Long-term rain attenuation probability and site diversity gain prediction formulas

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 7, JULY 2005 2307

Communications______________________________________________________________________

Long-Term Rain Attenuation Probability and Site DiversityGain Prediction Formulas

Athanasios D. Panagopoulos, Pantelis-Daniel M. Arapoglou,John D. Kanellopoulos, and Panayotis G. Cottis

Abstract—Simple models for long-term induced rain attenuation on aslant path and site diversity gain are presented in this work. As verifiedby numerous tests against the ITU-R databank and other data from the lit-erature, the proposed models exhibit a very good performance. The novelslant path rain attenuation prediction model compared to the ITU-R oneexhibits a similar behavior at low time percentages and a better root-mean-square error performance for probability levels above 0.02%. Moreover,comparing the proposed site diversity gain model with other widely ac-cepted models from the literature, an improved performance is observedfor distances less than 15 km, while the model performs equally well forgreater distances. Furthermore, a sensitivity test between the proposed andHodge’s formula with respect to the separation distance is also carriedout. While the lower limit of the proposed model is found to be =

1 7km, its extension covering large-scale site diversity is successfully com-pared with experimental data coming from Japan. The set of presentedmodels exhibits the advantage of easy implementationwith little complexityand is considered useful for educational and back of the envelope compu-tations.

Index Terms—Rain attenuation probability, satellite communications,site diversity.

I. INTRODUCTION

The Ka (20/30 GHz) and V (40/50 GHz) frequency bands arebecoming increasingly attractive for user oriented future commercialsatellite services, due to their large available bandwidths. However,they suffer more from rain fades in comparison to the almost congestedKu (12/14 GHz) band. Therefore, prediction models for annual rainattenuation, such as the ones developed by several research groupsover the past three decades [1] are required to provide guidance in thecourse of balancing availability requirements and cost [2].

To combat rain attenuation, several fade mitigation techniques havebeen developed such as diversity protection schemes, power controland adaptive processing techniques [3]. Among these techniques, themost efficient is site diversity (SD) [1]. SD takes advantage of the spa-tial characteristics of the rainfall medium by using two earth stationsto exploit the fact that the probability of attenuation due to rain oc-curring simultaneously on the alternative Earth-space paths is signifi-cantly less than the relevant probability occurring on either individualpath. Though the cost effectiveness of SD remains questionable, the in-terest on SD has been renewed (see for example the Iridium project [4]),due to the significant reduction of ground terminal antennas and otherhardware sizes. Nowadays, terminals can be installed in customers’premises and the use of public terrestrial networks to carry out sig-

Manuscript received July 16, 2004; revised December 28, 2004. This workwas co-founded by the European Social Fund (75%) and National Resources(25%)—Operational Program for Educational and Vocational Training II(EPEAEK II) and particularly the Program “PYTHAGORAS” 68/816.

The authors are with the Wireless and Satellite Communications Group, Divi-sion of Information Transmission Systems and Materials Technology, School ofElectrical and Computer Engineering; National Technical University of Athens,Zografou 15780, Greece.

Digital Object Identifier 10.1109/TAP.2005.850762

naling seems possible [1], [5]. Moreover, SD is considered for alterna-tive feeder links of a satellite network.

The models predicting rain attenuation and those concerning the per-formance of SD systems can be classified in two major categories: (a)regression models based on available attenuation statistics valid onlyfor a few specific locations [6] and (b) physical models based on theunderstanding of the rain process and the rainfall medium exhibiting agood performance globally [7]–[9].

The present paper proposes simple models that combine the best fea-tures of both categories for the estimation of slant path rain attenuationand site diversity gain. That is, the proposed models are easily imple-mented and can be used for educational and back of the envelope com-putations. They also have a satisfactory performance on a global basis,since they are based on a regression fitting analysis on numerical re-sults taken from physical models.

In Section II the Weibull distribution was selected to represent theannual rain attenuation [10] and a simple prediction model for singlesite attenuation was derived using an appropriate regression fittinganalysis. Recently released ITU-R recommendations concerninghighly reliable worldwide rain rate and rain height statistics [11], [12]provided some of the physical inputs for the fitting procedure. Fur-thermore, over 300 Earth-space experiments obtained from the ITU-RStudy Group 3 databank [13] and the data from the Advanced Com-munications Technology Satellite (ACTS) propagation experiment[14], [15] were used for a comparative test of the proposed methodagainst the current ITU-R procedure [16] with excellent results. Someserious problems of the ACTS databank have been reported [15],because of the effects of water on the receiving antenna terminal.For this reason, the corrected experimental long-term rain attenuationprobability density functions [15] have been used for the validation ofthe present model.

