machine precision. In summary, the Gabor transform is used as
a tool for the localization of the single-delay atoms, which are then approximated via standard techinques. This last
operation is possible because only one dominant delay is attributed to each atom.
4 Numerical results
We apply the proposed macromodeling technique to a set of measured scattering responses of a PCB interconnect
link (courtesy of C. Schuster, IBM). We present results for the return loss and for the insertion loss. Since we are
dealing with measured data, we apply a thresholding procedure to the Gabor transform and retain only the coefficients
such that Q(w, T)12 > c, in order to perform data denoising. This induces a reconstruction RMS relative errord in the
inversion of the Gabor transform, which amounts in present case tod 0.009 for S1,1 and tod 0.007 for S1,3. The
right panels in Fig. 3 depict the thresholded Gabor coefficients, showing the presence of three significant distinct delay
terms for each of the two responses. The automatic identification of these delays led toTk {0, 1.31, 2.86} ns for
Sl, and toTk {0.67, 2.01, 3.38} ns for S1,3. A VF-based rational fit resulted in{6, 8, 4} poles and{8, 10, 4}
poles for S1,1 and S1,3 single-delay atoms, respectively. A comparison between model and raw data is shown in
Fig. 3. The accuracy is excellent. Only a small deviation in the insertion loss magnitude is visible, due to the fact
that we constrained the model response to be vanishing for large frequency. Based on these preliminary results, the
proposed technique seems to be very promising for the generation of low-complexity models for long interconnect
links characterized by significant propagation delays.
References
[1] B. Gustavsen, A. Semlyen, "Rational approximation of frequency responses by vector fitting", IEEE Trans. Power
Delivery, Vol. 14, July 1999, pp. 1052-1061.
[2]S. Grivet-Talocia, H. M. Huang, A.E. Ruehli, F. Canavero,I. M. Elfadel, "Transient Analysis of Lossy Transmis-
sion Lines: an Effective Approach Based on the Method of Characteristics", IEEE T-ADVP, Vol. 27, N. 1, 2004,
pp.45-56.
[3] R. Mandrekar, M. Swaminathan, "Causality enforcement in transient simulation of passive networks through
delay extraction", 9th IEEE Workshop on Signal Propagation on Interconnects, 2005, pp. 25-28.
[4] Bing Zhong, Tao Hu, D. Fu, S.L. Dvorak, J.L. Prince, "A study of a hybrid phase-pole macromodel for transient
simulation of complex interconnects structures", IEEE T-CAD, Vol. 24, N. 8, Aug. 2005, pp. 1250-1261.
[5] B. Gustavsen, "Time delay identification for transmission line modeling", 8th IEEE Workshop on Signal Propa-
gation on Interconnects, 2004, pp. 103-106.
[6]I. Daubechies, Ten Lectures on Wavelets, Philadelpia: SIAM, 1992.
202


A study of a hybrid phase-pole macromodel for transient simulation of complex interconnects structures&quot;,

Causality enforcement in transient simulation of passive networks through delay extraction&quot;,

Rational approximation of frequency responses by vector fitting&quot;,

Ten Lectures on Wavelets,

Time delay identification for transmission line modeling&quot;,

Transient Analysis of Lossy Transmission Lines: an Effective Approach Based on the Method of Characteristics&quot;,

Delay-Based Macromodels for Long Interconnects via Time-Frequency Decomposition

GRIVET TALOCIA, STEFANO

PORTO@iris (Publications Open Repository TOrino - Politecnico di Torino)

