JOINT MATHEMATICS COLLOQUIUMUNIVERSITY OF IDAHOWASHINGTON STATE UNIVERSITY |
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Abstract |
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Causality verification and enforcement is of great importance for performance evaluation of high speed digital electrical interconnects. We present two techniques based on Kramers-Kronig dispersion relations, also called Hilbert transform relations, and construction of causal periodic continuations. The first method employes periodic polynomial continuations, while the second approach constructs Fourier continuations using a regularized singular value decomposition (SVD) method. Given a transfer function sampled on a bandlimited frequency interval, non-periodic in general, both approaches construct an accurate approximation on the given frequency interval by allowing the function to be periodic on an extended domain. This allows one to significantly reduce (for polynomial continuations) or even completely remove (for Fourier continuations) boundary artifacts that are due to the bandlimited nature of frequency responses. Using periodic continuations eliminates the necessity of approximating the transfer function behavior at infinity in order to compute Hilbert transform. We perform the error analysis of the methods and take into account a possible presence of a noise or approximation errors in data. The developed error estimates can be used in causality characterization of the given data. The methods can be employed to verify and enforce causality before the frequency responses are used for macromodeling. The performance of the methods is demonstrated on several analytic and simulated examples.
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