P Gilvin
email: phil.gilvin@ukhsa.gov.uk

M Caresana

J-F Bottollier-Depois

V Chumak

I Clairand

J Eakins

P Ferrari

O Hupe

P Olko

A Röttger

R J Tanner

F Vanhavere

E Bakhanova

V Bandalo

D Ekendahl

H Hödlmoser

D Matthiä

G Reitz

M Latocha

P Beck

D J Thomas

R Behrens

IMPACT OF NEW OPERATIONAL QUANTITIES EURADOS PROJECT ON THE IMPACT OF THE PROPOSED ICRU OPERATIONAL DOSE QUANTITIES

Following the publication of the joint ICRU/ ICRP report on new operational quantities for radiation protection, the European Dosimetry Group (EURADOS) have carried out an initial evaluation. The EURADOS

INTRODUCTION

The International Commissions on Radiation Units and Measurements (ICRU) and on Radiological Protection (ICRP) have published joint recommendations [START_REF]International Commissions on Radiation Units and Measurements. ICRU Report 95. Operational Quantities for External Radiation Exposure[END_REF] for new operational quantities for external exposure. The new quantities have been devised to address certain problems with the existing ones, including the need to cover a wider range of radiation types and energies. ICRP have carried out a consultation process and ICRU have revised the report in the light of comments received.

As part of its strategic research agenda [START_REF]Visions for Radiation Dosimetry over the Next Two Decades -Strategic Research Agenda of the European Radiation Dosimetry Group: Version 2020[END_REF] the European Radiation Dosimetry Group, EURADOS (www.eurados.org), seeks to contribute to the development and understanding of fundamental dose concepts, such as in the area of operational quantities. Accordingly, EURADOS has carried out a project to evaluate the impact of the proposed operational quantities and to make recommendations for their application. Our report [START_REF]Evaluation of the Impact of the New ICRU Operational Quantities and Recommendations for their Practical Application[END_REF] analyses the differences between the new and existing quantities before going on to examine their impact and application.

EXISTING AND NEW OPERATIONAL QUANTITIES

Purpose of Operational Quantities (OQs)

The system of radiological protection 4 is based on a set of protection quantities, such as effective dose and equivalent dose, that represent risk and detriment. However, the protection quantities are not measurable. Instead, a set of operational quantities is used to act as reasonable estimators of the protection quantities. Although the operational quantities are likewise not directly measurable, they do possess a closer relationship with the field quantities, which are directly measurable. The definitions of the operational quantities have also served as conceptual guidance for designers and manufacturers of measuring devices, whether active or passive: the definition at a specified depth makes the design of filters intuitive.

Deficiencies of Existing OQs

The current operational quantities [START_REF]ICRU 1985. Report 39. Determination of Dose Equivalents Resulting from External Radiation Sources[END_REF][START_REF]ICRU 1988. Report 43. Determination of Dose Equivalents from External Radiation Sources-Part II[END_REF][START_REF]International Commissions on Radiation Units and Measurements[END_REF][START_REF]ICRU 1993. Report 51. Quantities and Units in Radiation Protection Dosimetry[END_REF][START_REF]ICRU 1998. Report 57. Conversion Coefficients for use in Radiological Protection Against External Radiation[END_REF] were defined during the period 1985-98. Their definitions include references to notional spherical, cuboid and cylindrical phantoms of ICRU 4-element tissue, and to the positioning of a dosemeter on the human body. The conversion coefficients from the field quantities (fluence or air kerma) were calculated using the simplifying "kerma approximation", whereby it is assumed that kerma and absorbed dose are equal -that is, all of the energy released at a point is also absorbed at that point. This method was used because it is valid for most workplaces and because the numerical calculation capability was, at that time, limited.

It took decades for the current operational quantities to be completely adopted across the world, and in the USA they have never been fully adopted. Nevertheless, they provided a significant advance in radiation protection, allowing additivity for different radiation types and energies, and overall, they have provided a consistent and useful system. However, there have always been problems of overestimation, up to a factor of 5 at sub-80 keV photon energies, and even more at higher energies (Eph > 2 MeV), where secondary charged particle ranges can exceed 10 mm. Also, the current quantities only cover a limited range of particle types. In recent years these drawbacks have become more important, with growth in medical diagnostic and interventional techniques and in the use of particle accelerators.

New Operational Quantities

With modern advances in computing power, it has become possible to perform "full transport" Monte Carlo calculations, avoiding the need for the kerma approximation. Further, the new quantities 1 are defined, in vacuo, on the same anthropomorphic computational phantoms in which the protection quantities are now defined [START_REF]International Commission on Radiological Protection[END_REF][START_REF] Behrens | Dose Conversion Coefficients for Photon Exposure of the Human Eye Lens[END_REF][START_REF]Conversion Coefficients for Radiological Protection Quantities for External Radiation Exposures[END_REF] . Thus, the new operational quantities have a closer relationship with the current protection quantities.

Meanwhile, references in the current operational quantity definitions to fixed depths (10 mm, 3 mm, 0.07 mm) have provided some guidance for instrument and dosemeter manufacturers. With a wider range of radiation types and energies, this is no longer considered appropriate.

Table 1 summarizes the definitions of the current and new operational quantities. For full details readers are strongly recommended to read ICRU Report 95 1 , and our EURADOS report [START_REF]Evaluation of the Impact of the New ICRU Operational Quantities and Recommendations for their Practical Application[END_REF] .

EURADOS EVALUATION PROCESS

The present work was not part of the formal consultation. Rather, it looks ahead at practical matters concerning the introduction of the new quantities.

Topics Considered

As well as the main differences between the current and proposed operational quantities, the EURADOS task group considered impacts on:

• dosemeter and instrument design • radiation protection practices • calibration and reference fields • international standards • regulation (including dose registries etc.)

Benefits

If they can be measured well, the new quantities will bring with them improved estimates of detriment, especially in low-energy photon fields and high-energy fields around accelerators. They will also allow for a wider range of particle types and energies to be considered. The proposed switch to absorbed dose for tissue reactions -as is recommended -looks sensible. Note, however, that for eye lens doses, we await current ICRP deliberations on cataract formation, and their view on whether it constitutes a tissue reaction or a stochastic effect.

Unsurprising Changes

A number of changes will obviously follow from the introduction of new quantities, whenever they happen. For example, the large number of international standards covering type testing, calibration practice, proficiency testing and usage recommendations, will have to be revised in due course. Little or no change, however, will be needed to:

• dose records and registries. • regulations. • radiation protection practices. • calibration laboratory practice -but see below.

