phd. activity report: oct.90 - july 91 -...

PhD. Activity Report: Oct.90 - July 91

Jan Verspecht

Jan Verspecht bvba

Gertrudeveld 151840 SteenhuffelBelgium

email: [email protected]: http://www.janverspecht.com

Your Ref. Nr.: DTW/mp/U.10.d-X/90739 Our Ref Nr. : JV/1/28/92/001

ELEC at VUB

Phd. Activity report : Oct. 90 - July 91

Jan Verspecht

Jan Verspecht 2 of 35

0. General introduction 3

1. Calibration problem for measuring nonlinear devices 4

1.1 Linear DUT measurements 4

1.2 Nonlinear DUT Measurements 5

2. Calibration of sampling oscilloscopes 6

2.1 Introduction 6

2.2 Input channel characterization: “back-to-back” procedure 62.2.1 Introduction 62.2.2 Pulse generation during sampling 72.2.3 Pulse capture 82.2.4 Combining pulse generation and capturing to calibrate 92.2.5 Experimental results 10

2.3 Input channel characterization: “power measurement” procedure 122.3.1 Experimental set up 122.3.2 Results 13

2.4 “back-to-back” versus “power measurement” 142.4.1 Comparison of the results 14

3. Phase distortion uncertainty 16

3.1 Introduction 16

3.2 Mathematical theory 173.2.1 Mathematical consequence of the model 173.2.2 Useful lemma’s and notations 173.2.3 Prove of the limit 19

4. Time jitter compensation 23

4.1 Introduction 23

4.2 The median method 23

4.3 The compensated average method 244.3.1 Method described in literature (footnote 2) 244.3.2 The “non-stationary sigma” method (developed by myself) 25

4.4 Comparison of “non-stationary sigma” versus “median” 294.4.1 About the simulation software 294.4.2 Results: 30

4.4.3 Conclusions 34

Appendix 35

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0. General introductionThis annual report covers my activities at the ELEC department at the VUB from

October 1990 until July 1991. The first part of this report covers the results of the workI did directly related to the main topic of my research activities: the calibration prob-lem for measuring nonlinear events at microwave frequencies. This work is probablythe most interesting for the reader, as it covers new and even unpublished ideas, algo-rithms and theory.

The second part of this report is covered in the Appendix, it summarizes my activi-ties mainly during the first three months of my employment. During these months I hadto learn a lot of things only indirectly related with the topic of my PhD, such as manip-ulating a 8510 network analyzer, a 54120 sampling oscilloscope, learning about all kindof software tools like RMB, MDS,...

Two of the chapters in this report, concerning the phase distortion uncertainty andthe time jitter compensation, are being prepared for publication. Library searches indi-cate that the algorithms I developed are original.


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1. Calibration problem for measuring nonlinear devices

1.1 Linear DUT measurements

As an introduction to the proposed solutions for the calibration problem for meaing nonlinear microwave devices, I will first give a short and intuitive overview of tclassical calibration problem for measuring linear devices. For brevity this calibramethod will be called “the linear calibration”, in opposition to the calibration problefor measuring nonlinear devices, which will be called “the nonlinear calibratioPlease note that the last mentioned calibration will be called nonlinear because itbe used for measuring nonlinear devices, the method itself acts upon raw data aslinear transformation.

For simplicity I will cover the classical one-port network analyzer calibration prolem. The systematic errors of such a measurement instrument can be modeledlows:

Fig. 1

In this model “a” stands for incoming wave and “b” for reflected wave, index “mstands for “measured” and “d” for “at DUT (device under test)”. The systematic erare modeled by a linear two-port network (“Error Adapter”).

We want to know (reflection factor at DUT) as a function of

(measured reflection factor). It is easy to show that this function is actually a bilin

transform1. Such a transform is described by 3 complex unknowns. To find these 3knowns we can measure three calibration standards (devices with known reflectioefficient, typical load, short, open) and use the “cross ratio invariance”. This invaria

is expressed as follows: .

In this expression indices 1, 2, 3 stand for the respective calibration elementsstands for the DUT, “m” for measured and “r” for real. We know the left part becait’s our measured (raw) data and we know all of the right reflection factors assocwith calibration elements (provided by calibration element manufacturer). From thispression we thus can deduce the real (= calibrated) value of the reflection factor oDUT.

