1076.1 Table-driven Modeling Subcommittee November 8, 2010 – 8am PST – 5pm CET

1. 1076.1 Table-driven Modeling SubcommitteeNovember 8, 2010 – 8am PST – 5pm CET Joachim Haase

2. Agenda • Call to order • Volunteer for minutes • Approval of agenda • Review and approve minutes from the last meeting • Results of review and vote on the requirements' document • Discussion on combinations of interpolation and extrapolation methods • Next meeting • AOB • Adjourn

3. Approve minutes from October 11, 2010 meeting • Minutes available in the Table-driven modeling member-only areahttp://www.eda.org/twiki/bin/view.cgi/P10761/TDMMtgMinutesOct1110 • Thanks to Alain for preparing the minutes • Review of IEEE Patent policyhttp://www.eda.org/twiki/pub/P10761/WebHome/pat-slideset.pdf

4. Results of review and vote on the requirements' document • Eligible voters: 5 Votes returned by eligible voters: 4 Votes received by ineligible voters: 4 • The reviewers/voters were: David Bishop, John Shields, Subramanian Sivaramakrishnan, David Smith, Alain Vachoux, Arpad Muranyi, Joachim Haase, Ernst Christen • All returned votes were affirmative, some with comments. With this result the requirements for Table Driven Modeling have been approved. • Results of vote on requirements‘ document are availablehttp://www.eda.org/twiki/pub/P10761/ProjectTableDrivenModeling/TDM_Requirements_feedback.txt

5. Discussion on interpolation and extrapolation methods • Material for discussion of the 1-dimensional case preparedhttp://www.eda.org/twiki/pub/P10761/ProjectTableDrivenModeling/combinations_of_interpolation_and_extrapolation_methods.pdf • Starting point • The material summarizes combinations of different interpolation and extrapolation method. • The extrapolation case „ERROR“ (violation of the interpolation rangex0 x  xm ) on the left or on the right is not explicitely handled in the following.

6. Interpolation methods (x0 x  xm) 1see also „$table_model in Verilog-AMS“ by Ernst Christenhttp://www.eda.org/twiki/pub/P10761/ProjectTableDrivenModeling/VerilogControls.txt

7. Extrapolation on the left (x<x0) It seems that there are not possibilities to handle extrapolation on the left and on the right in a different way in Modelica and Matlab built-in functions.

8. Extrapolation on the right (xm<x) It seems that there are not possibilities to handle extrapolation on the left and on the right in a different way in Modelica and Matlab built-in functions.

9. Extrapolation in Verilog-AMS (LRM 2.3 – Section 9.20.2) • The constant extrapolation method returns the table endpoint value. • Linear extrapolation extends linearly to the requested point from the endpoint using a slope consistent with the selected interpolation method. • The user may also disable extrapolation by choosing the error extrapolation method. With this method, an extrapolation error is reported if the $table_modelfunction is requested to evaluate a point beyond the interpolation region.

10. Example (1) Verilog-AMS like with LastTwoPoints

11. Example (2)

12. Example (3)

13. Example (4)

14. Interpolation in Verilog-AMS • Verilog-AMS LRM 2.3 ?

15. Cubic spline with constant extrapolation => Complete spline1 Values are given. f‘(0) = 0 f‘(3) = -0.7421+2*(-0.7421 )+3*0.7942 = 0 First order derivatives at the endpoints are set to 0.0. 1 as used in Verilog-AMS

16. Cubic spline with linear extrapolation => Natural spline1 Method requiresthe values. f‘‘(0) = 0 f‘‘(3) = 2*(-0.5141)+6* 0.1714 = 0 Second order derivatives at the endpoints are zero. First order derivatives at end points f‘(0) = 0.9920 and f‘(3-0) = -0.9395determine the slope of the linear continuation. 1 as used in Verilog-AMS

17. Cubic spline interpolation – Not-a-knot spline1 Method requiresequal values. Third order derivatives at the first and last inner point are continuous. f‘‘‘(1-0)= -0.0104 = f‘‘‘(1) = -0.0104 f‘‘‘(2-0) = -0.0104 = f‘‘‘(2) = -0.0104 1 as used in Matlab

18. Problem – Piecewise linear extrapolation • Alternatives • Using the last two points (this was done in the prepared material). • Set second order derivative to zero at the end point for the interpolation and extrapolation interval. Requires continuous first order derivatives at the end point (this is done in Verilog-AMS). 1 2

19. Problem – Periodic Extrapolation (1) If periodic extrapolation is used on left or right values of the first and last point should be equal (y0 = ym).

