High Frequency Sampling Oscilloscopes used for Vector Network Analysis

Synthetic Vector Network Analyzer

Nathan Waivio ATE Systems Engineering

Northrop Grumman Rolling Meadows, IL USA

Abstract—This paper is about using sampling oscilloscopes to perform Vector Network Analysis measurements. This paper discusses algorithms and hardware required to make vector network measurements and then compares the measurements on a simple low pass filter to measurements taken by a vector network analyzer. Oscilloscopes and digitizers have been steadily increasing in bandwidth. At the same time Interchangeable Virtual Instrument drivers have made porting software between different scopes a simple matter. These technologies make investments in algorithm development more cost effective. The algorithms have a longer life, lower costs to maintain, and ease of porting the algorithms to future scopes.

Keywords-Oscilloscopes; Vector Network Analysis; Algorithms

I. INTRODUCTION This paper demonstrates how it is possible to construct a

synthetic Vector Network Analyzer (VNA) using an Oscilloscope and RF Generator. Synthetic Instruments (SI) have been previously used for Vector Signal Analysis (VSA) [1]. Interchangeable Virtual Instrument (IVI) classes have been defined for oscilloscopes and RF Generators and this greatly simplifies the porting of software to new instruments. Commercial off the shelf (COTS) oscilloscopes and digitizers have been greatly increasing in bandwidth. Some Oscilloscopes have input bandwidths up to 20 GHz. IVI classes have been defined for many instruments including Oscilloscope and RF Generators, although a standard has not been defined for VNA.

II. DISCRIPTION OF ALGORITHM A RF signal generator is used to generate a pulsed signal

and the output signal is split into a reference signal and a test signal. The reference signal is fed directly to an oscilloscope; the test signal is then passed through a Device Under Test (DUT) to measure the transmission parameters (S12, S21). The DUT could also be connected with a directional coupler to measure the reflection parameters (S11, S22). The two signals are converted to an analytic signal then windowed and Fast Fourier Transformed (FFT) to obtain the complex voltage spectrum. The test signal’s complex voltage spectrum into a reference impedance of 50 Ohms is divided by the reference signal’s complex voltage spectrum to get the scattering

parameters. The measurements are statistically analyzed to determine which frequency bins contain coherent signals and therefore valid scattering parameters. The scattering parameters are recorded. The RF signal generator is then set to another frequency and the analysis is repeated. The measurement process is conducted with the DUT removed from the path and with the DUT in the path. Calibration factors for S21 are calculated from the measurement with the DUT removed from the path. Then the calibration is applied to the measurement to remove fixed differences from the path.

A. Obtaining the Complex Voltage Spectrum For the input sequence x(t) that consists of real samples

from the oscilloscope [2]. The analytic signal is:

a(t) = x(t) + iH{x(t)} (1)

The analytic signal is complex and H{x(t)} is the Hilbert transform of x(t). The analytic signal is useful in calculating the envelope of the signal to measure pulse parameters. The analytic signal is windowed and the gain is adjusted to compensate for the window.

aw(t) = a(t)·w(t)/0.215578 (2)

The flat top window is used in this application because it spreads the signal power and phase into multiple frequency bins ensuring that the signal is detected. Fig. 1 and 2 depict the amplitude and frequency response of the flat top window [3].

The single sided amplitude spectrum is calculated by taking the FFT of the windowed analytic signal.

Aw(f) = FFT{aw(t)}/N (3)

The result of the FFT is the complex amplitude in units of Volts peak. The FFT is divided by the number of samples contained in the analytic signal. The single sided amplitude spectrum has the property that the magnitude of the positive frequency half of the FFT is the peak voltage and the negative frequency half of the FFT is zero. The negative frequency half

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of the FFT can be discarded because it contains no additional information.

B. Power Spectrum The complex voltage spectrum is then converted to a power

spectrum. The spectrum’s magnitude calculated by multiplying the complex voltage spectrum by its complex conjugate and taking the square root.

