3.1.9 Calibration of High Speed Oscilloscopes

Dominic F. Vecchia

Jack C.M. Wang

Statistical Engineering Division, CAML

Paul D. Hale

Optoelectronics Division, EEEL

The design of low cost lightwave communications systems requires accurate measurements of the response of optical to electrical converters in both magnitude and phase. The frequency range of interest is about 1 MHz to 50 GHz or more. To meet this need NIST is investigating methods to calibrate the frequency response of equivalent time sampling devices (both optical and electrical) with impulse or sinusoidal stimuli. Different methods will be used to cross check these calibrations.

In this work, a high-speed sampling oscilloscope automatically can produce histograms comprising thousands of quasi-random-time samples from input waveforms swept over many frequencies and power levels. The model for N random-time samples from a signal generator under test is given by

$$V_j = a_0 + a_1 \; \sin(2 \pi f t_j) + a_k \; \sin(2 \pi k f t_j + \phi_k) + e_j$$

where $\{t_j, j = 1, 2, \ldots, N\}$ are independently and uniformly distributed on [0, 1/f], and the e_j's denote white Gaussian noise with variance $\sigma^2$. The amplitude a₁ is the main parameter of interest, but the amplitude and phase, a_k and $\phi_k$, are to be estimated if the harmonic term is detected. We have obtained the first twelve moments of the sampling distribution of V for the most likely situations of a second or third harmonic term (k = 2 or 3). For deriving method-of-moments estimates of the parameters, we convert from moments to cumulants, since the latter quantities are simpler expressions than the former. For instance, the first five cumulants for the second harmonic model are: $\kappa_1 = a_0$, $\kappa_2 = \sigma^2 + (a_1^2 + a_2^2)/2$, $\kappa_3 = -(3/4) a_1^2 a_2 \sin(\phi_2)$, $\kappa_4 = -(3/8) (a_1^4 + a_2^4)$, $\kappa_5 = (5/2) (a_1^2 + 3 a_2^2/4) a_1^2 a_2 \sin(\phi_2)$. These expressions show that, in the event that a₂=0, an appropriate estimate of $\kappa_4$, when tranformed, will provide an estimate of a₁, the primary parameter of interest. An estimate of the noise variance can then be obtained from an estimate of $\kappa_2$, though it need not exist for each sample.

More generally, we use unbiased estimates $k_1, k_2, \ldots, k_{12}$ of the corresponding cumulants to estimate all of the parameters and, by propagation-of-error, their approximate standard errors. Estimates are obtained following an approximate test for the existence of the harmonic term (the procedure accounts for the possibility that, if the harmonic is in phase ($\phi_2=0$), testing for nonzero values of $\kappa_3, \kappa_5, \ldots$ is not sufficient to detect the harmonic). In the worst case ( $a_2 > 0, \phi_2 = 0$), the sixth cumulant is also needed in the derivations. Estimates of higher-order cumulants are required because the standard deviation of k_j is a function of $\kappa_{2j}$and lower order cumulants.

Figure 9: Histograms of 40,000 random time-samples from a 3 GHz waveform. The histogram in the top figure, by its symmetry, would suggest that a second harmonic, if present, is in phase with the fundamental. Asymmetry of the bottom histogram, which was obtained from a signal at a higher power level, is consistent with a second harmonic term. Measurable harmonic content is more likely as the power level is increased.

       

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