The accuracy of this method actually depends on the mechanical specificatiothe calibration elements. Once the geometry of a calibration element is known, itflection coefficient can be deduced by Maxwell’s equations within a certain errorpending only on mechanical specifications.

1. See “Measuring Nonlinear Systems”, doctoral thesis of Marc Vanden Bossche, page 25-26

Ideal Analyzer Error Adapter DUT

bm

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bm

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Γ1m Γ3m–( ) Γ2m Γdm–( )Γ1m Γ2m–( ) Γ3m Γdm–( )

-----------------------------------------------------------Γ1r Γ3r–( ) Γ2r Γdr–( )Γ1r Γ2r–( ) Γ3r Γdr–( )

------------------------------------------------------=

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1.2 Nonlinear DUT Measurements

When we want to characterize weakly nonlinear DUTs by a VIOMAP1 we need toperform several experiments in which we excite the DUT with several signals andmeasure the incoming and outgoing waves for each experiment. In contrast with thear case this measurement is no longer a relative one but is actually an absolutesurement! Consider for example a weakly nonlinear one-port excited by an incomwave with only one frequency component. The reflected wave now contains no loonly one frequency component but also several significant harmonics. For determVIOMAP we would have to know the absolute amplitude of all frequency componeand their phases relative to the phase of the exciting fundamental.

The model of the systematic errors of a “nonlinear one-port network analyzer” istually the same as it is in the linear case:

Fig. 2

The index “i” refers to the different frequency components of the signals. Becawe are interested in an absolute measurement we have to determine all four counknowns of the error adapter. It is easy to prove that by using the classical calibrelements only three independent equations can be found. This was enough for peing a calibration in the linear case, but is certainly not enough when we want to do msurements on nonlinear DUTs. To find a fourth equation three methods are proposthe doctoral thesis of Marc Vanden Bossche. The first method is based on the usnonlinear calibration element, a “golden diode”. Obviously this method would be vhandy in use. Main problem with this method is of course how to specify such anment. As accuracy for linear calibration elements is determined mainly by mechaspecifications this certainly is no longer true for nonlinear (read semi-conductor) dees, which characterization depends on many difficult to measure parameters. Thond method proposed several pieces of the nonlinear analyzer are measured sepbut we can never find out what the systematic error is due to the characteristic o

sample head2. In the third method we will use a “reference generator”. This is a genetor which generates a fundamental frequency and harmonics with accurately knphases relative to the fundamental and accurately known absolute amplitudes. Wwould have to know the generators output impedance. Similar to the first methodmain problem is how to specify such a reference generator. To do this we need anrate signal analyzer. This accurate signal analyzer could be a calibrated broadbandpling oscilloscope. The study of the calibration of sampling oscilloscope measuremwill be the main topic in what follows.

1. See “Measuring Nonlinear Systems”, doctoral thesis of Marc Vanden Bossche, page 93 and followin2. See “Measuring Nonlinear Systems”, doctoral thesis of Marc Vanden Bossche, page 38

Ideal Analyzer Error Adapter DUT

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2. Calibration of sampling oscilloscopes

2.1 Introduction

The sampling oscilloscopes mentioned above are of the so called “equivalentsampling” type. These oscilloscopes nowadays are available with a bandwidth u50 GHz (cfr. HP 54124T). Although this measurement tool is very complicated frotechnological point of view, the basic principle is very simple. The sampling head (aode triggered by a driving pulse) is able to take a sample with a very small apetime (picoseconds). The sampler needs a recovery time of about 1 ms. The princicapture periodic waveforms with a fundamental frequency of a few GHz neverthethe maximum sampling frequency is only about 1 MHz is very straightforward. As sas the sampler has recovered a precision broadband trigger circuit waits for a cevent on the signal. As soon as the trigger notifies the event he starts a counter wcertain variable delay. After the delay the driving pulse is activated and the sampleodes are triggered. The sampled value is digitally stored and associated with aequal to the delay. By repeating this process with several different delays the wholriod of the microwave signal can be scanned.

