INTRODUCTION
During the 1990s, technological and theoretical advances greatly simplified
the direct measurement of the complex largesignal response of devices, circuits,
and systems containing nonlinear elements. Nonlinear Vector Network Analyzer
(NVNA) is the newest and the most practical tool to complete the large signal
network analyzes (Remley and Sohreurs, 2007; Lin
et al., 2010). At this stage the NVNA faces on two large problems:
One is how to push the highest measurement frequency (50 GHz) to higher frequency
limit; another is how to increase the phase resolution of the phase calibration
(Moer and Rolain, 2006). This study is the initial attempt
in order to solve the second problem.
NosetoNose (NTN) calibration method is using two equivalent sampling oscilloscopes connected directly, one producing the kickout pulse as the excitation signal source, another as the receiver. We can get the response function by using the deconvolution method. But the frequency resolution can only achieve 250 MHz because the limit of the NTN technology.
The Agilent 86100 sampling oscilloscope has a limited data memory. When measuring
an impulse response, one has to meet two boundary conditions. On one hand, one
wants to keep the time window for the measurement as small as possible to allow
high time resolution and reasonable S/N ratio. On the other hand, one must keep
the window long enough to make sure that the impulse response is not truncated.
In the case of an NTN measurement, a kickout pulse is generated by one oscilloscope
and measured by a second oscilloscope. However, due to the imperfect internal
match of both oscilloscopes, some of the pulse is reflected back and forth between
both sampling oscilloscopes. After averaging the kickout pulses, a first reflection
is clearly visible at approx. 1 ns delay with respect to the main pulse. By
comparing the spectrum after averaging for different time window lengths, it
was found experimentally that one can measure up to the third reflection. The
time window for the measurement was therefore set to approx. 4 ns, so the frequency
resolution can only achieve 250 MHz (Degroot et al.,
2000).
As a contrast to the NTN calibration, sweptsine (frequencydomain) measurements
can be used to determine the magnitude of the frequency response of an oscilloscope.
The sweptsine calibration can be made at any frequency at which fundamental
microwave power standards are available, typically from 1 MHz to greater than
50 GHz. However, since the sweptsine calibration does not give phase information,
an oscilloscope calibrated using this technique alone is not adequately characterized
for timedomain metrology (Dienstfrey et al., 2006;
Wang et al., 2009).
The study describes a procedure for reconstructing the phase response of the oscilloscope response function from its magnitude. Phase response functions have the property that, in principal, the phase can be recovered from the KramersKronig (KK) transform of the logarithm of the magnitude. In practice, naive attempts to apply the standard theory can yield extremely large absolute errors in the computed phase. Although, the truncation error may be large in absolute scale, due to the localizing nature of the KK transform, this error is inherently low rank in the sense that using three customized basis functions can approximate it.
The study solves a linear least squares problem for the difference between the NTN measured values of the phase and the values computed via a phase assumption as an expansion in the specialized three basis functions by using the phase response of the NTN calibration and the magnitude of the sweptsine method. This expansion corrects the absolute size and coarse trends in the truncated phase approximation. Reconstruct the phase response of the Agilent 86100C equivalent sampling oscilloscopes. Then analyze the phase uncertainty of the oscilloscopes.
PHASE RECONSTRUCTION
An arbitrary frequencydomain response function can be factored as:
where,
(f) is the complex frequency response, τ is a real time offset and B (f)
is an “allpass filter”. One distinguishing characteristic of a phase
function is that its phase is determined by its magnitude via a KK transform.
There are several ways to express this, here we use the simplest:
exp (jφ (f)), then we can get:
The expression above is the KK transform. This equation is strict right when the integral ceiling is infinity. But infinity in frequency measurement is unattainable in actual operation and the signal generator cannot reach the infinite frequency either therefore we should cut off the integral ceiling. So a pervasive problem in the application of any KK analysis is to estimate the error due to the finite bandwidth of a measurement. The integral in (2) should be truncated:
where, Ω is the maximum frequency attainable by experiment. The problem
now is how to estimate the effect of Ω. Using the method of Dienstfrey
et al. (2006) we get Eq. 4 and 5:
where, {Ψ_{1} (f), Ψ_{2} (f), Ψ_{3}
(f)} are the orthonormal basis functions. And the condition number of the matrix
A_{mj} = Ψ_{j} (f'_{m}) can be reduced, so the
results will be stable when the test data change and the reliability will also
be enhanced. Normalization of orthonormal {Ψ_{1}, Ψ_{2},
Ψ_{3}}is shown for the case Ω = 1 in Fig. 1.Then
substitute f/Ω for f in Eq. 5 and the value of
is shown in Fig. 1.
DATA COLLECTION
We do the experimentation by using the Agilent 86100C equivalent sampling oscilloscopes.
First we connect the two same equivalent sampling oscilloscopes module directly
in order to get the amplitude and phase responses of the 86100 C from 150 GHz
using the NTN calibration and the frequency resolution is 250 MHz (shown in
Fig. 2, 3) (Qinghua
et al., 2008; Zhang et al., 2006).
In Fig. 2, the points are the amplitude response of the 86100
C measured by the NTN calibration by 0.25 GHz interval. And the points in Fig.
3 are phase response.

