Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

Minimum-Phase Calibration of Sampling Oscilloscopes



Andrew M. Dienstfrey, Paul D. Hale, Darryl A. Keenan, Tracy S. Clement, Dylan F. 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 gives rise to large errors in estimated phase, but that these errors may be approximated using a small number of basis functions. As an example, we obtain a minimum-phase calibration of a sampling oscilloscope in the frequency domain. This result rests on data obtained by an electrooptic sampling (EOS) technique in combination with a swept-sine calibration procedure. The EOS technique yields magnitude and phase information over a broad bandwidth, yet has degraded uncertainty estimates from dc to approximately 1 GHz. The swept-sine procedure returns only the magnitude of the oscilloscope response function, yet may be performed on a fine frequency grid from about 1 MHz to several gigahertz. The resulting minimum-phase calibration 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 at frequencies common to both measurements is determined as part of our analysis. A full uncertainty analysis is provided.
IEEE Transactions on Microwave Theory and Techniques


Hilbert transform, Kramers Kronig relation, linear response functions, minimum phase, mismatch correction, oscilloscopes


Dienstfrey, A. , Hale, P. , Keenan, D. , Clement, T. and Williams, D. (2006), Minimum-Phase Calibration of Sampling Oscilloscopes, IEEE Transactions on Microwave Theory and Techniques, [online], (Accessed June 21, 2024)


If you have any questions about this publication or are having problems accessing it, please contact

Created August 15, 2006, Updated November 10, 2018