20. Problem – Periodic extrapolation (2) • Special requirements are necessary for quadratic and cubic spline interpolation if one or both extrapolation methods are „Periodic“Proposal: • If slopes of the first and the last interval are unequal 0.0 and both are positive or negative set second order derivative to 0.0 at both end points. • Otherwise set first order derivatives at the end points to 0.0. 1 2

21. Problem – Interpolation with stepwise function Special rules for handling extrapolation on the rightif interpolation with with stepwise function is used.

22. Problem – Using Matlab‘s extrapval method Using Matlab‘s extrapval methods leads often todiscontinuities at the end points.

23. Conditions to specify the interpolation algorithms Number of data points • Linear interpolation requires at least two given data points. • Quadratic spline interpolation requires at least three given data points. • Cubic spline interpolation requires at least four given data points. Derivatives at end points • Quadratic spline interpolation requires that the first order derivativef’(xo) at x0 is given. • Cubic interpolation with complete splines requires the knowledge of the first order derivatives f’(xo) and f’(xm) at the first and last given data point. • Cubic interpolation with natural splines requires that the second order derivatives f’’(xo) and f’’(xm) at the first and last given data point are zero. General requirement • The interpolation shall meet the given data points.

24. Proposals to handle interpolation and extrapolation (1) • At least two data point should be given. • If less than three data points are given and quadratic spline interpolation is required, linear interpolation is used. • If less than four data points are given and quadratic spline interpolation is required, quadratic spline interpolation is used if three data points are available otherwise linear interpolation will be used. • The 1-dimensional table modelling function requires • the argument for which the function value has to be determined • arrays x and y with given values, • specification of the interpolation method, extrapolation method on the left and the right, • a value for continuation on the left if “Constant extrapolation with parameter for left value” extrapolation is required • a value for continuation on the right if “Constant extrapolation with parameter for right value” extrapolation is required

25. Proposals to handle interpolation and extrapolation (2) • If one of the extrapolation methods is “Periodic extrapolation” then first and last dependent values of the given knots must be equal. That means, y0 = ym. • If the interpolation method is • “Value of the nearest knot” or • interpolation with the “Step-wise function” then the extrapolation methods • “Description of the first (or last) last interpolation interval is used” • “Piecewise linear extrapolation” deliver the y value of the first or the last knot respectively.

26. Discussion (1) • Choice of interpolation and extrapolation methods : • Shall cubic spline interpolation with not-a-knot condition be supported? • Shall extrapolation with Matlab‘s „extrapval“ method be supported? • How linear extrapolation shall be carried out? • Are the proposals on the previous two slides accepted? • Additional conditions for cubic and quadratic spline interpolation: • Which condition shall be used in the subsequent extrapolation cases • “Periodic” (see proposal on one of the previous slides) • “Description of the first/last interpolation interval is used”(proposal: same condition as for linear extrapolation) • “Error”(proposal: same condition as for linear extrapolation)

27. Discussion (2) • Questionnaire concerning VHDL-AMS • Data representation • Specification of interpolation and extrapolation methods • Default methods • Interface of the interpolation function in the 1-dimensional case • Error handling • Handling of discontinuities in time domain simulation • Can increasing x values be assumed

28. Action items • Update of the requirements‘ document • Summarize handling of interpolation and extrapolation for theone-dimensional case • Proposal for the interface of a real-valued function for the one-dimensional case

29. Next meeting • Monday, December 6, 2010 – 8am PST (UTC-8) = 5pm CET (UTC+1)