05 August 2020POLITECNICO DI TORINORepository ISTITUZIONALEDelay-Based Macromodels for Long Interconnects via Time-Frequency Decomposition / GRIVET TALOCIA, Stefano. -STAMPA. - (2006), pp. 199-202. ((Intervento presentato al convegno IEEE 15th Topical Meeting on ElectricalPerformance of Electronic Packaging (EPEP 2006) tenutosi a Scottsdale, AZ (USA) nel 23-25 Oct. 2006.OriginalDelay-Based Macromodels for Long Interconnects via Time-Frequency DecompositionPublisher:PublishedDOI:10.1109/EPEP.2006.321228Terms of use:openAccessPublisher copyright(Article begins on next page)This article is made available under terms and conditions as specified in the  corresponding bibliographic description inthe repositoryAvailability:This version is available at: 11583/1709962 since: 2018-02-16T17:11:58ZIEEE2132006 IEEE Electrical Performance of Electronic PackagingDelay-Based Macromodels for Long Interconnectsvia Time-Frequency DecompositionsS. Grivet-TalociaDip. Elettronica, Politecnico di Torino, C. Duca degli Abruzzi 24, 10129 Torino, ItalyPh. +39 011 5644104, Fax +39 011 5644099 (e-mail grivet@polito. it)Abstract: We introduce a new macromodeling scheme for electrically long interconnects characterized by tabulatedfrequency responses. The transfer function of the interconnect is modeled as a superposition of multiple single-delayatoms, which are identified via a selective inversion of a Gabor transform of the raw frequency data. Each atom is thenapproximated by a delayed rational function, leading to a highly-efficient SPICE-ready macromodel.1 Introduction and motivationsMacromodeling techniques are now a standard tool for fast system-level simulation of interconnects. The inter-connect structure of interest is first characterized via full-wave simulation or direct measurement in time or frequencydomain, leading to the matrix transfer function of the structure. The latter is then processed by a suitable fitting algo-rithm in order to provide a closed-form approximation that can be synthesized into SPICE equivalent for system-leveltime-domain analysis.Several macromodeling techniques are available according to different classes of structures. Interconnects that areelectrically small at the highest frequency of interest can be approximated by lumped blocks characterized by rationaltransfer functions. Vector Fitting (VF) [1] is the standard tool for these structures. Long transmission lines can also betreated in this way since the propagation delay is known explicitly and can be extracted [2]. A complex interconnectlink made of a chain of various blocks (Fig. 1) can be dealt with by first segmenting the structure into its basic parts,and then by macromodeling each part independently. Unfortunately, this procedure may lead to high-complexity globalnetwork which may take long simulation time for the computation of an eye diagram of the interconnect.In this work, we explore another possibility based on a global low-complexity macromodel based on delayed ra-tional approximations. We include multiple delay terms in the macromodel, which results then consistent with theelectromagnetic behavior of the physical structure. Also, we suggest a procedure for the estimation of the macromodelparameters starting from external port characterization, including direct measurements.We consider an arbitrary interconnect, as depicted in Fig. 1, made of a chain of cascaded blocks. Each of theseblocks can be a transmission-line structure, a lumped block, or an electrically-long 3D interconnect (e.g., a connector,which cannot be described as a standard tramsmission line). The topology of the network can be even more complexthan in Fig. 1, including ramifications, stubs, etc. The basic formulation applies to these more complex cases as well.Let the interconnect structure be described by its scattering matrix S(s) defined at its external accessible ports.The internal ports are hidden in this description, but any discontinuity which is induced by the junctions of the variousblocks has an effect. In time-domain, and with a suitable pulse excitation, this effect is visible as a reflected/transmitteddistorted and attenuated echo of the excitation pulse, appearing at the external ports after a well-defined propagationdelay. These considerations motivate the functional form which we assume for a global macromodel of the structure.Denoting as S(s) the scattering matrix of the model, we assume(k)(,)-sTk Q(k) i ) with Sij(jw) Sijjw) (1)over a prescribed frequency band, and where N (k) (9 (s) are polynomials of a suitable (small) order. It can beshown that the above formulation is exact when only ideal transmission lines and lumped blocks are present. If lossylines and more complex structures are included, some approximation is obviously involved due to a finite number ofechoes and to the rational approximation of the coefficients Q() (s).Similar representations have already been suggested in the macromodeling literature. The approach in [3] extractsonly one dominant delay term and disregards higher order echoes. However, these can be quite important in the overallapproximation, as we will demonstrate on practical examples. Also, this technique may fail when more the one delayis present in the raw data. Another approach (see [4] and references therein) and denoted as HPPM (Hybrid Phase-Pole Model) is a particular case of (1). In this work, only simple structures are considered, the delays are assumed1-4244-0668-4/06/$20.00 ©2006 IEEE 199214Lumped T-line Connector T-line Connector T-line LumpedFigure 1: Interconnect