In Section III, a simple model estimating the site diversity gainbased on the convective raincells physical model [7] is presented.Since emphasis is placed on the verification of the model throughexperimental data, 76 site diversity experiments, that is almost theentire ITU-R Study Group 3 database, have been used to test the modelagainst the Hodge [6], EXCELL [8], and Matricciani [9] models. Theresults showed that the proposed model performs better for separationdistances less than 15 km and equally well beyond this limit. Moreover,extending the physical site diversity model in [7], using the logarithmiccorrelation coefficient proposed by Paraboni–Barbaliscia [17], [18], anew formula for large-scale site diversity systems is presented. Theformula is validated using recently published experimental data fromJapan [19]. Finally, in Section IV some useful conclusions are drawn.

II. SINGLE SITE ATTENUATION

It is widely acknowledged that the most accepted models for de-scribing point rainfall rate statistics are the lognormal and gamma dis-tributions. The lognormal model provides a good approximation in re-gions of low rainfall rates, while the gamma model is more accurate inregions of high rainfall rates. On the other hand, the Weibull distribu-tion provides a very good approximation for both high and low rain-fall/attenuation values. The analytical derivation of the attenuation ex-ceedance probability adopting the Weibull distribution and taking intoaccount Crane’s model for the vertical structure of rainfall yields [10]

Pr[A � AS ] = exp [�wA(AS cos �)m ] (1)

0018-926X/$20.00 © 2005 IEEE

2308 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 7, JULY 2005

Fig. 1. Mean � , standard deviation � , and RMS � of the error testing variable V . The number i of experiments included in this study of error for eachprobability level is given in the parenthesis.

where AS is the threshold exceeded by the rain attenuation A on theslant path and � is the slant path elevation angle. In addition, wA andmA are the statistical parameters of the Weibull distribution dependingon the corresponding parameters wr , mr of the rainfall rate distribu-tion as well as on other geometrical and electrical parameters of thesatellite link (see the Appendix). Since the calculation of these param-eters is generally not an easy task, a regression fitting analysis wasfollowed employing the Levenberg-Marquadt algorithm for nonlinearleast squares fitting [20]. The rain rate statistics and rain height inputsnecessary for the fitting procedure were provided by the novel ITU-Rprediction methods [11], [12].

The resulting simple rain attenuation model for a satellite slant pathis presented as the product of two functions

AS(p) = S(p; f; �; � ) � C(R0:01;�): (2)

In (2), the function S is the factor expressing the dependence of rainattenuation AS (decibels) exceeded for p% of an average year on thefrequency of operation f (gigahertz), the elevation angle of the slantpath � (degrees) and the polarization tilt angle � (degrees). In otherwords, S incorporates the system characteristics of the satellite link in-fluencing the annual rain attenuation distribution. On the other hand,

the function C is the factor expressing the dependence of rain atten-uation on the rainfall rate for 0.01% of an average year R0:01 (mil-limeters/hour) and on the absolute latitude value j�j (degrees) of thespecific region, which means that, incorporated in C , are the climaticcharacteristics influencing the radio wave propagation. The analyticalexpression of both functions is different for regions with absolute lati-tudes above and below 30� and is given by

S =

2:461pL [�1:072 + exp(0:032f)]

�[�0:01 + ��0:741][�0:21 + 141:9� ] j�j � 30�

�0:231pL [�1:091 + exp(0:036f)]

�[0:001 + ��0:971][0:021� 13:5� ] j�j < 30�

(3)

where

L=�0:86�0:045 ln p+0:052 ln f�0:163 sin � j�j�30�

�1:176�0:087 ln p+0:102 ln f�0:304 sin � j�j<30�(4)

and

C =0:019� 2:564 exp 0:047R0:01 +

j�j15 411

+2:583 exp(0:047R0:01) j�j � 30�

3:479 + 1:712 exp(0:028R0:01) j�j < 30�

(5)


Fig. 2. Geometrical configuration of a site diversity scheme.