EFFECTS ON DOSIMETRY

Conversion Coefficients

Charged Particle Equilibrium (CPE) and the Kerma Approximation

One problem with the new quantities is that they are effectively defined in vacuo. In practice, this condition is not realizable. When, for example, a photon beam passes through air, it is accompanied by secondary charged particles -in this case, electrons. This contamination of the primary beam is unavoidable but controllable. When the point is reached at which the number of secondary particles being generated is equal to the number being absorbed, a state of charged particle equilibrium (CPE) is reached. By deliberately ensuring CPE, calibration laboratories can standardize conditions, avoiding any problems with variable beam contamination. CPE, i.e. equilibrium conditions, can be achieved by ensuring sufficient source-to-target distance and beam diameter in air, or, much simpler and for photon energies above the X-ray range practically only, by means of a polymethyl methacrylate (PMMA) buildup layer [START_REF] Behrens | Secondary charged particle equilibrium in 137 Cs and 60 Co reference radiation fields[END_REF] . The latter is recommended by ISO 4037-3 [START_REF]Radiological protection -X and gamma reference radiation for calibrating dosemeters and doserate meters and for determining their response as a function of photon energy -Calibration of area and personal dosemeters and the measurement of their response[END_REF] .

The CPE condition therefore currently applies to all calibrations and type-test exposures -in other words, everything that is needed for using instruments and dosemeters. Furthermore, if one wishes to calibrate a device in non-equilibrium conditions, i.e. without CPE, one needs to separately determine the spectra of the different radiation type contributions, e.g., photons and electrons, by means of precise spectrometry or detailed Monte Carlo transport simulations, and fold the spectra with the corresponding conversion coefficients with full transport to obtain the new operational quantities. To circumvent this, CPE conditions are always used. Also, CPE is justified as it is assumed that it exists in many workplace situations as usually a significant amount of material is located between the source and the person to be monitored -such as the source encapsulation, shielding, walls and air.

Conversion Coefficients in ICRU 95

By its nature, CPE is associated with the conditions required for the kerma approximation to be valid. Therefore, the approximation will be valid in most situations in which instruments and dosemeters are used. Recognizing this, ICRU [START_REF]International Commissions on Radiation Units and Measurements. ICRU Report 95. Operational Quantities for External Radiation Exposure[END_REF] have provided, in addition to the primary sets of conversion coefficients calculated with full transport, a supplementary set for photons, based on the kerma approximation. The energy range for these conversion coefficients is limited to the energies for which CPE can be assumed. In practice, it will be this supplementary set of conversion coefficients that is used in almost all photon applications.

Example: Extremity Dosemeters

It is just as well that the kerma-approximation conversion coefficients are available. Our analysis shows that changing the design of, for example, singleelement extremity dosemeters to measure the new operational quantities, using the full-transport coefficients, would be extremely difficult. Figure 1 shows the relative responses of a typical fingerstall extremity thermoluminescence dosemeter (TLD), featuring a single-element of LiF:Mg,Cu,P, in terms of the current and proposed quantities. It can be seen how the Dp local skin rod response compares with the Hp(0.07) response (circles), when Dp local skin rod is calculated either with the full transport (squares) or the kerma-approximation (triangles) conversion coefficients. Because extremity dosemeters must respond correctly to both photons (solid symbols) and electrons (open symbols), the comparison is shown for both types of radiation. The minimum electron energy considered is 70 keV, which is the lowest energy for which hp(0.07) conversion coefficients are provided [START_REF]ICRU 1998. Report 57. Conversion Coefficients for use in Radiological Protection Against External Radiation[END_REF] , though dp local skin rod data are now given 1 down to 50 keV. The curves are normalized to the corresponding 137 Cs photon response appropriate for each dataset.

In each case we can estimate the ranges of response from the minimum to the maximum, considering both photon and electron curves. The results for Hp(0.07) and Dp local skin rod (kerma approximation) are very similar, with a factor of just under 5 between the minimum and maximum responses. On the other hand, the result for Dp local skin rod (full transport) shows a much greater response range: the factor here is more than 20, chiefly owing to the altered shape of the photon response curve.

The consequences are as follows. First, current extremity dosemeters are likely to be able to measure the new quantities without much modification, provided the kerma-approximation conversion coefficients are used. Second, if the full-transport conversion coefficients were used instead, it would be very difficult indeed to design an extremity dosemeter to measure the new quantities. This clearly illustrates why the extra, kermaapproximation, coefficients were needed.

Effect on Existing Instruments and Dosemeters

Redesign

To optimize measurements of the new operational quantities, many types of passive dosemeters and instruments will need a measure of redesign. It is in the redesign, testing and roll-out of modified or new dosemeters that significant costs will arise. The options are, in order of increasing severity: • adjust the calibration, adjust the instrument, or apply a constant correction factor. • change the dose algorithm (for multi-filter dosemeters). • change the filtration, e.g. replace a plastic filter with a suitable metal one.

•

replace the system (it may be impractical to adapt some dosemeter types/ sensitive materials).

Example: LiF (Mg,Ti)

A material that is likely to be badly affected is "conventional" lithium fluoride, that is, doped with magnesium and titanium. LiF (Mg, Ti) is widely used in thermoluminescence dosimetry systems. This concern is illustrated in Figure 2, which shows three ratio curves.

The first shows a typical energy response characteristic for an extremity dosemeter using that material in terms of the current operational quantities. Because extremity dosemeters in general employ very thin, and usually tissue-equivalent, filters, the dosemeter response in this case closely matches that of the material itself. The second curve shows the ratio of the current conversion coefficients, hp (10), to the new, hp; and the third shows the product of the two: the response for the material in terms of the new quantities. In the sub-80-keV energy range, the material already over-responds in terms of the current quantity; but since the corresponding values of the new quantity are significantly smaller, the over-response of the material will in future be much worse. The problem may be tractable for multi-element designs, where differing filtrations can be used, but is unlikely to be so for simple one-or two-element dosemeters.

Areas for Further Work

The conclusions given above are, at this stage, broad. Detailed simulation/ feasibility work is needed on:

• instrument and dosemeter re-design and simulations -filtration, calibration • dose algorithms • evaluation of energy and angle dependence under the new quantities • evaluation of measurement uncertainties • determination of radiation type contributions in typical workplace situations. These must be followed by physical type tests to confirm the new properties. New solutions may be found, but this process must be well advanced before new / revised international and national standards can be completed.

Other Matters

Phased Introduction of Limited Value

Not all of the switch to the new quantities, in terms of dosemeter and instrument design, has to be done at once. Indeed, the roll-out of new instrument and dosemeter designs will have to be phased. For instruments, manufacturers will need to support both old and new instruments / devices until most users have adapted or replaced their existing models. For individual monitoring services, because dosemeters are used for varying periods and are returned at varying rates, there will inevitably be a period over which both old and new dosemeters, which will require different treatment, are being returned. Besides this, a phased introduction will allow suppliers and their customers to spread the significant financial investments that will be necessary.

However, a phased introduction will, for instruments or dosemeters, carry increased complexity, and therefore increased risk of errors. Any phased transition should therefore be relatively short, taking investment costs into account.

Doses at Diagnostic X-ray Energies

As we have seen, in this energy range the current quantity Hp (10) significantly overestimates effective dose E. The new quantity Hp will be much closer to E, and occupational doses will appear to be lower (see the ratio of conversion coefficients in figure 2). But radiation professionals should be clear that the values of E will not change at all. Therefore, protection measures should not be relaxed.

