Minimum-phase calibration of sampling oscilloscopes
Andrew Dienstfrey, Paul D. Hale, Darryl A. Keenan, Tracy S. Clement, Dylan Williams
We describe an algorithm for determining the minimum phase of a linear, time-invariant response function from its magnitude. The procedure is based on Kramers-Kronig relations in combination with auxiliary direct measurements of the desired phase response. We demonstrate that truncation of the Hilbert transform operator required by the theory gives rise to large errors in estimated phase. Although large in magnitude, we claim that these errors may be approximated using a small number of specialized basis functions. As an example, we obtain a minimum-phase calibration of a digital high-speed sampling oscilloscope in the frequency-domain. This result rests on data obtained by the electro-optic sampling system (EOS) developed at the National Institute of Standards and Technology in combination with a microwave swept-sine calibration procedure. The EOS calibration yields magnitude and phase information over a broad bandwidth, yet has degraded uncertainty estimates at lower frequencies, in this example from DC to less than one GHz. The swept-sine procedure returns only the magnitude of the oscilloscope response function, yet may be performed on a fine frequency grid from DC to several GHz. Applying the algorithm described in this paper results in a minimum-phase calibration that spans frequencies from DC to 110 GHz, and is traceable to fundamental units. The validity of the minimum-phase character of the oscilloscope response function over a large bandwidth is determined as part of our analysis. A full uncertainty analysis is provided.
IEEE Transactions on Microwave Theory and Techniques
, Hale, P.
, Keenan, D.
, Clement, T.
and Williams, D.
Minimum-phase calibration of sampling oscilloscopes, IEEE Transactions on Microwave Theory and Techniques, [online], https://doi.org/10.1109/TMTT.2006.879167
(Accessed December 7, 2023)