Spectrally Accurate Causality Enforcement Using SVD-Based Fourier Continuations for High-Speed Digital Interconnects

We introduce an accurate and robust technique for accessing the causality of network transfer functions given in the form of band-limited discrete frequency responses. These transfer functions are commonly used to represent the electrical response of high-speed digital interconnects used on chip and in electronic package assemblies. In some cases, small errors in the model development lead to noncausal behavior that does not accurately represent the electrical response and may lead to a lack of convergence in simulations that utilize these models. The approach is based on Hilbert transform relations or Kramers-Krönig dispersion relations and a construction of causal Fourier continuations using a regularized singular value decomposition method. Given a transfer function, nonperiodic in general, this procedure constructs highly accurate Fourier series approximations on the given frequency interval by allowing the function to be periodic in an extended domain. The causality dispersion relations are enforced spectrally and exactly. This eliminates the necessity of approximating the transfer function behavior at infinity and explicit computation of the Hilbert transform. We perform the error analysis of the method and take into account a possible presence of a noise or approximation errors in data. The developed error estimates can be used in verifying the causality of the given data. The performance of the method is tested on several analytic and simulated examples that demonstrate an excellent accuracy and reliability of the proposed technique in agreement with the obtained error estimates. The method is capable of detecting very small localized causality violations with amplitudes close to the machine precision.