Another point concerns the practice, which is sometimes used in homogeneous fields, of controlling eye lens dose by controlling whole body dose. Important in this case is the ratio of the new quantities Hp/Dp,lens. This ratio is below unity for all photon energies, whereas the corresponding ratio for the current quantities, Hp(10)/Hp(3), is around unity above approximately 40 keV. With the new quantities, then, it will no longer be feasible to control eye lens dose by controlling whole body dose, because Hp is smaller than Dp,lens.

Air and Space Crews

The new operational quantities cover a wider range of particles and energies, and this will make them useful in high-energy applications, e.g. in particle accelerator environments. Note, however, that new operational quantities are not needed for aircrew dosimetry, because the existing dose programs calculate effective dose E directly. Such programs are currently validated by wide use of tissue-equivalent proportional counters (TEPCs) calibrated for H* (10). This works well and should not be changed.

The new operational quantities, like the current ones, are not for space crew dosimetry.

COST, RESOURCE AND OTHER IMPACTS

Costs and Resources

Funds and resources will need to be dedicated to:

Other Implications

As indicated above, large numbers of international standards will need changing. Depending on the performance attainable by instruments and dosemeters in terms of the new quantities, some requirements may need to be relaxed. Calibration laboratories will see changed practices but, if the kerma-approximation conversion coefficients are used, changes will be relatively small.

Lengthy Transition

Judging from experience, different countries will adopt the new quantities at different rates. Concerns about costs could particularly affect countries whose radiation protection resources are limited, and full adoption might not be achieved until the late 2030s. Inevitably there will be a loss of harmonization during the transition, with different countries using different quantities.

RECOMMENDATIONS

•

We echo ICRU's view that the process of adopting the new operational quantities should be carried out over a suitably long timescale, e.g. decades. This will allow for mature consideration of the changes, and for account to be taken of the parallel development of new ICRP recommendations, for example concerning whether cataract formation should be treated as a tissue reaction or a stochastic effect.

•

While the new quantities bring benefits, it is not clear at this stage that they will outweigh the associated costs. National competent authorities should be aware of the potential costs that would be borne by manufacturers and undertakings in changing to the new quantities. They should prompt stakeholders to ensure that the costs are evaluated at an early stage.

•

Stakeholders, including instrument suppliers, dosemeter suppliers, individual monitoring services and calibration laboratories, should begin work immediately to evaluate and make initial plans for the change.

•

To fully support the training and information processes, a planned communication programme will be required, both internationally and nationally.

Figure 1 .

 1 Figure 1. Relative response characteristics of a single-element extremity dosemeter, in terms of the current operational quantity and the new.

Figure 2 .

 2 Figure 2. How the new conversion coefficients affect the relative photon response of an example "conventional" lithium fluoride (LiF (Mg,Ti)) dosemeter together with the ratio of the current and the new quantities for personal whole-body monitoring.

Table 1 . Summary of the definitions of the existing and new operational quantities. CURRENT QUANTITIES

 1

	Dose Quantity	Symbol	Definition	Quantity defined in…	Exposure conditions
	Ambient dose equivalent	H*(10)	Dose equivalent at 10 mm depth		In expanded and aligned field
	Directional dose equivalent (eye lens dose) Directional dose equivalent (local skin dose)	H'(3,Ω) H'(0.07,Ω)	Dose equivalent at 3 mm depth Dose equivalent at 0.07 mm depth	ICRU sphere	In expanded field
	Personal dose
	equivalent (whole-body	Hp(10)	Dose equivalent at 10 mm depth	Human body (calculated using
	dose)			geometric phantoms of	Beneath "a specified
	Personal dose equivalent (eye lens dose)	Hp(3)	Dose equivalent at 3 mm depth	ICRU tissue that represent parts of the body, e.g. trunk, head, extremities,	point on the body" (when worn, usually below the dosemeter)
	Personal dose equivalent (local skin dose)	Hp(0.07)	Dose equivalent at 0.07 mm depth	for different energies and angles of incidence)
			NEW QUANTITIES
	Dose Quantity	Symbol	Definition	Quantity defined in…	Exposure conditions
			All are defined as:
			CC relating particle fluence
	Ambient dose	H*	to the maximum value of effective dose, Emax, for
			various irradiation conditions
	Directional absorbed dose in the lens of the eye	D′lens(Ω)	CC relating particle fluence to the value of absorbed dose in the lens of the eye	Calculational phantoms: • anthropomorphic for the human
	Directional absorbed dose in local skin	D′local skin(Ω)	CC relating particle fluence to the value of absorbed dose in the local skin	body 10 ; • stylized for the lens of the eye 11 ; • stylized for the local	Radiation field at a specific point in the area or on the body
			CC relating particle fluence	skin 1
	Personal dose Personal absorbed dose in the lens of the eye	Hp Dp lens	to the value of effective dose E CC relating particle fluence to the value of absorbed dose in the lens of the eye	(personal doses calculated for different energies and angles of incidence)
	Personal		CC relating particle fluence
	absorbed dose in	Dp local skin	to the value of absorbed dose
	local skin		in local skin

Particle fluence × conversion coefficient (CC)

*Corresponding author: phil.gilvin@ukhsa.gov.uk

ACKNOWLEDGEMENTS

The authors would like to express their thanks to our colleagues at ICRU for their helpful co-operation, and in particular to Thomas Otto for his valuable comments on our report; to other EURADOS colleagues who have provided help and support; and to those manufacturers and suppliers who have kindly provided us with relevant information.


Member, IEEE Mihai Telescu

Member, IEEE Riccardo Trinchero

Student Member, IEEE Nastaran Soleimani

Noël Tanguy

Senior Member, IEEE Igor Simone Stievano

Noel Tanguy

Stochastic Time-Domain Mapping for Comprehensive Uncertainty Assessment in Eye Diagrams

Keywords: signal integrity, high-speed interconnects, smart textiles, wearable electronics, eye diagram, uncertainty quantification (UQ), kernel-machine regressions

published or not. The documents may come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

Benefiting from constant interest from both industry and academia, stochastic analysis evolved significantly over the past years. Various techniques are currently available, allowing designers to evaluate the impact of system uncertainties for various circuit applications, packaging and interconnects. Historically, the most common approach to stochastic analysis was Monte Carlo (MC). Featuring several improved versions, the technique is straightforward. Data is collected via repeated simulation and allows various metrics to be computed, quantifying the performance and robustness of a system subject to uncertainties related either to the manufacturing process or environment parameters. [START_REF] Spence | Tolerance Design of Electronic Circuits[END_REF], [START_REF] Zhang | Development of robust interconnect model based on design of experiments and multiobjective optimization[END_REF]. The main drawback of MC is its computational cost. This prompted researchers to look for alternative solutions to the computationally expensive brute-force MC sampling, allowing them to infer the statistical characteristics of a system from limited data. Many of the first successful approaches were based on polynomial chaos expansion (for an overview of this family of algorithms see [START_REF] Manfredi | Stochastic transmission line analysis via polynomial chaos methods: an overview[END_REF]). These were followed by other machine learning inspired solutions falling mainly into the category of kernel machine M. Telescu and N. Tanguy are with the Univ Brest, CNRS, Lab-STICC, F-29238 Brest, France. R. Trinchero, N. Soleimani and I. S. Stievano are with the Department of Electronics and Telecommunications, Politecnico di Torino, Torino, Italy (corresponding: {riccardo.trinchero@polito.it}).

