This paper presents an evaluation method for a 1 mm coaxial calibration kit that can be used from DC to 110 GHz. The analytical model for the calibration kit was revisited and verified by comparing it with the electromagnetic High-Frequency Structure Simulator (HFSS). We also proposed a method to measure or appropriately estimate the physical parameters of the analytic model. This approach calculates the uncertainty based on the physical parameters, so that the uncertainty can be appropriately propagated to different measured quantities based on the covariance between all frequencies, including the real and imaginary parts. To verify the proposed method, a commercially available 1 mm calibration kit was evaluated, and the impedance of a device under test was measured using the evaluated kit. We compared the measured results with those of the National Institute of Standards and Technology (NIST) and confirmed that they agreed well with each other within the uncertainty. Additionally, the multiple reflections caused by the impedance mismatch between the signal source and the instrument was corrected, and its calibrated uncertainty was obtained in the time domain. Thus, the uncertainty of the impedance measurement in the frequency domain was properly propagated to the time domain.
A novel method is proposed for the calibration of channel mismatch in a time-interleaved real-time digital oscilloscope (RTDO). A simple simultaneous equation is derived from the Fourier transform of the time-interleaved signals. Thus, it only requires a transfer function of time-interleaved ADCs (TIADCs), while most of previous works have employed additional filters. A measurement method for the transfer function of a commercial RTDO is also proposed. The accuracy of the calibration method is determined by the noise produced after the interleaving process. To validate the proposed method, we measure two-tone signals using a commercial RTDO. The calibrated results clearly show signals at spurious frequencies are substantially reduced.
Impairments of analog-to-digital converters (ADCs) used in digital real-time oscilloscopes (DRTO) have caused inevitable signal distortions in measurements. To calibrate these errors with traceability, we propose a novel method that consists of two steps. First, each transfer function of the ADCs is measured using pulse trains from a photodiode calibrated up to 110 GHz. Each data set of the ADCs is superimposed to convert the repetitive pulse to a single pulse to solve the under-sampling problem of the separated data depending on each ADC. Then, the signals of the device under test (DUT) are also separated and superimposed depending on the ADCs, and they are calibrated in the frequency domain based on the measured transfer functions. After a calibration process, the data set is reconverted to the time domain to achieve traceable calibration. To verify our method, we have measured the output of another 70 GHz photodiode with a calibrated DRTO. In terms of results, time-interleaved errors are suppressed by more than 24 dB up to the bandwidth of the DRTO.
In this paper, the propagation of uncertainty on a nonlinear measurement model is presented using a higher-order Taylor series. As the derived formula is based on a Taylor series, it is necessary to compute the partial derivatives of the nonlinear measurement model and the correlation among the various products of the input variables. To simplify the approximation of this formula, most previous studies assumed that the input variables follow independent Gaussian distributions. However, in this study, we generate multivariate random variables based on copulas and obtain the covariances among the products of various input variables. By applying the derived formula to various cases regardless of the error distribution, we obtained the results that coincide with those of a Monte-Carlo simulation. To apply high-order Taylor expansion, the nonlinear measurement model should be continuous within the range of the input variables to allow for differentiation, and be an analytic function in order to be represented by a power series. This approach may replace some time-consuming Monte-Carlo simulations by choosing the appropriate order of the Taylor series, and can be used to check the linearity of the uncertainty.
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