Causality Verification Using Polynomial Periodic Continuations for Electrical Interconnects

We present a method for checking causality of band-limited tabulated frequency responses. The approach is based on Kramers-Krönig relations and construction of periodic polynomial continuations. Kramers-Krönig relations, also known as dispersion relations, represent the fact that real and imaginary parts of a causal function form a Hilbert transform pair. The Hilbert transform is defined on an infinite domain, while, in practice, discrete values of transfer functions that represent high-speed interconnects are available only on a finite frequency interval. Truncating the computational domain or approximating the behavior of the transfer function at infinity causes significant errors at the boundary of the given frequency band. The proposed approach constructs a periodic polynomial continuation of the transfer function that is defined by raw frequency responses on the original frequency interval and by a polynomial in the extended domain, and requires the continuation to be periodic on a wider domain of a finite length and smooth at the boundary. The dispersion relations are computed spectrally using fast Fourier transform and inverse fast Fourier transform routines applied to periodic continuations. The technique does not require the knowledge or approximation of the transfer function behavior at infinity. The method significantly reduces the boundary artifacts that are due to the lack of out-of-band frequency responses, and is capable of detecting small, smooth causality violations. We perform the error analysis of the method and show that its accuracy and sensitivity depend on the smoothness and accuracy of data and a polynomial continuation. The method can be used to verify and enforce causality before the frequency responses are employed for macromodeling. The performance of the method is tested on several analytic and simulated examples.