On Checking Causality of Tabulated $S$-Parameters

Abstract

The existing formulation for the causality testing of tabulated S-parameters can be oversensitive to noncausality, in that it can report causality violations for data that may not result in an inaccurate transient simulation. Existing numerical procedures that implement the present formulation can result in false-positive test outcomes (i.e., causal data being declared as noncausal). A mechanism to reduce the oversensitivity in the formulation and new testing procedures that minimize false positives are proposed. Oversensitivity is reduced significantly if the target fitting error (of the macromodel to tabulated data) is chosen as the minimum amplitude of the noncausality to be detected. Sources of false-positive outcomes are identified, and four procedures that minimize false positives are proposed. These procedures differ in resolution, accuracy, and computational requirements. Numerical results demonstrating the proposed modifications to the formulation and numerical results comparing and contrasting the different procedures are presented.

Reliable detection of causality violations in tabulated scattering parameters through filtered dispersion relations

“…5 we reported the lowest fitting error we could achieve when applying VF to the samples. As in previous example, the causality error from the proposed method is closer to the VF error than the error returned by previous techniques [4], [7], [9].…”

“…3. We notice that the proposed causality error (13) is closer to the VF error than the causality error provided by previous techniques [4], [7], [9]. This happens because these techniques use a generalized form of dispersion relations with anchor points.…”

“…By verifying if Kramers-Krönig relations are respected, one can test the causality of a given set of frequency samples [4]- [7]. This practice is useful since non-causal frequency samples can impede their conversion into an accurate model for transient simulations [2], even when this step is performed with robust macromodeling techniques like Vector Fitting [8].…”

“…The resulting truncation error can severely bias the verification of causality, unless rigorously estimated as in [4], [7], [9]. These works, however, rely upon a generalized form of Kramers-Krönig relations which, although powerful, is not straightforward to implement.…”

“…Thanks to the presence of a filter, the new relations allow for a precise control of truncation error, and lead to a robust method to test causality. Compared to state-of-the-art solutions [4], [7], the new method is easier to implement, more intuitive, and provides a better estimate of the detected violation. This information is useful to quantify the impact of the violation on subsequent modeling steps, as numerical examples will demonstrate.…”