regression (e.g., [START_REF] Trinchero | Machine Learning and Uncertainty Quantification for Surrogate Models of Integrated Devices with a Large Number of Parameters[END_REF]- [START_REF] Ma | A Fast Optimization Method for High-Speed Link Inverse Design With SVR-AS Algorithm[END_REF]) and artificial neural networks (ANNs) (e.g. [START_REF] Kushwaha | Comparative Analysis of Prior Knowledge-Based Machine Learning Metamodels for Modeling Hybrid Copper-Graphene On Chip Interconnects[END_REF]- [START_REF] Ma | Uncertainty Quantification of Signal Integrity Analysis for Neuromorphic Chips[END_REF]).

This paper focuses on kernel-machine regression techniques, such as: the support vector machine (SVM) regression [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF], the least-squares support vector machine (LS-SVM) [START_REF] Suykens | Least Squares Support Vector Machines[END_REF] regression and the more recent vector-valued kernel ridge (KRR) [START_REF] Soleimani | Compressed complex-valued least squares support vector machine regression for modeling of the frequency-domain responses of electromagnetic structures[END_REF], [START_REF] Soleimani | Bridging the Gap Between Artificial Neural Networks and Kernel Regressions for Vector-Valued Problems in Microwave Applications[END_REF], [START_REF] Mauricio | Kernels for vectorvalued functions: A review[END_REF] which have been proven particularly effective for various microelectronics and radiofrequency applications [START_REF] Trinchero | Machine Learning and Uncertainty Quantification for Surrogate Models of Integrated Devices with a Large Number of Parameters[END_REF], [START_REF] Trinchero | Modeling of eye diagram height in high-speed links via support vector machine[END_REF], [START_REF] Nguyen | Comparative Study of Surrogate Modeling Methods for Signal Integrity and Microwave Circuit Applications[END_REF]. Indeed, kernel-machine regressions provide an interesting alternative to the otherwise more flexible ANNs, especially well suited for problems in which a "relatively small" set of training data is available [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF], [START_REF] Suykens | Least Squares Support Vector Machines[END_REF], [START_REF] Rudi | Falkon: An optimal large scale kernel method[END_REF]. Unlike ANNs, the linear model structure adopted by kernel regression (i.e., the model unknowns appear linearly) has the key advantage of heavily simplifying the training phase, which reduces to the solution of a standard convex optimization problem, thus leading to several advantages in terms of training time and accuracy w.r.t. the number of training samples [START_REF] Soleimani | Compressed complex-valued least squares support vector machine regression for modeling of the frequency-domain responses of electromagnetic structures[END_REF], [START_REF] Kushwaha | Comparative Analysis of Prior Knowledge-Based Machine Learning Metamodels for Modeling Hybrid Copper-Graphene On Chip Interconnects[END_REF].

In this specific context, the present paper explores a new methodology for uncertainty quantification in the field of signal integrity. The inner boundaries of an eye-diagram are approximated by a polygon. Several kernel machine regression algorithms are then used to build a surrogate model capable of mimicking the impact of system uncertainties on this polygon, resulting in a "stochastic map". A particularly challenging example was selected in order to demonstrate the technique: a flexible differential digital link for smart textiles. The implementation uses copper yarn and is subject to various sources of uncertainties, including high tolerances during manufacturing [START_REF] Cottet | Electrical characterization of textile transmission lines[END_REF]. It is important to observe that unlike previous approaches, which predicted the impact of uncertainties on specific metrics derived from the eye diagram (e.g., vertical opening, eye SNR, etc), stochastic time-domain mapping fully models inner-eye geometry. Several metrics may thus be computed in post-processing, the use of masks is also possible. The aim is to allow a statistically meaningful design leading to more robust systems. The approach is particularly interesting for cases where high tolerances are an issue in the design flow. These include emerging technologies with non-standard manufacturing processes, or low-cost products for mass consumers where high tolerances are a result of financial considerations. Note, however, that the stochastic mapping approach is not application-oriented but general and very adaptable.

The paper is organized as follows. An example related to wearable electronics, based on published data, is presented in Section II. A description of how the eye-diagram is processed follows in Section III. Section IV provides an in-depth description of the algorithms used to perform stochastic mapping. Numerical results relative to the selected example are collected in Section V and are followed by a brief Conclusion.

II. TEST CASE

In the past two decades significant progress has been made in the field of wearable electronics and wearable computing. Fabrics integrating sensors, processors, energy harvesters, lights and batteries, so called "smart textiles" or "e-textiles", have attracted the interest of various private and public actors [START_REF] Cottet | Electrical characterization of textile transmission lines[END_REF]- [START_REF] Wagih | Microwave-Enabled Wearables: Underpinning Technologies, Integration Platforms, and Next-Generation Roadmap[END_REF]. The defense industry has a particular interest for this field: providing the soldier on the modern battlefield with apparel integrating sensor hubs, processors and various interfaces has numerous advantages [START_REF] Nayak | Electronic textiles for military personnel[END_REF]. Civilian applications also exist and are either health-oriented or focus on various specialized garments designed for exploration, hazardous environments or extreme sports. In a more general context, etextiles may also be used in non-wearable applications (e.g. smart bed-sheets) or even in applications which are not humanoriented (geo-textiles) [START_REF] Shi | Smart Textile-Integrated Microelectronic Systems for Wearable Applications[END_REF].

This new generation of electronics brings new challenges in terms of signal and power integrity. While the main design objective is always the transmitting electrical signals with as little distortion as possible, the specific context of etextiles magnifies the impact of mechanical factors and forces designers to carefully consider aspects such as environment stability and service life [START_REF] Agcayazi | Flexible Interconnects for Electronic Textiles[END_REF]. Flexible data busses are characterized by much higher manufacturing imperfections than the ones in regular PCBs and suffer greater impact from environment parameters such as temperature and humidity. External mechanical action, both in terms of compression and stretching is also an issue.