Fig. 1: 
Plot of orthonormal basis functions 

Fig. 2: 
Amplitude response of the NTN calibration 

Fig. 3: 
Phase response of the NTN calibration 
Use the IFFT method in order to get the timedomain response of the equivalent sampling oscilloscopes from the frequency response which we get using the NTN calibration. Then do Ztransform to analyze the zeropole point distribution of the system. In order to know whether we can use the KK transform to analyze the oscilloscopes. The result is shown in Fig. 4.
In Fig. 4, the circles show the positions of the zero points of the transfer function of the Agilent 86100C sampling oscilloscope and the crosses show the positions of the pole points. From the Fig. 4 we can see all the zeropole points of the frequency response of the 86100C equivalent sampling oscilloscopes are in the unit circle. It verifies that the KK transform can be used to get the phase response through the amplitude response.
Second get the magnitudes of the frequency response of the 83484A module of the oscilloscopes using sweptsine measurements. Use the Agilent E8257D PSG Analog Signal Generator (250 KHz40 GHz) to be the trigger source and the AV1487 Wideband Synthesis Frequency Sweep Signal Generator (made by CETC No. 41 institute 0.0140 GHz) to be the signal source. Both the trigger source and the signal source use the same reference (10 MHz). The calibration kits are HP 437B Power Meter and Agilent 8487A Power Sensor (50 MHz50 GHz 1μW100mW). The mean of the 83484A background noise is 566.03 μV and variance is 10.74 μV.
The magnitude collection is in three intervals, they are 1200 MHz 1MHz interval; 101000 MHz 10 MHz interval; 0.240 GHz 0.2 GHz interval, totally 476 sampling frequencies. Finally unite three sweptsine measurements to an arbitrary frequency grid data using the merging method. The blue, pink and red points in Fig. 5 represent the three different amplitude response of the 86100 C measured by the sweptsine measurements.
This study selects the 140 GHz magnitude data of NTN calibration and the 1MHz40 GHz magnitude data of the sweptsine measurements to data merging. The global data set is formed by computing the average of the magnitude measurements at all frequencies where experiments overlapped. The average is formed by weighting each measurement by the inverse of its associated variance, i.e., the square of its standard uncertainty. In Fig. 6 the red points are the amplitude response of the 86100 C by merging the results of NTN calibration and sweptsine measurements. Analyze the data with the 130 GHz phase response of the NTN calibration.
THEORY AND RESULT OF THE PHASE FITTING
The frequencies at which the phase is measured
need not be the same as those of the magnitude measurements ,
nor even as dense.

Fig. 4: 
The zeropole point distribution of 86100C 

Fig. 5: 
Magnitude of the sweptsine measurements 

Fig. 6: 
Magnitude of merging the NTN calibration and the sweptsine
measurements 
The frequencies for which we have magnitude data should “cover” the
frequencies associated with the phase measurements, or it will make the data
divergent.