It should be noted that R0:01 can be obtained either from the rainmaps given in [11], or preferably for greater accuracy, from long-termmeasurements acquired at the specific site.

To verify the agreement of the proposed method with experimentalresults, more than 350 experiments contained in the ITU-R databankof Study Group 3 [13] and in the ACTS campaign [15] were exam-ined. The restrictions imposed by the input parameters of the proposedmethod (j�j � 60�, � � 10�, f � 10 GHz, p � 5%) indicatedthat around 330 experiments could be taken into account to comparethe proposed method with the existing ITU-R methodology [16]. Thecomparison was based on the algorithm proposed in RecommendationP.311-11 [21] concerning rain attenuation predictions. The procedureis as follows:

1. Calculate the ratio Si of the predicted attenuation ASi to themeasured attenuation Ami for each radio link

Si =ASi

Ami

: (6)

2. Calculate the corresponding test variable

Vi =lnSi; for Ami � 10 dBA

10

0:2lnSi; for Ami < 10 dB:

(7)

3. Repeat the procedure for each time percentage.4. Calculate the mean �V , the standard deviation �V , and the

RMS value �V of the Vi values for each time percentage

�V = �2

V + �2

V

0:5: (8)

Results of the �V , �V , and �V values of the proposed and the ITU-Rmethods, taking into consideration the 0.001%–5% annual probabilityrange, are given in Fig. 1. The two models exhibit a similar error be-havior, although at low time percentages, the ITU-R method achieveslower mean errors. However, the superiority of the proposed method isnoticeable, firstly in terms of stability when comparing the two stan-dard deviation curves, and, secondly, in terms of the RMS error abovethe 0.02% probability level.

Concluding, the proposed model is: (a) easy to apply; (b) agrees sat-isfactorily with experimental results in various regions; (c) does not de-pend significantly on the technique used to obtain the rainfall rate data;and (d) yields a physical insight. Thus, it satisfies all the criteria posedby ITU-R to become widely applicable. It should be again emphasizedthat the proposed model is also useful for educational purposes and canbe used by the satellite communications system engineers for back ofthe envelope computations.

III. SITE DIVERSITY GAIN

A site diversity scheme employs a master and a remote station sev-eral kilometers apart in order to take advantage of the inhomogeneity ofintense rainfall, which occurs within localized areas (raincells) havinga diameter of a few tens of kilometers [22]. This inhomogeneity results

in a decorrelation of the rain attenuation on the alternative paths. Con-sequently, if the signal is received via different paths, a possible deepfade is unlikely to occur on both of them. The jointly received signalsare sent to the master station where they are further processed based oneither signal selection, switching, or combining [23]. A diversity con-trol unit coordinating the signal flow and a signal processing unit mustbe incorporated at the master and the other earth station, respectively.

The commonly used performance measure for SD is the site diversitygain (SDG) GSD defined as the difference between the single site AS

and the joint attenuation AD , both expressed in decibels, for the sameexceedance probability level p%

GSD(p) = AS(p)� AD(p): (9)

Since the calculation of the diversity gain based on the convective rain-cells model [7] requires the solution of a complicated transcendentalequation, a regression fitting procedure on the numerical results ob-tained from the latter model was followed. During this procedure, thedependence of the diversity gain on climatic conditions was investi-gated and was found negligible, since for the same value of AS thevalues of GSD are almost identical for regions with different rainfallcharacteristics. Also, the observed overestimation of the available di-versity statistics by the underlying model [7] was taken into account,resulting in a novel and more accurate expression of the simple form

GSD = GA �GD �G� �Gf �G�: (10)

In the above relationship, GA , GD , G� , Gf , G� are factors ex-pressing the dependence of the site diversity gain on the single siteattenuation AS (decibels), the site separation distance D (kilome-ters), the common elevation angle of both slant paths �(degrees),the frequency of operation f (gigahertz), and the orientation of thebaseline between the two earth stations � (degrees). The geometricalparameters of an SD configuration mentioned above are shown inFig. 2. As far as the attenuation threshold AS(p%) of an averageyear is concerned, one can resort to either the ITU-R model or to theone discussed in Section II. Each dependence factor is given by thefollowing:

GA =8:19A0:0004S + 0:1809AS � 8:2612 (11)

GD = ln(3:6101D) (12)

G� =1:2347(1� ��0:356) (13)

Gf = exp(�0:0006f) (14)

G� =1� 0:0006�: (15)

It should be noted that the proposed form for GSD is the same as thatsuggested by Hodge [6] but the dependence of the factors GA , GD ,G� , Gf , G� on the various parameters of the problem is different.