The results obtained in this paper are based on the serial link shown in Fig. 1 which has been implemented and simulated in LTSPICE [START_REF]LTspice Information Center |Analog Devices[END_REF]. It consists of a differential data communication channel implemented by conductive copper wires weaved in a textile structure [START_REF] Telescu | Worst-Case Optimization of a Digital Link for Wearable Electronics in a Stochastic Framework[END_REF]. The flexible interconnect is described using the classic multiconductor transmission line theory and the per unit length parameters computed from the electrical and geometrical information of the structure [START_REF] Paul | Analysis of multiconductor transmission lines[END_REF], [START_REF] Paul | Computation of the Transmission Line Inductance and Capacitance Matrices from the Generalized Capacitance Matrix[END_REF]. Note that conductive fibers remain, as of 2023, the most popular technological solution for interconnects in e-textiles [START_REF] Dyson | E-Textiles and Smart Clothing Markets 2023-2033: Technologies, Players, and Applications" idtechex[END_REF]. The link features a differential driver at the near end and its paired receiver at the far end. In order to increase maximum datarates, a passive equalizer is inserted between the driver and the line. It is assumed that the equalizer and buffers are implemented either using conventional electronics or semiflexible substrates. They are affected by uncertainties, albeit to a lesser extent. The center section of the interconnect (identified by a transmission line segment with length L 2 in the scheme) accounts for a potentially altered geometry. This can either take the form of compression due to external mechanical action or to the very nature of the application (e.g. bending at knees or elbows) or on the contrary stretching of the material. Either situation alters the distance between the conductors on a specific segment of the link. The circuit model accounts for the resulting discontinuity.

L 1 + L 2 + L 3 = 33 cm,
NRZ encoding is used at a data rate of 2 Gbps. At this stage of the investigation, the driver and the receiver are modeled using compact, linear models, very similar to the ones implemented in IBIS-AMI simulations. Specifically, an equivalent Norton-based voltage controlled circuital representation of the differential drivers and receivers is used, according to the method developed in [START_REF] Siviero | Constructive Signal Approximations for Fast Transient Simulation of Coupled Channels[END_REF]. This representation enables the accurate simulation of coupled channels including both differential-and common mode signals. The driver and the receiver currents are defined as a function of the pertinent output and input port voltages, respectively, via a constant conductance matrix. The dynamic behavior of the devices is represented by a dynamic linear time-invariant submodel. Timevarying current source terms are instead used to represent the switching activity of the driver in terms of a time varying analog waveform. Device parameters are representative of the application at hand and the involved data rate. For a detailed presentation of the buffer models see equation ( 2) and Fig. 2 in [START_REF] Siviero | Constructive Signal Approximations for Fast Transient Simulation of Coupled Channels[END_REF] and the explanations therein.

However, the methodology described in this paper can be used as is with non-linear buffer models. For applications where such models are required for increased accuracy, they may readily be used [START_REF] Siviero | Constructive Signal Approximations for Fast Transient Simulation of Coupled Channels[END_REF]- [START_REF] Stievano | Reliable eyediagram analysis of data links via device macromodels[END_REF]. It is important to note that the geometrical specifications of the interconnect, including the fabrication tolerances, are based on the experimental work published in [START_REF] Cottet | Electrical characterization of textile transmission lines[END_REF]. The selected structure is labelled as GSSG-1 (see Fig. 3 in the previously referenced paper) and consists in a differential pair between two ground conductors.

Without loss of generality, in our application, 11 uniformly distributed uncertain parameters are considered. The distance between the signal lines is 411 µm ± 7.3 %. The distance between each signal line and the adjacent ground line is 481 µm ± 6.7 %. Conductor diameter is 228 µm ± 11.1 %. The total length of the link is 33 cm ± 1cm (mainly due to sizing). The section under mechanical stress varies in length from 2 to 5 cm and the compression [stretching] factor is 25 %

III. EYE APERTURE AND DATA LINK QUALITY

Throughout this paper, the performance of the communication via the data link is quantitatively assessed using the eye aperture on the receiver side, as obtained by observing the differential voltage response v d (t).

It is common practice to use the eye-diagram when designing high-speed interconnect systems. Associating eye-diagram analysis to uncertainty quantification and stochastic analysis has been investigated in recent years [START_REF] Reuschel | Efficient Prediction of Equalization Effort and Channel Performance for PCB-Based Data Links[END_REF], [START_REF] Cho | Signal Integrity Design and Analysis of Differential High-Speed Serial Links in Silicon Interposer With Through-Silicon Via[END_REF]. In addition, thorough theoretical research has consolidated eye computation and processing, as well as jitter analysis making them highly reliable, standard tools [START_REF] Oh | Hybrid Statistical Link Simulation Technique[END_REF], [START_REF] Kim | Analytical Eye-Diagram Determination for the Efficient and Accurate Signal Integrity Verification of Single Interconnect Lines[END_REF]. Machine learning techniques, such as the ones used in the paper, have also been used in the past in relation to eye-diagram analysis [START_REF] Trinchero | Modeling of eye diagram height in high-speed links via support vector machine[END_REF], [START_REF] Beyene | Application of Artificial Neural Networks to Statistical Analysis and Nonlinear Modeling of High-Speed Interconnect Systems[END_REF], [START_REF] Ambasana | Eye Height/Width Prediction From S -Parameters Using Learning-Based Models[END_REF]. The novelty of the present work comes from the way in which machine learning is used, from the meta-modelling strategy itself. Instead of focusing on specific parameters derived from the eye-diagram and computed as scalar quantities (eye-height, horizontal opening etc), the clean inner eye is literally mapped by a polygonal representation. The metamodelling process focuses on the vertices of the polygon. With this approach, most of the metrics commonly used in data-link design can be readily computed a posteriori, in a stochastic framework.

It should be noted that the methodology described in the present paper may be used regardless of the way in which the eye pattern is obtained. Post processing the image of the eye is one option, using sampled recorded responses is another. The former alternative usually relies on both in-house or commercial solutions enabling the fast simulation of a very large number of bits (on the order of millions) [START_REF] Siviero | Constructive Signal Approximations for Fast Transient Simulation of Coupled Channels[END_REF], [START_REF] Oh | Hybrid Statistical Link Simulation Technique[END_REF]. The latter is usually preferred when the problem has a dominant nonlinear nature, usually due to the buffers. In this case, a classic transient simulation is carried out by means of a circuit solver like SPICE where transceivers are replaced by physics-based transistor-level models or by faster behavioral nonlinear surrogates, yielding a sampled transient response of the pertinent variable of interest (e.g., v d (t)) [START_REF] Stievano | Reliable eyediagram analysis of data links via device macromodels[END_REF]. The inherent limitation in this case is the CPU time required to run SPICE and usually this is feasible when a relatively limited number of bits is considered, at most on the order of tens or hundreds of thousands.

A possible implementation of a compact routine allowing to compute the polygonal shape of the aperture and parameters such as the eye area is detailed in the appendix. This pseudocode is made available in order to provide readers with a fully documented and self contained set of tools for both the computation of the inner eye, as described in this section and the surrogate modeling, as described in the next section. Figure 2 shows an example eye diagram which can be obtained for the considered test case, together with its inscribed inner polygon (see the dark thick line) obtained with the proposed procedure. The diagram is generated from the data link response computed for a given sample of the uncertain parameters. The blue circles in the figure are the minimum and the maximum vertical positions of the clean inner eye area observed at a given set of points in the normalized unit interval. The green circles, instead, represent the jitter and can be used to define the horizontal eye aperture arising from the zero-crossing points, leading to the left and right vertices of the inner polygon.