structures under investigation, including an arbitrary number of transmission-line segments,lumped discontinuities, and more complex 3D interconnect blocks such as connectors.to be analytically known, and the estimation of poles/residues of the approximation requires some prior time-domainsimulation of the structure. Here, we rely only on tabulated frequency-domain samples at the external ports, and weprovide a sound numerical procedure for the estimation of all macromodel parameters.We will denote the elements Q(k) (s)e-Tk in (1) as single-delay atoms. Our proposed method first provides adecomposition of Sij (s) into single-delay atoms based on an adaptive inversion of the so-called Gabor transform of theraw data. This is the subject of Sec. 2. Then, each atom is processed by extracting the (unique) delay and computingthe rational approximation via standard techniques. Details are provided in Sec. 3. The numerical results of Sec. 4illustrate the feasibility of the proposed approach for real interconnect structures. No details on SPICE implementationwill be provided here since it requires standard methods.2 Time-Frequency decompositionThe mathematical formulation is performed here on a generic scalar transfer function H(w). We define the Gabortransform [6] ofH asQ(w,T) = J HQ()WWTQ((9d&: Ww,T(() = W($ w)e Jdr W(4) = ~7-4e 42W2 (2)-00The auxiliary function W((), namely a Gaussian with a normalization such that W 2 1, serves as a templateto define the set of "basis" functions WW,T (i). These are defined in (2) as an amplitude-modulated (parameter Tis proportional to the number of oscillations) and translated (parameter w is the center of the translation) version ofW(Q). If W(() were taken to be identically one, the definition in (2) would become (up to a normalization constant)exactly the inverse Fourier transform of H(Q), which is the system impulse response h(T). Hence, the variable T hasthe physical meaning of time or time-delay. The Gaussian window W(y) in (2) plays the role of a sharp bandpassfilter. Therefore, T(w,T) can be regarded as the inverse Fourier transform of H(Q), but retaining only those frequencycomponents located in a frequency band centered around w. For this reson, g(w, T) belongs to the class of the so-calledtime-frequency transforms, since it provides a localization of the various components of H both in frequency w andtime T.The transform Q(w, T) is highly redundant, since it is obtained by mapping a function of one variable H(s) into afunction of two variables. Nonetheless, an inversion formula is availableH(s) + J J Q(w, T)Ww,T()dWdT (3)allowing to recover the original function from its coefficients. The theory of Gabor and more general time-frequencydecompositions is well-known and many mathematical properties have been proved (see [6] and references therein).The main result that we will exploit in this work is that Q(w, T) 2, sometimes called spectrogram, represents the time-frequency decomposition of the energy of H(S). Also, we will use the fact that the reconstruction formula (3) can becombined with a time-frequency localization process in order to split H(s) into separate components. More precisely,K H() I ffQw )wT~dd, K k 2(4H() Z kH ( k),2()= II(Qk T)Wg,T(()dodT U Qk = X (4)ki kl~~~~2In other words, the entire time-frequency plane (w, T) can separated into disjoint subsets Qk. Integration over each ofthose regions leads to a partial reconstruction Hk (i). The superposition of all partial components leads exactly to theoriginal function H(Q,.200215IG(coj)12, constantcoefficients I G(c,o)12, rational coefficients rational coefficients80 80 80r 60 60 60&E EEE5 40 40 -40 PoH 20 _ F 20 .._ 200 0 0 L-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 0 0.05 0.1 0.15Frequency (normalized) Frequency (normalized) E(c)1/2Figure 2: Time-frequency energy distribution Q(w, T)2 for the synthetic examples of Sec.2. Constant coefficients(left) and rational coefficients (middle). Right panel is the RMS w-average of the coefficients in middle panel.The reason for the introduction of the Gabor transform for present application is illustrated by two simple examples.First, we consider a synthetic response of the formKS(s) S Ake-sTk, with K = 4, Ak= (0.5)k-1, Tk= 10 + 20k. (5)k=1over a normalized unitary frequency band. This example represents the transmission response of a mismatched idealtransmission line with resistive terminations. The (normalized) time-frequency energy distribution 9 (w, T) 2is de-picted in the left panel of Fig. 2. The delays are well resolved by the transform, which also shows that the frequencybehavior of the single-delay atoms is constant, as expected. Conversely, the amplitude of the single-delay atoms isdecreasing for increasing delay, coherently with (5).The second example is similar, but we replace the constant coefficients in (5) with rational frequency-dependentcoefficients Ak (s) constructed as fourth order rational functions with randomly-selected poles and residues. The cor-responding energy distribution Q9 (w, T) 2 is depicted in the middle panel of Fig. 2. It should be noted that although thecoefficients are now frequency-dependent, the localization of the time delays is still excellent. This fact will enable theseparation of the single-delay atoms, as described in Sec. 3.A final remark on discretization issues. The transform in (2)-(3) is continuous and defined on an infinite domain.Therefore, some discretization and truncation must be performed in order to apply this tool to finite-bandwidth sampledresponses. The natural