S-Parameter Sampling in the Frequency Domain and its Time-Domain Response

Abstract

S-parameter characterization of high-speed interconnect components is prone to numerical, modeling, and/or measurement error, due in part to the sampled nature of the data. This is a problem in modern package design where extensive signal integrity simulations are required to validate a system's performance. This paper presents a new method of treating sampled S-parameter data where a criterion is developed to clearly state the minimum number of S-parameter frequency points required to adequately represent an analog physical system in the frequency domain. This criterion is akin to the Nyquist principle when sampling in the time domain. Based on this principle, the proper time domain representation of S-parameters can be obtained using the inverse Fast Fourier Transform (IFFT). However, in order to use the IFFT, the bilinear transformation is applied to the vector fitted Sparameters to place them in the z-domain. From the z-domain, the proper discrete impulse response is obtained. A lower bound, based on the discrete Heisenberg principle, is also offered to deal with frequency-time resolution of the S-parameters. The proposed method is successfully tested in measured and simulated S-parameter data. The usefulness of this method is to accurately represent the time domain behavior of the physical system S-parameter data, i.e. time delay causality, and therefore facilitates the applications of well-developed digital signal processing (DSP) techniques in S-parameter sampled data.

Numerically Efficient Miniaturization-Oriented Optimization of an Ultra-Wideband Spline-Parameterized Antenna

“…The reflection shown in Fig. 8(a) has been obtained through conversion of the window-corrected timedomain signal to the frequency spectrum [51], [52]. For more information on the approach see [49]- [52].…”

Experimental radar data for monitoring brain atrophy progression

“…The data can be transformed to a time-domain using transformation algorithms such as IFFT [4] . It can be loaded using MATLAB, Python, or any other language.…”

Analysis of Analog Sampled S-Parameters Data Using DSP Techniques

“…Polynomial periodic continuations [15,16] in conjunction with the Hibert Transform have been also used to check for causality. However, most of these algorithms use S-parameter sampled data without considering the restrictions posed by the sampling frequency condition [17], leading to false positives, i.e., declaring causal when the S-parameters are not or vice versa.…”

“…In a previous paper, the authors presented a technique emphasizing a sampling theorem in the frequency domain, which sets the minimum separation of two frequency point intervals to extract meaningful time domain information from S-parameter data [17]. Then vector fitting and bilinear transformation were used to appropriately place the analog Sparameter data in the z-domain obtaining a true impulse response.…”

“…In this paper, we start with the analysis of Sparameters using concepts from circuit theory. This allows us to re-derive the sampling criterion in the frequency domain discussed by the authors in [17] while emphasizing the analog nature of the S-parameters. The VF algorithm [18][19][20][21][22][23][24] facilitates the representation of S-parameters as ratio of two polynomials in the Laplace domain, aiding the conversion, through the bilinear transformation, of this ratio, into an equivalent ratio in the z-domain of an infinite impulse response (IIR) filter [25][26][27], where one can easily take the IFFT.…”