When we look at the systematic errors that occur in such a system mainly two ses can be distinguished. First we have the characteristic of the input channels (froput connector to sampler head) and the effect of the limited aperture time ofsamplers. A second source of error is the so called “time jitter”. This is the name fofact that the delay time between signal trigger event and signal sampling event is aly a stochastic (noisy) variable due to noise in the broadband trigger circuitry. Latewill be demonstrated that the presence of timing jitter causes a systematic error corable with a low-pass filtering whenever signal averaging is applied to eliminate thfects of the white noise. Both problems need to be tackled when we want to usampling oscilloscope as an accurate signal analyzer.

As a remark I also would like to mention the new so called “Microwave TransitAnalyses” (HP 71500A) as a tool to measure a reference generators characteristicthough this instrument does not have the problem of timing jitter when locked wisynthesizer (however some phase noise will always be present) it will be much hto measure the characteristics of the input channels because of the IF part (whichity was one of the hardest nuts to crack in designing the instrument).

As for now promising methods are developed to tackle the input channel chara

ization of sampling oscilloscopes1 as well as the timing jitter problem2. This is the rea-son why I concentrated on the use of sampling oscilloscopes and not on the uinstruments similar to the 71500.

2.2 Input channel characterization: “back-to-back” procedure

2.2.1 Introduction

Before going into the mathematical model of this calibration process I will first ga short intuitive explanation of the calibration procedure. Whenever a sampling diofired and takes a sample there will flow a small current into the so called holding caitor. This current flows during the aperture time of the sampler (picoseconds). This

1. See “Characterizing high-speed oscilloscopes”, Ken Rush, IEEE Spectrum September 19902. See “2.3 Compensation for Time Jitter Error”, Jan Verspecht, Annual Report: Oct.90 - July 91

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rent causes a pulse to be launched towards the input connector. The pulse appearinput connector and contains information as well on the characteristics of the ichannel from connector to sampler as on the shape of the sampling pulse. Theidea is now to sample this pulse with another oscilloscope (theoretically identicaconnecting two input channels together. The pulse shape measured with the ssampling oscilloscope will be the pulse that appears at the input connector of theoscilloscope distorted by the input characteristic of the second oscilloscope. By asing that both oscilloscopes are identical the characteristics of one scope can bemined by the shape of the measured pulse.

2.2.2 Pulse generation during sampling

The equivalent model of the oscilloscope input channel:

Fig. 3

In this model we distinguish the input connector and the transmission line tosample head, a 50 Ohm load for matching, a time variant resistor, modelling thepling diode itself which acts like a switch, and the holding capacitor Cp. For the follow-ing analysis to be true the following assumption has to be made: (25 + Zmin).C>>Tap. Inthis expression Zmin is the minimum resistance of the switch and Tap is the switch aper-ture time. This assumption holds in any case with the usual sampling heads be25.C>>Tap (cfr. 54120). The shape of Z(t) will be such that it is different from infinionly during the aperture time.

The sampling oscilloscope I had at my disposal was a 54120. This scope offerpossibility of putting an initial charge on the holding capacitor (this is automaticadone when we use the “offset” function). The question is what pulse appears at thenector when the sampling diodes are fired. First we will take a look at the pulse coout of the sampling head. The equivalent scheme is the following:

In this scheme we see that the transmission line is replaced by a 50 Ohm loacause we assume that no wave is going to the right and that the holding capacitorplaced by a voltage source with initial voltage Voffset because of the assumption that thtime constant of the system is much larger then the aperture time of the switch.

50Ω

Zc 50Ω=

connector

Z t( )

C

sampling head–

Voffset

Z t( )

50Ω

50Ω

i t( )

v t( )

Ref Nr. : JV/1/28/92/001


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this equivalent scheme we calculate the shape of the pulse launched towards th

nector: . (Eq. 1)

For later use we will define the Fourier transform of s(t) as S(f). The shape ofpulse (p(t)) appearing at the connector will be the convolution of the pulse created adiode and the impulse response of the transmission path diode-connector (noh(t)): . (Eq.2)

In the frequency domain this relation will be: . (Eq.3)

2.2.3 Pulse capture

The problem is the relation between the signal p(t) at the input connector andsignal appearing on the screen as it is digitized by the scope. To do this we will firsscribe a model in which we take a sample at time Ts of the signal a(t), which appears athe sampling head. We define our time such that Z(t) corresponds to a sample taTs=0. Taking a sample at time Ts is then described by the function Z(t-Ts).