Vp(f) = √(Aw·Aw*) (4)

The result of (4) is a real magnitude with units of Volts peak. To calculate power we need to convert the units to Volts RMS. To convert from Voltage peak to Volts RMS the quantity is divided by the √2. The DC component of the Power spectrum is not divided by the √2 because only sinusoidal waves are divided by the √2 to convert to RMS.

VRMS(f) = Vp(f)/√2 for all f≠0 (5)

VRMS(f) = Vp(f) for f=0 (6)

To get the power spectrum in units of dBm the RMS voltage spectrum must then converted to milliwatts. The reference impedance is 50 ohms so the power is RMS voltage squared divided by the resistance.

PmW = 1000·VRMS2/50 (7)

The final step in converting to dBm is to take the logarithm and multiply by 10.

PdBm = 10·log10(PmW) (8)

C. Calculating the Scattering Parameters To calculate the scattering parameters the complex voltage

spectrum of the test signal is divided by the complex voltage spectrum of the reference signal [4].

sTest(f) = ATest(f)/ARef(f) (11)

The magnitude of the scattering parameter can be displayed as a log magnitude by taking the logarithm and multiplying by ten similar to (8). This same method is used for both the calibration factor and measurement of the DUT. The result from (11) contains both valid scattering parameters and noise. The scattering parameters must be separated from the noise by detecting the coherent signals.

D. Signal Detection The measurement routine takes 25 measurements to gain a

statistically significant sampling of the signal. Each frequency bin of the scattering parameter is statistically analyzed to determine if a coherent signal is present. A coherent signal is defined as one in which the phase relationship is constant between the test signal and the reference. The phase is converted to the conventional units of degrees, when displaying the measurement in the graphical user interface. The phase is calculated from the complex scattering parameters by using the arg function.

φ(f) = arg(s(f)) (9)

Here the units of phase are in radians. The phase should be in the interval of (-π, π]. To convert to degrees the phase must be multiplied by 360 and divided by 2π.

φ(f) =360·φ(f)/2π (10)

A circular averaging method is used to average the phase [5]. The circular mean is necessary for quantities like phase that wraps at +/–180 degrees. The sine and cosine of the angle is calculated and the mean of the 25 measurements is computed. The circular variance is the magnitude calculated by the Pythagorean Theorem of the mean sine and cosine of the angles. The average angle is calculated by the two argument atan2 function.

Figure 2. Frequency Response of Flat Top Window

Figure 1. Time Domain Amplitude of Flat Top Window

Varn = √((mean(sin(φ(fn))))2+(mean(cos

Phasen = atan2(mean(sin(φ(fn))),mean(c

If the circular variance of the phase is 1 thbin is considered to have a perfectly coherennoise. If the circular variance is 0 then thdoesn’t exist. For our purposes we only look fvariance of greater than 0.999, this ensuressignal is measured with a high signal to nscattering parameters are then recorded for thea coherent signal.

E. Sweeping the Span The measurement process described ab

across the entire span. The span is divided insteps. Using a pulsed RF signal is beneficial bspectral lines in the frequency domain serve asand allow many data points to be measured istimulus could also be a pulse generator waveform generator so long as the source isspectral lines in the frequency range of osweep. The span is divided into a prime numthat the spectral lines will not overlap and mpoints will be measured.

F. Applying Calibration The final step is to apply the calibration

calibrated scattering parameters. Due to the nsetup the measurement can only be conductedwith the normalization method of calibration.coupler was used in the setup to measucomplicated error model could be used and ithe accuracy of the measurements. The methused in this paper is closest related to Transmcalibration method for vector network analyzecalibration factors are the scattering paramwhen the path is connected without the DUTparameters for the DUT are divided byparameters without the DUT.

sDUT(f) = sTest(f)/ sCal(f)

This result will closely match the Transmission Response calibrated vector netwo

III. TEST SETUP AND MEASURMEN

Fig. 3 and Fig. 4 show a diagram of the tessetup consists of a Tektronix MSO 4104 scoGS/s and 1 GHz bandwidth, the sample rameasurements is 1 GS/s.