To test the agreement with experimental results, the proposedmethod was compared against 76 experiments referred to in the ITU-RStudy Group 3 databank [13] along with the Hodge’s formula [6], anda similar simple model predicting diversity gain adopted by the ITU-R


TABLE IMAIN PARAMETERS OF EXPERIMENTS WITH SEPARATION DISTANCES �15 km USED IN THE DIVERSITY GAIN MODEL TESTING

in Recommendation P.618-8. The comparative test was carried outseparately for two cases: experiments concerning separation distancesD � 15 km and experiments for D < 15 km, respectively. The 15km threshold is the limit below which SD is referred to as short-dis-tance [8]. Separation distances larger than 15 km lead to a greaterdecorrelation of the rain attenuation on the alternative slant paths and,therefore, to higher diversity gains.

This specific grouping of experiments made it possible for modelssuch as the EXCELL raincell model [8] and Matricciani’s extension ofthe two-layer rain model [9] to be incorporated in the comparison. Asfor the error calculations, the concept of the relative diversity gain gintroduced in [8], [9] and defined as

g =GSD

AS(16)

was adopted yielding a smaller dependence on single site attenuationthan GSD. Then, the percentage error was evaluated using

ep = [gep � gmp]� 100 (17)

where gep, gmp are the estimated and measured relative gains, re-spectively, for a given probability p. Also, the prediction errors wereweighted with respect to the duration of the experiment, since statis-tical uncertainties arise whenever the measurement period is short. Thedetailed parameters of the SD links taken into account for the groupof experiments concerning D � 15 km (35 experiments) are given inTable I, whereas a similar table concerning the case D < 15 km (41experiments) is given in Table I of [8]. The results of the test in termsof the mean and RMS errors are summarized in Table II, testifyingthe superiority of the proposed method in comparison to the rest ofthe models in case of SD systems of short separation distance. Since


TABLE IIRESULTS OF THE COMPARATIVE TEST BETWEEN THE PROPOSED METHOD AND

OTHER MODELS CONCERNING SITE DIVERSITY GAIN

TABLE IIIMEAN AND RMS ERROR OF THE PROPOSED AND OTHER MODELS

the EXCELL model is recommended for short distance SD systems,the proposed model was compared only with the Hodge formula andMatricciani’s model forD � 15 km. As shown in Table III, an equallygood performance is observed.

The overall improved performance of the proposed model with re-spect to the Hodge procedure is in our opinion related to the sensitivityof the model with respect to the site separation distance D, which isthe factor that has the most serious influence on site diversity gain. Theabsolute sensitivity is defined as

SG

D=@GSD(D)

@D(18)

and it is generally used to check when a parameter of a model has itsgreatest effect.

In Fig. 3, the absolute sensitivity of the Hodge and the proposedformulas with respect to site separation distance are drawn for typicalvalues of the rest of operational parameters. The influence of the siteseparation distance on the predicted value of SDG obtained from theHodge model is very small and almost negligible for D > 10 km,while the proposed model is more sensitive on the variations of D.

Another classification of SD systems is micro scale SD [5], which in-volves earth stations separated by less than 3 km. Such SD systems areinvestigated as a way to improve the cell coverage in mobile satellitecommunications or narrow spot beams in broadcast/multicast services.In [5], micro scale SD simulated experimental data for 31.6� and 30.6�

slant paths at 19.77 GHz are reported. The Synthetic Storm Technique(SST) has been applied to a reliable rain rate database obtained at Fu-cino, Central Italy and at Gera Lario, Northern Italy. The results comingfrom Fucino are presented in Fig. 4 for different levels ofAS . The levelD = 1:7 km constitutes a lower bound above which the model pro-posed in (10) is valid, while the exact same lower bound for the validregion was indicated in [5] after comparing experimental results withthe Hodge model. In [5], it is also reported that GSD remains almostconstant for distances 0:5 km � D � 2 km providing a rather signifi-cant value for GSD of a micro scale site diversity satellite network.