It is important to point out that the optimal number of vertices of the inner polygon shown in Fig. 2 depends on the shape of the eye and thus is strongly related to the specific application at hand. In this paper, the points have been selected to guarantee a good accuracy of the approximated eye area. If needed, an automatic procedure can be devised, based on a heuristic step-by-step greedy algorithm. Basically the number of points along the abscissa, which represent the horizontal coordinates of the vertices, would gradually be increased in the range [0,1] of the unit interval until the variation of the resulting eye area becomes negligible. A relative threshold mechanism can be set. Also, one should avoid vertical probing in the area affected by time-domain jitter (e.g., the UI points 0.1 or 0.9 in the example eye). As already mentioned, horizontal probing is used to model this type of uncertainty.

IV. SURROGATE MODELS

A. Problem Statement

Let us consider the problem of approximating the variability of the vertices of a polygon approximating a generic eye diagram collected in the vector y = [y 1 , . . . , y D ] T ∈ Y with Y ⊆ R D as a function of a set of parameters of the link collected in the vector x = [x 1 , . . . , x p ] T ∈ X with X ⊆ R p . For the example eye of Fig. 2, the entries of vector y (i.e., D) correspond to positive and negative vertical positions of the blue circles calculated at 7 time points [0.2, 0.3, . . . , 0.8] T defined in a normalized unit interval, and the horizontal position of the green circles associated to the jitter, respectively. The overall number of output components is D = 16, while the number of input parameters p = 11. Within the machine learning framework, the modeling problem stated above can be reformulated as the problem of building a generic vector-valued surrogate model M : X → Y, starting from the information available on the training set D = {(x l , y l )} L l=1 , where L represents the total number of training pairs, such that x l ∈ X corresponds to the l-th configuration of the input parameters and y l = [y (1) l , . . . , y

(D) l
] T ∈ Y is the corresponding output vector.

B. Scalar-Output SVM and LS-SVM Regressions

The above vector-valued learning scenario can be reinterpreted as the problem of learning D independent scalar regressions (i..e, one for each output component), each of them considering the training set

D (i) = {(x l , y (i) l )} L l=1 , such that D = ∪ D i=1 D (i) , in which y (i) l
is i-th component of the output vector associated to the l-th input configuration x l . This allows to reformulate the overall vector-valued learning problem in terms of D independent scalar-output regressions, which can be suitably tackled with standard kernel machine approaches, such as the SVM and LS-SVM regressions [START_REF] Trinchero | Machine Learning and Uncertainty Quantification for Surrogate Models of Integrated Devices with a Large Number of Parameters[END_REF].

Specifically, each training set

D (i) = {(x l , y (i)
l )} L l=1 can be used together with the SVM and LS-SVM regression to train the corresponding surrogate model [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF], [START_REF] Suykens | Least Squares Support Vector Machines[END_REF], such that:

M (i) SV M/LS-SV M (x) = L l=1 β (i) l k (i) (x l , x) + b (i) , (1)
where

β (i)
l and b (i) are the regression coefficients and bias term estimated during the model training and k (i) (•, •) : R p×p → R is a scalar kernel function associated to the i-th output component.

Several kernel functions k(•, •) have been proposed in the literature. Hereafter in this paper, we will adopt the Gaussian radial basis function (RBF) kernel, which writes [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF]:

k(x i , x) = exp - ∥x i -x∥ 2 2σ 2 (2)
where σ 2 is the kernel hyper-parameter.

The SVM and the LS-SVM regression share the same dual space representation in (1). However, for the SVM regression the coefficients β (i) l and the bias terms b (i) l are estimated by minimizing the ε-insensitive loss function computed between the model predictions and training outputs, while for the LS-SVM regression the above unknowns are estimated as the ones minimizing a squared loss function [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF], [START_REF] Suykens | Least Squares Support Vector Machines[END_REF]. A Tikhonov regularizer and its corresponding hyperpameter are used by both methods to suppress over fitting and to improve their generalization on the test set, by minimizing the model variance and thus its sensitivity to the training set [START_REF] Ghojogh | The theory behind overfitting, cross validation, regularization, bagging, and boosting: Tutorial[END_REF].

Concerning the training algorithms, the SVM regression is built by numerically solving a quadratic optimization problem [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF], and thanks to the ε-insensitive norm, it can lead to a sparse solution in which some of the coefficients β

(i) l = 0.
On the other hand, the training of the LS-SVM regression can be done in a closed-form as the solution of a linear system of equations. Concerning the model hyperparameters, the SVM regression with Gaussian RBF kernel requires the tuning of 3 hyperparameters: the shape factor σ in (2), the width of the ε-insensitive zone and the regularizer hyperparameters. On the other hand, the LS-SVM regression with Gaussian RBF kernel requires the tuning of 2 hyperparameters: the shape factor σ in (2) and the regularizer hyperparameters. For each regression the above hyperparameters are tuned via cross-validation [START_REF] Ghojogh | The theory behind overfitting, cross validation, regularization, bagging, and boosting: Tutorial[END_REF]. The Statistics and Machine Learning Toolbox and the LS-SVMLab Toolbox version 1.8 [START_REF] Ls-Svmlab | version 1.8[END_REF] have been used for the SVM and the LS-SVM regression, respectively.

It is important to remark that this type of modeling scheme for vector-valued problem, based on single-output regressions, requires the training and tuning of the hyperparameters of D regression models, thus compromising its efficiency in regression problems with a large number of output components [START_REF] Manfredi | A data compression strategy for the efficient uncertainty quantification of time-domain circuit responses[END_REF].

C. Vector-Valued KRR

An alternative modeling approach consists of applying a vector-valued formulation of the kernel ridge regression (KRR) which allows to directly account for the vector-valued nature of the regression problem at hand [START_REF] Soleimani | Compressed complex-valued least squares support vector machine regression for modeling of the frequency-domain responses of electromagnetic structures[END_REF], [START_REF] Soleimani | Bridging the Gap Between Artificial Neural Networks and Kernel Regressions for Vector-Valued Problems in Microwave Applications[END_REF], [START_REF] Mauricio | Kernels for vectorvalued functions: A review[END_REF]. Specifically, the vector-valued KRR can be directly adopted to train a single vector-valued surrogate model M KRR : X → Y, which according to the represented theorem for vector-valued regression problems presented in [START_REF] Micchelli | On learning vector-valued functions[END_REF], writes:

M KRR (x) = L l=1 K(x, x l )c l , (3)
where A separable structure consisting of the product of scalar kernel function is considered for the matrix kernel function K(x, x ′ ) [START_REF] Mauricio | Kernels for vectorvalued functions: A review[END_REF], such that:

M KRR (x) = [M (
[K(x, x ′ )] [d,d ′ ] = k x (x, x ′ )k o (d, d ′ ), (4)
where k x and k o are scalar kernels acting independently on the input space (i.e., k x : X × X → R) and on the output dimensions (i.e., k o : {1, . . . , D}×{1, . . . , D} → R). Without loss of generality, the Gaussian RBF kernel in (2) will be used hereafter for the kernel k x and k o , respectively. For the considered model structure, the regression coefficients collected in the vectors c l can be estimated in a closedform via the solution of a discrete-time Sylvester equation (additional mathematical details are provided in [START_REF] Soleimani | Compressed complex-valued least squares support vector machine regression for modeling of the frequency-domain responses of electromagnetic structures[END_REF]), which writes:

K x CB + λC = Y, ( 5
)
where λ is the hyperparameter associated to the Tikhonov regularizer, K x is a L × L Gram matrix computed from the input samples {x l } L l=1 (i.e., Thanks to the properties of the Kronecker product, the above matrix equation can be vectorized and reformulated in terms of a standard linear system with (LD) equations in (LD) unknowns [START_REF] Horn | Topics in Matrix Analysis[END_REF] with a computational cost proportional to O(L 3 D 3 ) for a plain inversion or to O(KL 2 D 2 ), where K is the number of iteration, for a conjugate gradient descent algorithm [START_REF] Soleimani | Compressed complex-valued least squares support vector machine regression for modeling of the frequency-domain responses of electromagnetic structures[END_REF]. On the other hand, this paper will use a more efficient implementation based on a diagonalization procedure, inspired by [START_REF] Baldassarre | Multi-output learning via spectral filtering[END_REF], for which the computational cost for the model training reduces from O(L 3 D 3 ) to O(L 3 +D 3 +L 2 D+ LD 2 ), thus leading to beneficial effect on the training time when the product (LD) is large (additional details can be found in [START_REF] Soleimani | Efficient Implementation of the Vector-Valued Kernel Ridge Regression for the Uncertainty Quantification of the Scattering Parameters of a 2-GHz Low-Noise Amplifier[END_REF]).

[K x ] ij = k x (x i , x j )), B is a D × D Gram matrix computed on the output dimensions {1, . . . , D} (i.e., [B] ij = k o (d i , d j )), C = [c 1 , . . . , c L ] T ∈ R L×D
Different from modeling strategies based on scalar-output regressions in Sec. IV-B, the training of the proposed implementation of the vector-valued KRR requires the tuning of 3 hyperparameters only. Such tuning is done via a 3fold cross validation. Moreover, the vector-valued formulation allows to learn possible correlations in both the input and output dimensions, thus improving the model reliability and robustness to noise.

V. NUMERICAL RESULTS

The simulation setup described in Sec. III was used to generate the training and test sets used to construct and assess the accuracy of a surrogate models constructed via the SVM, LS-SVM and vector-valued KRR regressions presented in Sec. IV. These training and test sets were generated by randomly varying the considered 11 parameters defining the link, according to the range of variation discussed in Sec. II, via a latin hypercube sampling (LHS) scheme [START_REF] Mckay | A comparison of three methods for selecting values of input variables in the analysis of output from a computer code[END_REF]. For each set of parameters, the received differential voltage v d (t) was recorded and the eye aperture was computed according to the procedure in Sec. III. All performance figures in the present section were computed using 5000 test samples, i.e. 5000 different responses of the received voltage v d along with its corresponding eye plot and inner polygon. Also in this case, the input configurations of the test were generated by randomly varying the considered 11 parameters of the link according to the range of variation discussed in Sec. II via a LHS scheme.

Table I provides a quantitative picture of the accuracy of the considered three surrogate modeling approaches for an increasing number of training samples L from 60 to 300 (see first column). Both the eye apertures in mV , computed at the different location pairs labeled from one to seven in Fig. 2, and the eye width in ps corresponding to the eighth pair in the same figure are considered (second column). The remaining columns collect the average absolute errors computed as the mean absolute deviation of the eye apertures in mV and eye width in ps predicted by the considered surrogate models, and the corresponding ones obtained from the 5000 reference eye realizations in the test set. Bold text is used to highlight the lowest errors among the different modeling methods. This comparison clearly highlights the capability of all the considered surrogate models to learn the actual information provided by the training set. Among them, in most of the considered modeling scenarios, the vector-valued KRR shows an improved accuracy compared to the SVM and LS-SVM regression, thanks to its capability of learning possible correlation among the output components. Moreover, as shown in Fig. 3, the vector-valued KRR approach also offers a faster training time with respect to the other plain scalar-output kernel regressions, since its training CPU time seems to be less affected by the number of training samples (i.e., L). It is important to remark that the CPU times reported in Fig. 3 refer to the training time only, and do not include the computational cost relative to the generation of the training set.

The generation of 300 training samples required approximately 11 hours. This expensive data-collection phase must be carried out once. After the model training, its evaluation on the 5000 test samples required few seconds, while the corresponding SPICE simulations required 4 days.

Figure 4 shows all the possible eye apertures computed for the test samples (see gray cloud) superimposed by the best, median and worst profiles of the inner area computed using the reference responses (top panel, black lines) and, respectively, the proposed vector-valued KRR surrogate model (bottom panel, blue lines). The median curve corresponds to the 50% threshold between all the larger and the smaller eye apertures. This additional and less conventional performance figure stresses the flexibility of the proposed approach in possibly providing the information of the probability (e.g., in terms of a quantiles) associated to a given eye aperture.

Similarly, Fig. 5 plots three possible eye apertures defined by their corresponding polygons associated to three sets of uncertain parameters. The figure compares the prediction obtained using the proposed vector-valued KRR model trained with 300 training samples with the reference profiles, thus highlighting the very good accuracy of the proposed surrogate modeling approach.

As an additional and final validation, the eye quality is assessed by a single scalar quantity such as the eye aperture computed from the proposed polygonal approximation of the inner eye area. For the same test case, Fig. 6 compares the probability density functions (PDFs) computed from the same 5000-sample test set, used as reference, with the corresponding ones obtained from the surrogate model built via the LS-SVM regression, the SVM regression and the proposed vectorvalued KRR by using L = 300 samples. The three panels show the PDFs associated to the eye area (left panel), eye height computed in the region of the fourth polygonal edge pairs of Fig. 2,i again defined between pairs (8 r ) -(8 l ) (right panel). The eye area is computed as briefly outlined in the fourth item of the procedure described in the appendix. The above comparison highlights a good agreement among the PDFs predicted by the proposed surrogate models and the ones computed from 5000 test samples, especially on the tails of the distributions.

VI. CONCLUSIONS

The main novelty of the present paper is the stochastic timedomain mapping approach. For a given eye pattern, the inner eye is mapped using a suitable polygonal approximation. After an initial training stage, a parametric surrogate model is built, predicting the effect of system uncertainties on the shape of this polygon. This allows designers to effectively compute the metric of their choice or use eye-masks in a stochastic context. The procedure was demonstrated in the case of a flexible data link for wearable applications, with uncertain electrical and geometrical parameters, using machine learning regression to build the surrogate model. Data was collected by simulating the link in a differential configuration operating at 2 Gbps. Three types of regressions were used, showing that for the problem at hand, a vector formulation of the so-called kernel ridge regression is the preferable choice. A nice additional feature is that the latter naturally embeds the possible correlation which exists among the vertices of the eye polygon. This also has a beneficial effect in terms of training time, model reliability and robustness.