The equivalent scheme is:

Fig. 5

This time the initial charge on the capacitor is zero. The voltage on the capacitoter the diodes have been fired will be equal to the integral of the current andbe digitized by the scope before being displayed on the screen at time Ts. This dis-played value will be equal to:

In this expression K is a constant associated with the oscilloscope, the functionintroduced for later use. This result can be achieved by using the following equivascheme:

Fig. 6

Because of the assumption concerning the system time constant the capacitorreplaced by a short.

s t( ) 2525 Z t( )+-----------------------Voffset=

p t( ) s t( ) h t( )∧=

P f( ) S f( )H f( )=

50Ω

Zc 50Ω=

connector

Z t Ts–( )

C

sampling head–

a t( )p t( )

i c t( )

vosc Ts( ) K ic t( )dt

∞–

∞

∫ K a t( ) 125 Z t Ts–( )+----------------------------------dt

∞–

∞

∫ K a t( )g Ts t–( )dt

∞–

∞

∫= = = (Eq.4)

Z t Ts–( )50Ω

50Ω i c t( )2a t( )

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Eq.4 together with Eq.1 leads to the following expressions:

We note that the sign of the argument of Z in Eq.5 is negative, this means thaconjugate of S has to be taken in the frequency domain. The relation between thea(t) appearing at the sampling head and the wave p(t) appearing at the connector wa convolution of p(t) with the impulse response of the connector and the impetransmission line towards the sampling head. It can be shown that this impulssponse function will be equal to the h(t) mentioned in Eq.2 (reciprocity theory). Tleads to the following equations:

2.2.4 Combining pulse generation and capturing to calibrate

To perform the actual calibration two scope input channels are connected togeone scope continues sampling and generates pulses which are captured by a seccilloscope. The scheme:

Fig. 7

What will be the pulse shape as measured by our receiving scope is now thelem to solve. The pulse shape p(t) appearing at the connectors will be given byThe signal appearing on the scopes screen will then be given by Eq.7. In the frequdomain we will use Eq.3 and Eq.8. This results in the following equation for V(f),Fourier transform of the measured pulse:

vosc Ts( ) a Ts( ) g Ts( )∧ a Ts( ) K25 Z Ts–( )+------------------------------∧= =

Vosc Fs( ) A Fs( )Sconj Fs( ) K25Voffset---------------------=

(Eq.5)

(Eq.6)

vosc Ts( ) p Ts( ) h Ts( )∧( ) K25 Z Ts–( )+------------------------------∧=

Vosc Fs( ) P Fs( )H Fs( )Sconj Fs( ) K25Voffset---------------------=

(Eq.7)

(Eq.8)

50Ω

Zc 50Ω=Z t Ts–( )

C

a t( )

50Ω

Zc 50Ω=

connectors

Z t( )

C

receiving scopesending scope

p t( )

Vosc Fs( ) S Fs( )H Fs( )H Fs( )Sconj Fs( ) K25Voffset---------------------= (Eq.9)

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Eq.9 can be written as follows:

As we see in Eq.8 the scope is fully characterized by the product of H(F), S(F) aconstant factor. This constant factor can easily be determined by a DC-measureExcept for the phase of S(f) we find all frequency dependent information on the sccharacteristics in Eq.10. It is sufficient to take the square root out of Eq.10.

Of course calculating the square root of a complex function care has to be tabout the phase. Before taking the square root, the phase function needs to bwrapped to eliminate phase ambiguity! This can easily be achieved by delayingmeasured pulse in time domain such that the linear part of the phase is eliminatedEq.15) and then taking into account the fact that the phase function is a continfunction everywhere except in those frequency points where the amplitude functiona zero or a pole, which will be never the case in practice.

The left part of Eq.11 can be measured, the right part contains all the amplitudformation on the scopes response together with the phase information of H(f).phase of S(f) has disappeared and can never be determined by this method. In hliminary application note Ken Rush (HP R&D project manager Colorado Springs Dmentions “computer modeling suggests that the phase of H(f) is much greater thaphase of S(f) and that it dominates the oscilloscope’s phase response”. In a laterter I will give some more comments on this statement.

