Low- and High-Frequency Extrapolation of Band-Limited Frequency Responses to Extract Delay Causal Time Responses

Abstract

Frequency responses, such as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">S</i> parameters, can be utilized to analyze the time-domain characteristics of circuit components and systems using the hybrid time–frequency-domain approach [1]. However, most frequency responses have a limited frequency range and no low-frequency data due to the limitations of the bandwidth of instruments and the computational resource of simulators. Using these frequency responses, the accuracy and delay causality of the extracted time responses cannot be guaranteed, occurring an inaccurate time-domain analysis. Therefore, this article presents a low- and high-frequency extrapolation method to extract the delay causal time responses from the band-limited frequency responses. Unlike previous methods, the proposed extrapolation method can accurately estimate a propagation delay for improving the accuracy of the overall extrapolated frequency response, even if the applied band-limited frequency response has no low-frequency data. Moreover, in this method, the coefficients of both the low- and high-frequency extrapolation functions can be extracted precisely without an intensive iteration. To verify the proposed method, a two-port network using a multisection band-stop filter and a four-port differential channel using coupled transmission lines were designed. In both cases, the proposed method had a higher accuracy than previous methods, such as the delay-based macromodeling scheme and noniterative extrapolation for the high-frequency range. Furthermore, the computational efficiency of the proposed method was also higher than the delay-based macromodeling while maintaining a similar efficiency as with the noniterative extrapolation method.