choice is to constrain w = moo and T= mTu. This discretization leads to technical complica-tions, since a pair of dual window functions {W(), W(() } must be used for the transform and its inverse, respectively.The actual shape of the dual window depends on the resolution of the time-frequency discretization lattice. The theoryofframes provides all mathematical details [6]. Of course, all the results in this work have been produced with thediscrete Gabor transform, although the theoretical formulation adopts the continuous transform for simplicity.3 Separation and approximation of single-delay atomsWe now describe how the single-delay atoms can be separated based on the Gabor coefficients. We average thetime-frequency energy distribution via8(T)=(w,T) dw. (6)This quantity presents well-pronounced maxima corresponding to the delays Tk, as clearly seen from the right panelof Fig. 2. The minima between each pair of maxima, which we denote as Tk, can be used to separate the single-delayatoms. Therefore, we defineQk {(w,T) : T C (Tk,Tk+1), Vw} (7)as horizontal strips in the (w, T) plane. The single-delay atoms Hk (() are then obtained by applying Eq. (4).Each local maximum Tk of S(T) provides a good estimate for the delay associated to the corresponding single-delayatom. However, we refine this estimate by applying a procedure similar to [5]. Assuming that the true delay is Tk, atentative delay-free atom is defined as Hk( )e±J+Tk. Then, VF is applied to compute its rational approximation, andthe VF approximation error k (Tk) is computed. The optimum delay Tk is determined by minimizing the error k (Tk)over a suitable (small) interval centered at Tk.201216Magnitude 1 Angle 0 S1 IG(w ,)121 lll4 ll 4X 100.8 Data 2 330.6 Model 200.4 - Dataw 10.2 -2 Model_0-4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~0.5 1 1.5 2 0 0.5 1.5 2 -2 -1 0Frequency, MHz x 104 Frequency, MHz 104 Frequency, MHz x 104S1 3' Magnitude S1 3' Angle 103 S1 3' G(o),1\l\\\4 4Dat0.8 Data 2 Moe 30.6 ~~~~~~~~Model l~ lm0-C0.4 i0.2 2 0-0 -410 0.5 1 1.5 2 0 0.5 1.5 -2 -1 0Frequency,MHz x 104 Frequency, MHz 104 Frequency, MHz x 104Figure 3: Model validation for Sl,1 (top row) and Sl,3 (bottom row). Right panels depict the Gabor transform energies.The above procedure results in a delayed rational approximation of each single-delay atom, as formulated in Eq. (1).The algorithm is fully automatic and is applicable to "black-box" frequency responses, such as direct measurements.The only approximation error which is involved in this global macromodeling scheme is due to the rational fit, sincethe inversion of the Gabor transform is exact up to machine precision. In summary, the Gabor transform is used asa tool for the localization of the single-delay atoms, which are then approximated via standard techinques. This lastoperation is possible because only one dominant delay is attributed to each atom.4 Numerical resultsWe apply the proposed macromodeling technique to a set of measured scattering responses of a PCB interconnectlink (courtesy of C. Schuster, IBM). We present results for the return loss and for the insertion loss. Since we aredealing with measured data, we apply a thresholding procedure to the Gabor transform and retain only the coefficientssuch that Q(w, T)12 > c, in order to perform data denoising. This induces a reconstruction RMS relative errord in theinversion of the Gabor transform, which amounts in present case tod 0.009 for S1,1 and tod 0.007 for S1,3. Theright panels in Fig. 3 depict the thresholded Gabor coefficients, showing the presence of three significant distinct delayterms for each of the two responses. The automatic identification of these delays led toTk {0, 1.31, 2.86} ns forSl, and toTk {0.67, 2.01, 3.38} ns for S1,3. A VF-based rational fit resulted in{6, 8, 4} poles and{8, 10, 4}poles for S1,1 and S1,3 single-delay atoms, respectively. A comparison between model and raw data is shown inFig. 3. The accuracy is excellent. Only a small deviation in the insertion loss magnitude is visible, due to the factthat we constrained the model response to be vanishing for large frequency. Based on these preliminary results, theproposed technique seems to be very promising for the generation of low-complexity models for long interconnectlinks characterized by significant propagation delays.References[1] B. Gustavsen, A. Semlyen, "Rational approximation of frequency responses by vector fitting", IEEE Trans. PowerDelivery, Vol. 14, July 1999, pp. 1052-1061.[2]S. Grivet-Talocia, H. M. Huang, A.E. Ruehli, F. Canavero,I. M. Elfadel, "Transient Analysis of Lossy Transmis-sion Lines: an Effective Approach Based on the Method of Characteristics", IEEE T-ADVP, Vol. 27, N. 1, 2004,pp.45-56.[3] R. Mandrekar, M. Swaminathan, "Causality enforcement in transient simulation of passive networks throughdelay extraction", 9th IEEE Workshop on Signal Propagation on Interconnects, 2005, pp. 25-28.[4] Bing Zhong, Tao Hu, D. Fu, S.L. Dvorak, J.L. Prince, "A study of a hybrid phase-pole macromodel for transientsimulation of complex interconnects structures", IEEE T-CAD, Vol. 24, N. 8, Aug. 2005, pp. 1250-1261.[5] B. Gustavsen, "Time delay identification for transmission line modeling", 8th IEEE Workshop on Signal Propa-gation on Interconnects, 2004, pp. 103-106.[6]I. Daubechies, Ten Lectures on Wavelets, Philadelpia: SIAM, 1992.202

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Delay-Based Macromodels for Long Interconnects via Time-Frequency Decomposition

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