The stimulus used is an Agilent N821Analog Upconverter with a frequency range f20 GHz. The signal used in this case was stespan and had a pulse width of 2 µs and a pulse4 µs. The span is divided into 29 steps anparameters are measured at each step.

The splitter used to divide the signal intotest signal was a Technical Research and

s(φ(fn))))2) (11)

cos(φ(fn)))) (12)

hen the frequency nt signal with no he circular mean for signals with a

s that a coherent noise ratio. The e bins that contain

bove is repeated nto 29 frequency because the many s coherent signals in one step. The

or an arbitrary s able to produce our measurement mber of steps so

more unique data

n to calculate the nature of the test

d unidirectionally, . If a directional ure s11 a more it would increase hod of calibration mission Response ers. The through

meters calculated T. The scattering y the scattering

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results from a ork analyzer.

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DR285, 2-way power divider with One output of the splitter was connof the scope, the other output of tDUT and then to channel 2 of the sc

For demonstration purposes thCircuits model SLP-300, low pascutoff. The measurement results tacompared with an Agilent 8720Analyzer with a Transmission Respo

Data was gathered from the Agsynthetic method and is displaymagnitude) and Fig. 6 (displaying Source 29 times through the span rof 6650 unique measurements of thAgilent VNA data consists of 201 VNA results did not change signpoints on the display was increased.

Figure 4. DUT Measu

Figure 3. Calibra

a range of DC to 18 GHz. nected directly to channel 1 the splitter connects to the

cope.

e DUT used was a Mini-ss filter with a 300 MHz aken with the scope will be 0ES S-Parameter Network onse calibration.

gilent 8720ES and from the yed in Fig. 5 (displaying

phase). Stepping the RF esulted in the measurement

he frequency response. The measurement points. The

nificantly if the number of

urement Setup

ation Setup

Several errant points in the synthetic data appear 330 MHz and above, these points are due to aliasing in the calibration measurement data, with the use of an anti-aliasing filter or other techniques these anomalies will disappear.

The maximum standard deviation of the magnitude data measurement points was 0.3dB for the 25 measurements taken, yielding a standard error of 0.06dB.

Many random and systematic measurement error sources exist just as they do for standard VNA systems. Some systematic errors may be greater for an oscilloscope than a VNA like error in the load match for the 50 Ohm termination into the scope, as well as limits on the dynamic range due to the use of 10 bit analog to digital converters in COTS oscilloscopes.

IV. CONCLUSIONS This paper demonstrates that it is possible to accurately

calculate thru scattering parameters using an oscilloscope and an RF signal generator. This approach has benefits because it adds more functionality to a common measurement instrument.

Future steps to further develop this technique would be to develop a better error model. A more detailed error model should be able to eliminate many of the linear systematic error sources. Another possible path would be to utilize a directional coupler for measuring reflection and transmission scattering

parameters simultaneously. A solution to the aliasing issue should be developed by either increasing the sample rate, inserting an anti-aliasing filter or investigate undersampling techniques that may solve the aliasing issue. It should also be possible to optimize the measurement speed, by using more multithreading and/or using a source that has spectral characteristics that do not require many steps to evenly sample the scattering parameters across the span.

Techniques like the one described in this paper, along with others described in Ref. [1], should be able to converge RF instrumentation to a synthetic instrument solution.

REFERENCES [1] Lowdermilk, W., Harris, F. “Vector Signal Analyzer Implemented as a

Synthetic Instrument”, Autotestcon, 2007 IEEE [2] “The Fundamentals of FFT-Based Signal Analysis and Measurement in

LabVIEW and LabWindows/CVI”, NI Developer Zone, National Instruments, 2010, July 10, 2010.

[3] “Window function (flat top).png” and associated Matlab M file, Wikimedia Commons, Wikimedia Foundation, Inc. December 17, 2005

[4] M. Hiebel, “Fundamentals of Vector Network Analysis,” Rohde & Schwarz, 2005, ISBN: 978-3-939837-06-0.

[5] “Mean of Circular Quantities”, Wikipedia, Wikimedia Foundation, Inc. June 19, 2010

Figure 5. DUT Magnitude Measurements

Figure 6. DUT Phase Measurements