Available at http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=6871413&isnumber=6999020 Humphreys, D.A.; Harris, P.M.; Rodriguez-Higuero, M.; Mubarak, F.A.; Dongsheng Zhao; Ojasalo, K., "Principal Component Compression Method for Covariance Matrices Used for Uncertainty Propagation," Instrumentation and Measurement, IEEE Transactions on , vol.64, no.2, pp.356,365, Feb. 2015. doi: 10.1109/TIM.2014.2340640. Crown Copyright 2014. 1 Abstract — We investigate a principal component analysis approach for compressing the covariance matrices derived from real-time and sampling oscilloscope measurements. The objective of reducing the data storage requirements to scale proportional to the trace length ࢔ rather than ࢔ ૛ is achieved, making the approach practical for representing results and uncertainties in either the time or frequency domain. Simulation results indicate that the covariance matrices can be represented in a compact form with negligible error. Mathematical manipulation of the compressed matrix can be achieved without the need to reconstruct the full covariance matrix. We have demonstrated compression of datasets containing up to 10 000 complex frequency components. Index Terms — Covariance matrix, Fourier transforms, frequency-domain measurements, measurement uncertainty, principal component analysis, time-domain measurements, timing jitter, uncertainty propagation, timebase drift. I. INTRODUCTION AVEFORM measurements are important for many applications and the traceable underpinning is provided by Electro-Optic Sampling systems in national measurement institutes [1]. In the past, parametric measures, such as transition duration [2], have been sufficient for many applications. Recently, the primary standard systems and their associated uncertainty budgets have improved to allow full- waveform characterization [3] and [4]. Measurement uncertainties of a waveform in the time or frequency domain show structure that can be captured using a This work was supported in part by EMRP under the Joint Research Project IND16 “Metrology for ultrafast electronics and high-speed communications,” and by the authors’ respective national metrology programs. The EMRP is jointly funded by the EMRP participating countries within Euramet and the European Union D A. Humphreys and P M. Harris are with the National Physical Laboratory, Teddington, TW11 0LW, UK (david.humphreys@npl.co.uk and peter.harris@npl.co.uk). M Rodríguez-Higuero is with INTA (Instituto Nacional de Tecnica Aeroespacial), Spain (rodriguezm@inta.es ). F Mubarak and D Zhao are with VSL, Delft, The Netherlands (FMubarak@vsl.nl and DZhao@vsl.nl ). Kari Ojasalo is with MIKES (Centre for metrology and accreditation), 02151 Espoo, Finland (Kari.Ojasalo@mikes.fi ). covariance matrix approach [5], which involves the matrix storage and manipulation of the variances and covariances that summarize the uncertainties associated with vector-valued quantities as outlined in the Guide to the Expression of Uncertainties in Measurement [6]. The covariance matrix approach provides a representation of the measurement uncertainties that can be expressed in either the time-domain or the frequency-domain and transformed between them. This is particularly important where different parts of the measurement are carried out with time-domain or frequency-domain instrumentation. An example would be characterizing an oscilloscope: The main measurement is made in the time-domain but the S-parameter measurements, required for impedance match correction, are carried out in the frequency domain. Both these measurements will contribute to the final uncertainty budget for the calibration. The frequency domain covariance matrix is formulated in terms of the real and imaginary components so that all the correlation relationships are maintained. The difficulty with this approach is that the size of the covariance matrix grows as the square of the trace-length and also that the number of measured waveforms required to generate a full-rank covariance matrix also increases with the trace length. Such a matrix would be suitable for Monte-Carlo propagation of the uncertainties using a Cholesky decomposition method [7]. This work is part of a larger Euramet Joint Research Project whose aim is to provide improved traceability and full waveform uncertainties for high-speed and high-frequency measurement instrumentation [8]. Here the motivation is to provide a compact and robust waveform uncertainty algorithm that can be disseminated and used in industry, with the aim of allowing the instrument calibration and uncertainties to be transferred to the user’s measurements. To achieve this goal, the covariance matrix must be stored in a format that grows proportional to the trace-length ሺሻ, rather than the square of the trace-length ሺ ଶ ሻ, and the ability to mathematically manipulate the waveform and its uncertainty is a prerequisite. We use the arrangement of real and imaginary covariance components developed in previous work that shows the structure of the real and imaginary components [9] - [11]. In Principal Component Compression Method for Covariance Matrices Used for Uncertainty Propagation David A. Humphreys, Senior Member, IEEE, Peter M. Harris, Manuel Rodríguez-Higuero, Faisal Mubarak, Member, IEEE, Dongsheng Zhao, Member, IEEE, and Kari Ojasalo W