4.3.3 Interpolation of Sampled Input Signals

The input signals for a transient simulation are specified at instances t_k. For in between simulation steps t_n with t_{k - 1} < t_n < t_k the input signals have to be interpolated appropriately. As has been mentioned in Section 4.3.1 it is necessary to exactly match the instances. This increases the total number of required time steps. Therefore the number of specified instances should be as small as possible to take full advantage of the quadratic time step estimation. To reduce the number of specified instances necessary for a sufficient representation of nonlinear input signals a higher order interpolation method can be used.

In MINIMOS-NT the Akima-interpolation [36] is used for input signal interpolation. This is a sub-spline interpolation method which is built from continuously differentiable, piecewise third order polynomials. Since it is a local interpolation there is no need to solve large equation systems. Therefore it can be computed very efficiently and has minimum memory requirements. An other important advantage of this method is that it does not lead to oscillations in the interpolated input signal.

At each specified instance the first derivative is estimated from the data at this point and from two neighboring points on each side. Using the input data and the estimated first derivative at the end points of each interval in between values can be interpolated by a third order polynomial.

Figure 4.10: Extrapolation of additional points for non-differentiable input signals.

Additional to the time and value for each instance information has to be provided whether the signal is differentiable at the instance. Between two instances at which the signal is differentiable a third order interpolation is used. For intervals where the input signal is differentiable only at one end point the interpolation is reduced to second order. When the input signal is not differentiable at both instances a linear interpolation is performed which enables the exact representation of piecewise linear input signals.

At the first and last specified data points and at points where the interpolated function is not differentiable additional instances have to be estimated from the input data to calculate the estimated first derivative. For these intervals the interpolation function reduces to second order (see Fig. 4.10).

The calculation of the coefficients of the interpolation polynomial is described in detail in Appendix B.