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3.2.4 Characterization of High Speed Oscilloscopes

C. M. Wang and K. J. Coakley
Statistical Engineering Division, ITL

P. D. Hale and T. S. Clement
Optoelectronics Division, EEELD. C. DeGroot
Radio-Frequency Technology Division, EEEL

The frequency-dependent phase and magnitude response of a device are required to determine its time-domain response and are used in optoelectronic device metrology, nonlinear device metrology, and high speed digital circuit design. Frequency-domain network analyzers measure device amplitude and phase responses relative to a reference tone, but cannot measure the total phase relationship of the frequency components in a broadband waveform. Here time-domain methods are required. We are investigating the use of a high speed sampling oscilloscope, which has been calibrated using the nose-to-nose method, to measure the phase response of fast optical detectors and electrical comb generators.

Actual measurements of the impulse response of a sampling oscilloscope are affected by various non-ideal properties of the hardware. Calibration of an oscilloscope's frequency response requires estimation and correction of these effects, which include distortion, drift, and jitter components in the time-base, and mismatch. Time-base distortion (TBD) is a deterministic error in the delay generator that triggers a sample. Drift and jitter are random variations in the sample time which occur on a long and short time scale relative to one complete sweep of the display. Many waveforms must be averaged to achieve a low noise level. Before averaging, the waveforms are corrected for drift. Relative drifts are estimated from cross-correlation analysis of all distinct pairs of signals. A manuscript on alignment of noisy signals has been submitted to IEEE Transactions on Instrumentation and Measurement.

Error due to TBD must be compensated to give good corrections above 15 GHz. We developed an efficient least-squares algorithm for estimation TBD. The method requires measurements of sinusoidal signals at multiple phases and frequencies. It can accurately estimate the order of the harmonic model that is used to account for the amplitude nonlinearity of the sampler. This work appears in the 1999 December issue of IEEE Transactions on Instrumentation and Measurement.

To completely characterize data acquisition channels, the additive and jitter errors must also be estimated. The additive and jitter errors are used in the weighting of the TBD estimation procedure. It's found, from a simulation study, that the reduction in root-mean-square error of TBD estimate by using the appropriate weighting is about 20%. Therefore, it's important to obtain an accurate estimate of additive and jitter errors. The signal model is given by

\begin{displaymath}y_i = g(t_i + \tau_i) + \epsilon_i

where yi the measured signal at time i is a function of actual time of sampling plus the additive noise. The actual sampling time consists of two parts; ti is the sum of the ideal sample time and TBD, and $\tau_i$ is the jitter. We assume that $\tau_i$ and $\epsilon_i$ are independent zero-mean random variables with variances $\sigma^2_{\tau}$ and $\sigma^2_{\epsilon}$respectively. Making a first-order approximation of $g(\cdot)$, we can write

\begin{displaymath}y_i \approx g(t_i) + g'(t_i)\, \tau_i + \epsilon_i

and obtain

\begin{displaymath}\sigma^2_y \approx \left(g'(t_i)\right)^2 \sigma^2_{\tau}+\sigma^2_{\epsilon}.

This allows us to estimate $\sigma^2_{\tau}$ and $\sigma^2_{\epsilon}$by solving a simple linear regression problem. In fact, this is the most popular method for obtaining estimates of additive and jitter variances.

Under the assumptions of Gaussian jitter errors and negligible harmonic distortion, we have shown that

\begin{eqnarray*}\sigma^2_y &=& \frac{A^2}{2}\left( 1 + e^{-8\pi^2 f^2\sigma^2_{...
...-8\pi^2 f^2\sigma^2_{\tau}}}
{4\pi^2 f^2\sigma^2_{\tau}}\right)

where A is the amplitude and f is the frequency of the sinusoidal signals. The result can be used to adjust for the bias of additive and jitter variance estimates obtained by the first-order approximation. These biases can be large if $\sigma^2_{\tau}$ is not small.

The jitter and additive noises also appear in the pulse signals from detectors. The effect of jitter on an averaged signal is that of a lowpass filter. We are currently investigating nonparametric methods for estimating these error variances. Once $\sigma^2_{\tau}$ is obtained, the estimated frequency-domain representation of the impulse response of the detector is then multipled by $\exp(\sigma^2_{\tau} \omega^2/2)$ over the frequency range of interest to deconvolve the jitter effects.

Preliminary results of repeated measurements on the magnitude and phase response of an ensemble of three 50 GHz oscilloscope plug-ins will be presented in the 55th ARFTG Conference on June 2000.

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Date created: 7/20/2001
Last updated: 7/20/2001
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