Stochastic mapping takes variability analysis one step further. It is of particular interest for applications where either the level of uncertainties or their impact is particularly high. These include flexible or structural electronics, ultra-low power systems, systems designed to operate in various hostile environments, but also very-low cost applications where high tolerances are a consequence of financial considerations.

APPENDIX

The proposed algorithm for the computation of the eye aperture through a polynomial approximation is described below. The procedure is outlined, including the essential steps using a metalanguage description and a MATLAB-like code. It is assumed that the sampled response of v d (t) is stored in the vector vd. It collects the values of the received differential voltage at the time samples stored in vector t. Also, due to the differential communication scheme, a zero-threshold (threshold=0 in the code) is assumed for computing the state (i.e., bit) transitions.

The main steps of the algorithm are: 1) the estimated crossing times are store in vector tz and are determined via linear interpolation:

>> vd = vd-threshold; >> iz = find(diff(sign(vd))); >> a = (vd(iz)-vd(iz-1))./(t(iz)-t(iz-1)); >> tz = (-vd(iz)+a. * t(iz))./a;

2) the time axis and the crossing times are wrapped into the unit interval defined by the bit time (TBIT); also, the jitter width (JITW) is computed by considering the largest and the smallest wrapped (i.e., normalized) transition times:

>> tzoffset = mod(tz(1),TBIT); >> tzw = mod(tz-tzoffset,TBIT); >> JLEFT = TBIT-min(tzw(tzw>TBIT/2)); >> JRIGHT = max(tzw(tzw<TBIT/2)); >> JITW = JLEFT+JRIGHT;

3) within the unit interval, a number of sampling points is set (e.g., POINTS=[0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8]) and the minimum and the maximum values of the received signal observed at these sampling points are computed, v d (t = (POINTS(kk)+(n-1)) * TBIT). In the above notation, the index kk varies from one to the max number of items in vector POINTS, and n ranges from one to the maximum number of bits composing the bitstream. The procedure in this step is suitably modified to avoid probing the differential signal in the jitter region. It is important to point out that the above mentioned max and min values define, for a given point in the unit interval, the position of the vertical eye height (e.g., see the blue circles in Fig. 2). Also, the green circles in on the left and on the right side of the eye correspond, in the x-axis, to JRIGHT/TBIT and 1-JLEFT/TBIT, respectively; 4) based on the inner polygonal shape built in the previous steps, different eye parameters (such as the eye width, height and area) can be readily computed. Among the previous parameters, the area can be computed via embedded functions such as polyarea in MATLAB or via the superposition of the 2D basic shapes (i.e., triangles or trapezoids) defined by the different vertices of the inner eye polygon like the one shown if Fig. 2.

Fig. 1 .

 1 Fig. 1. System under test: digital link for wearable applications. The nominal values of the equalizer and of the line lengths are: Re = 70 Ω, Ce = 3.2 pF L 1 + L 2 + L 3 = 33 cm,

[ 10 %

 10 ]. Tolerances of 1 % are assumed for the elements of the equalizer. Each eye diagram is computed for 4000 bits.

Fig. 2 .

 2 Fig. 2. Example eye diagram obtained for the data link of Fig. 1 for a given set of uncertain parameters. The plot also collects the information about the location of the polygonal edges associated to inner eye aperture, where the eye vertical apertures are labelled through the pairs from 1 to 7 and the eye width is indicated via the label pair number 8.

1 )

 1 KRR (x), . . . , M (D) KRR (x)] T is a vector collecting the model prediction for any x ∈ X , K(•, •) : R p×p → R D×D is a multi-output kernel matrix and c l = [c 1,l , . . . , c D,l ] T ∈ R D are column vectors collecting the regression unknowns.

  is a matrix collecting the model unknowns and Y = [y 1 , . . . , y l ] T is a L × D matrix associated to the training output.

Fig. 3 .

 3 Fig. 3. Comparison of the training time of the proposed regression techniques for an increasing number of training samples. The CPU times do not include the computational cost for the generation of the training set.

  .e., (4 u ) -(4 d ) (central panel) and eye width,

Fig. 4 .

 4 Fig.4. Data link analysis carried out by computing the best, the worst (dashed and solid thin lines) and the median (thick) inner eye aperture. The above eye apertures superimposed to the complete set of eye diagrams obtained for the test set (light gray cloud). The blue and the green clouds of circles play the same role of the corresponding colored circles in Fig.2. Top panel: reference; bottom panel: vector-valued KRR.

Fig. 5 .

 5 Fig. 5. Eye aperture computed for three sets of uncertain data link parameters. The eye opening polygons obtained by the proposed vector-valued KRR model are compared to the reference ones.

Fig. 6 .

 6 Fig. 6. Comparison between the PDFs of the eye area, height and width computed from the predictions of a surrogate model built via the LS-SVM regression, the LS-SVM regression and the vector-valued KRR with L = 300 training samples and the corresponding ones calculated on 5000 test samples.

TABLE I ABSOLUTE

 I AVERAGE ERROR COMPUTED FROM THE PREDICTIONS OF THE SURROGATE MODELS BUILT VIA THE REGRESSION TECHNIQUES PRESENTED IN SEC. IV FOR AN INCREASING NUMBER TRAINING SAMPLE (I.E., L = 60, 150, AND 300) ON A TEST SET WITH 5000 EYE REALIZATIONS.

		Methods	SVM	LS-SVM	Vector-Valued KRR
	L	pairs	Ave Error	Ave Error	Ave Error
		1u-1 d (mV )	6.21	6.18	5.97
		2u-2 d (mV )	7.49	7.25	6.73
		3u-3 d (mV )	6.65	6.68	6.91
	60	4u-4 d (mV )	7.49	7.17	7.29
		5u-5 d (mV )	10.13	6.80	6.68
		6u-6 d (mV )	4.03	3.83	3.79
		7u-7 d (mV )	4.25	4.07	4.03
		8r -8 l (ps)	3.62	3.22	3.34
		1u-1 d (mV )	4.73	4.61	4.52
		2u-2 d (mV )	4.16	4.27	4.20
		3u-3 d (mV )	4.93	4.92	4.99
	150	4u-4 d (mV )	9.99	4.86	4.75
		5u-5 d (mV )	4.22	3.78	3.79
		6u-6 d (mV )	3.06	2.72	2.73
		7u-7 d (mV )	2.72	2.66	2.65
		8r -8 l (ps)	2.67	2.63	2.67
		1u-1 d (mV )	3.50	3.44	3.51
		2u-2 d (mV )	3.25	3.19	3.13
		3u-3 d (mV )	3.75	3.71	3.72
	300	4u-4 d (mV )	4.05	3.96	4.03
		5u-5 d (mV )	3.08	2.96	2.99
		6u-6 d (mV )	2.38	2.40	2.33
		7u-7 d (mV )	2.32	2.31	2.20
		8r -8 l (ps)	2.23	2.15	2.12