Abstract-We introduce an accurate and robust technique for accessing causality of network transfer functions given in the form of bandlimited 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 non-causal 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 KramersKrönig dispersion relations and a construction of causal Fourier continuations using a regularized singular value decomposition (SVD) method. Given a transfer function, non-periodic 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 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.
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.
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