Fig. 7: 
Compare the phase of NTN with the result of reconstruction 
φ_{NTN} (f'_{m}) is the NTN calibration result. The result
is a dense matrix K (f'_{m}, f_{n}) of order MxN = 30x476 (Dienstfrey
et al., 2006). Here data of the NTN calibration is 30; data of the
magnitude is 476, suppose:
where, h is the vector of logarithms of the magnitude response measurements. Then solve for the undetermined coefficients α_{1}, α_{2}, α_{3} in a least squares sense. Finally, given sufficiently low residual in the least squares fit, an indicator of the validity of the phase assumption, we compute the phase of the oscilloscope response function as:
In this equation, the domain of the KK operator consists of the same frequencies as the magnitude measurements {f_{n}}, the target frequencies f are some arbitrary desired frequency grid. In our experiment α_{1} = 0.4175 α_{2} = 1.1156 α_{3} = 0.4798, standard deviation of the 30 frequency points (130 GHz) is 1. 26. We compare the phase calculated with the phase of the NTN calibration which we observe as the true oscilloscope phase response function in Fig. 7. The red stars are the phase response measured by NTN calibration; the blue rounds are the reconstructed phase.
In Fig. 8, the red line is the phase response of fine frequency grid reconstruction and its frequency resolution achieves 1 MHz, the purple points are the phase response measured by NTN calibration by 1 GHz interval.

Fig. 8: 
Phase response result of oscilloscopes (1 MHz40 GHz) 
In the process of this analysis, we observe that the true equivalent sampling oscilloscope phase response function as measured by the NTN calibration is indistinguishable from the reconstructed phase response from 1 to 40 GHz.
UNCERTAINTY ANALYSIS
The uncertainty of the phase get from the KK transform consists of three parts: the uncertainty of the NTN calibration: u^{2} [φ_{NTN}]; the uncertainty of the frequency sweep method: u^{2} [h] and the uncertainty from the transform matrix and the fitting.
As the phase is getting from KK transform of the amplitude information:
We can get the expressions (9) by using some mathematical transform.
where, K_{m.n} = KB_{n} (f'_{m}) is the mth row and
the nth column element of the matrix. K is the operator and B_{n} is
the piecewise linear function, as shown in Eq. 1012.
Given a scalar random variable X with mean E (X) = X_{0} and variance var (X) = u^{2} [X] and variable Y = F (X), where F is sufficiently differentiable, we assume that:
Using the rules for linear propagation of errors, we find that:
where,
is the diagonal matrix containing the uncertainties (systematic and random)
of the vector h. (In this experiment it is diagonal matrix of order 476x476
) We can obtain that the standard uncertainty of our sweptsine data is typically
between 0.0040.17 dB by using the method of Lin et al.
(2006), Williams et al. (2005) and Williams
et al. (2007).
The main diagonal elements of the matrix u^{2} [φ_{Ω}] are the uncertainty of φ (f'_{m}) which we need. Since the NTN calibration and sweptsine measurements are independent, the uncertainties u^{2} [φ_{NTN}] and u^{2} [φ_{Ω}] can be added in quadrature.
It is at this point that we benefit from having preorthogonalized the natural set of expansion functions to instead form the orthonormal basis {Ψ_{1} (f), Ψ_{2} (f), Ψ_{3} (f)}. It reduces the condition number of the matrix A_{mj} = Ψ_{j} (f'_{m}), therefore the calculation errors of the vector α = {α_{1}, α_{2}, α_{3}}, decrease observably.
Here we assume that the relevant random variables are h and the set {α_{1},
α_{2}, α_{3}}, by this argument, are independent.
The error propagation through the integral operator and the three pointwise
multiplications obey their respective linear propagation formalisms and the
resulting uncertainties of each are added in quadrature (Jargon
et al., 2010).

Fig. 9: 
Phase response uncertainty 
The final result of the phase uncertainty form 100 MHz30 GHz is shown in Fig. 9. The phase uncertainty of the oscilloscopes is defined by 2σ and σ is the standard deviation. In Fig. 9 the line is the phase response uncertainty of the reconstructed phase.
In the process of this analysis, we analyze the uncertainty of the NTN calibration, the frequency sweep method and the transform matrix and the fitting. Since the NTN calibration, sweptsine measurements and the algorithm are independent with each other, the uncertainties can be added in quadrature. Then we get the uncertainty of the reconstructed phase response.
ACKNOWLEDGMENTS
The authors are grateful to Dr. Paul D. Hale for his kindly help during the CPEM 2008 conference and the fruitful discussions on the NTN calibration, EOS (electrooptic sampling) calibration and the uncertainty analysis.
CONCLUSION
In this study we verify the equivalent sampling oscilloscopes reconstruct the phase response of 86100 C by using the KK transform in combination with the harmonic phase response measurement of the NTN calibration and the magnitude of the sweptsine measurements. In the process of this analysis, we observe that the true oscilloscope response function as measured by the NTN calibration is indistinguishable from the reconstructed phase response over a very large bandwidth. Analyze the uncertainty of the reconstructed phase response.