Finally, besides micro, short and long scale diversity, large scale di-versity satellite systems have also been proposed in the literature, whichutilize Earth stations separated more than 50 km and up to 1000 km,

Fig. 3. Absolute sensitivity of the proposed diversity gain model and theHodge model in terms of the site separation distance D. Hodge model

. Proposed model .

Fig. 4. Testing the diversity gain prediction method against data comingfrom the Fucino Plain in Italy for values of D less than 3 km (micro scaleSD). An elevation angle and frequency of 31.6 and 19.77 GHz, respectively,are supposed. The start of the valid region (at D = 1:7 km) is denotedby the vertical dashed line. Experimental data . Proposed method

.

aiming at further increasing the time a satellite terminal is available.Assuming the long-term rain rate statistics at the two terminals similarand after incorporating the logarithmic correlation coefficient proposedby Paraboni–Barbaliscia [17], [18] for D > 50 km

�n = 0:94 exp �D

30+ 0:06 exp �

D

500

2

(19)

in the analytical physical model [7], a simple formula for large scalediversity gain GLS

SD was found

GLS

SD = �0:002 + 0:012A1:02

S (26:3 + lnD)

�(1 + �0:06) exp(�0:004f): (20)


Fig. 5. Testing of the large-scale site diversity gain prediction model againstdata coming from Japan for different values of single-site attenuation andD =

300 km (large scale SD). The elevation angle and frequency of the slant pathsare 39.56 and 14 GHz, respectively. Experimental data � � � �. Proposedmethod .

The above correlation coefficient has been assessed through exten-sive data in Italy. The above formula is compared in Fig. 5 against ex-perimental data of such a large scale site diversity configuration oper-ating at 14 GHz between Tohuku Gakuinn Electric University (TGU)and Waseda University (WU) in Japan [19]. The distance between thetwo stations is 300 km. Although the predicted curve falls quite close tothe reported data, much more experimental results and comparison testsare needed to definitely conclude on the suitability of the expression in(20), since this specific experiment lasted for only six months (May toNovember 1995). The difference between the proposed model and theexperimental results were expected, since Japan’s climate, belongingto subtropical areas is quite different in comparison to Mediterraneanareas (Italy). Therefore, the large-scale site diversity gain model of (20)should be used with precaution.

IV. CONCLUSION

Two new simple prediction methods were introduced in the previoussections. The formula proposed for the long-term rain attenuation prob-ability on a slant path is based on the Weibull distribution and incor-porates recent ITU-R recommendations concerning the necessary rainrate and rain height inputs. The error behavior of the model was testedwith the leading rain attenuation prediction model recommended byITU-R for a large number of experiments at different probability levelsaccording to the procedure given in Rec. P.311-11. The results are veryencouraging. As a general remark, due to its simplicity, the proposedmodel can be used for educational purposes and safely for at least backof the envelope computations.

Furthermore, simple formulas for the site diversity gain were pre-sented, providing also some physical insight due to their reliance onthe convective raincells model. The models concerning diversity gainunderwent extensive testing, taking into account 76 experiments cate-gorized in two groups. For SD systems separated by less than 15 km,the proposed method performed better compared to the Hodge, Matric-ciani and EXCELL models. For SD systems separated by more than15 km, the proposed method performed equally well to the Hodge and

Matricciani models. It has also been shown, that the proposed model ismore sensitive to the site separation distanceD compared to the Hodgemodel, which is the current ITU-R procedure for predicting site diver-sity performance. Moreover, extending the convective raincell diversityprocedure including Paraboni–Barbaliscia’s correlation coefficient re-sulted in a new simple model for large-scale site diversity networks(D > 50 km).

APPENDIX AEVALUATION OF THE STATISTICAL PARAMETERS OF RAIN

ATTENUATION wA AND mA

In the ITU-R recommendation P.837-3 [11], the rainfall rate annualexceedance percentage for every geographical longitude and latitude ofEarth is provided. Assuming that the rainfall rate is approximated bythe Weibull distribution, one can obtain its statistical parameters wr ,mr through appropriate regression fitting analysis. For a satellite linkwith effective length L the parametermA can be calculated by solvingthe following transcendental equation [10]

� 2

m+ 1

�2 1

m+ 1

=H1 � �

2b

m+ 1 � (H1 � L2) � �2 b

m+ 1

�2 b

m+ 1 � L2

:

(A-1)The parameter wA is then calculated

wA = exp �mA � lna � w

�

r � � b

m+ 1 � L

� 1

m+ 1

(A-2)

where a and b are the parameters of the specific rain attenuation for thecorresponding link [24] and �( ) is the well known gamma function[25]. In the above formulas H1 is defined as

H1 = 2LG sinh�1 L

G+ 2G2 1�

L

G

2

+ 1 (A-3)

where the rainfall medium spatial factor G is

G =

1; j�j � 23�

1:5; 23� < j�j � 50�

1:75; j�j > 50�:

(A-4)

ACKNOWLEDGMENT

The authors would like to thank Prof. C. Riva from Politecnico diMilano for providing some additional experimental results on site di-versity. They are grateful to all propagation experts, whose past effortsproduced the extensive list of experiments used to testify the validity ofthe models proposed in this paper and are indebted to the anonymousreviewers, whose creative comments and suggestions helped signifi-cantly to improve the original version of this communication.

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A New Excitation Technique for Wide-Band ShortBackfire Antennas

RongLin Li, Dane Thompson, John Papapolymerou, Joy Laskar, andManos M. Tentzeris

Abstract—A new excitation technique is developed to improve theimpedance bandwidth and to lower the manufacturing cost of a shortbackfire antenna (SBA). The new excitation structure consists of a planarmonopole and a microstrip feed line, both of which are printed on thesame dielectric substrate. By splitting the printed monopole with a slot, awide-band performance can be achieved. The new split-monopole-excitedSBA achieves an impedance bandwidth of about 15% [voltage standingwave ratio (VSWR 2)] while maintaining good radiation perfor-mance. As an example, an SBA configuration with the new excitationtopology was designed and measured at the 5 GHz UNII band, and goodagreement was observed between the simulation and experiment. Theeffects of the geometric parameters of the excitation structure on theimpedance performance are investigated and the operating mechanism ofthe split-monopole-excited SBA is discussed. Being a low-cost, high-gain,and wide-band directional antenna, the new SBA can find applicationsin various wireless systems, such as LMDS, WLAN, and the emergingWiMAX networks.

Index Terms—Excitation technique, low-cost antenna, short backfire an-tenna, wide-band antenna, wireless applications.

I. INTRODUCTION

In recent years, there has been an increasing need for high-gain wide-band directional antennas in wireless applications [1], such as the localmultipoint distribution service (LMDS) systems [2] and the millimeter-wave wireless local area networks (WLAN) [3]. In particular, WiMax(world interoperability for microwave access), a technology based onan evolving standard for broadband point-to-multipoint wireless net-working, is becoming a hot spot in wireless industry [4], [5]. For aWiMax system, it is typical to use fixed, externally mounted (usuallyon rooftops or external walls) directional subscriber antennas to com-municate with base stations which are connected to the Internet. Sinceone of the major goals for wireless systems is to offer a less expen-sive infrastructure than a wired one (such as that based on a T1, DSL,or cable connection), the cost-effectiveness of a wireless deploymentis of primary concern. Also for the reasons of system flexibility andinteroperability, the ability for a wireless antenna to operate at a widefrequency band, covering more than one standard, is highly desirable.

The short backfire antenna (SBA), developed first in the 1960s [6],[7], may become one of the most competitive candidates for these wire-less application because of its low profile, high gain, lightweight, andhigh isolation from surroundings. However the widespread adoptionof this antenna is likely dependent on improvements in its impedancebandwidth and manufacturing costs. The SBA has been widely usedin mobile/maritime satellite communications, tracking, and telemetry[8]–[10], due to its excellent radiation characteristics (a gain on theorder of 13–15 dBi, with sidelobes of at least �20 dB and a backlobelower than �30 dB) [11], its compact structure (�0:5�0 in height, �0

Manuscript received September 7, 2004; revised January 16, 2005.This work was supported in part by the Georgia Electronic Design Center

(GEDC), in part by the National Science Foundation under NSF CAREERAward ECS-9984761 and NSF Grant ECS-0313951, and in part by the NSFPackaging Research Center.

The authors are with the Georgia Electronic Design Center, School of Elec-trical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA30332-0250 USA (e-mail: [email protected]).

Digital Object Identifier 10.1109/TAP.2005.850764

0018-926X/$20.00 © 2005 IEEE

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