2.2.5 Experimental results

During one week I had the opportunity to have a second oscilloscope in the labto perform the sophisticated measurements. The practical experimental set up ispeculiar. The receiving oscilloscope is set in TDR mode. The step generator is incrated in channel 1 and is connected by an SMA-cable of length 1m to the trigger opulse sending oscilloscope, which is in histogram mode. The two “channel 3”’s arenected by one 3.5mm precision female-female connector. The time axis of the reing oscilloscope will be referenced relative to the instants the step generator is firewe will see on channel 3 of this scope what signal appears on the channel 3 inputother scope, a certain time after the step generator was fired. Now we’ll take a lowhat happens for the pulse sending oscilloscope. At an instant of approximat5 nsec relative to the TDR pulse this scope will be triggered instantly because ohistogramming mode, hereby generating the desired pulse, coming out of his inpusampled by our receiving scope. All this is explained in more detail in the applicanote of Ken Rush, attached to this document. A few important remarks about thistical set up are mentioned in the following.

A first thing important to know is that the mentioned method can only be used toibrate channel 3 of the test set, this because of actual hardware limitations (this ised to the fact that this channel is used for transmission measurements in TDR mo

Another thing I did not mention in the theory above but very important when redoing measurements is the feedthrough of the sampling diodes driving pulse atcreation. This means that the actual pulse coming out of the input connector of a sas it has taken a sample is actually the sum of the driving pulse feedthrough an

Vosc Fs( ) H2 Fs( ) S Fs( ) 2 K25Voffset---------------------= (Eq.10)

Vosc Fs( )L

--------------------- H Fs( ) S Fs( )= (Eq.11)

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pulse mentioned in the theory above. To compensate for this effect we first do asurement with a positive Voffset and afterwards one with a negative Voffset (cfr. 2.2.2).When we now subtract the two measurements the feedthrough will be eliminatedcause its sign has not changed, in contradiction to the sign of the sampling pulse (EIn the following graph an example of such a measurement is given:

Fig. 8

On the x-axis we have a time axis in nsec, the y-axis gives voltage in mV.On this graph we can easily see that the two pulses created with a positive re

tively negative offset have a significant common mode (see after 38.40 nsec.) duefeedthrough of the sampling diodes driving pulse. To eliminate this common modejust have to subtract the negative offset pulse from the one with an opposite sign.

The resultant pulse has the following form:

Fig. 9

On this plot we have an x-axis in psec (there is a delay compared to Fig.8) andaxis in mV. We note that the resultant pulse has a duration of less then 200 psec. Ttermine the characteristic of the oscilloscope (both amplitude and phase) we nowto take the square root out of the Fourier transform of this pulse (Eq.11).

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The result for the magnitude is shown on the following graph:

Fig. 10

On the x-axis we have frequency in GHz, on the y-axis we have the transfer ftion (relative to DC) in dB. We note that the -3dB point is actually at about 30GHz.have a crosscheck on these experimental results we compared with another mThis method will be based on the measurement of the incident power compared tpower measured by the scope. The practical implementation and experimental rwill be discussed in the following chapter.

2.3 Input channel characterization: “power measurement” procedure

2.3.1 Experimental set upThe idea of this method is very simple. We inject a wave with a precisely kno

power into the channel to test and we compare this power with the power that issured by the oscilloscope. To be able to do this from DC until 40 GHz I developed sware to perform the following procedure.

First the output of our synthesizer (a 86340A) was calibrated with a working sdard of HP-Brussels until 40 GHz. Then I connected the synthesizer output directan input channel of the test set (with no cable and one single adapter). A certain pwas set on the synthesizer (using the previously measured calibration coefficieThen the frequency axis was scanned and for every frequency the power measuthe scope relative to the corrected power set on the synthesizer was stored. To bemeasure this power correctly until 40 GHz I injected noise into the trigger circuitryused the Vrms function on the acquired data. It is very important to remark that you cnever measure the actual power measured by the sampling head by using the triggconventional way. This due to the following: you have to extract some energy out osignal coming out of the synthesizer to trigger the scope, hereby loosing power accy. Secondly you have timing jitter problems (see later chapters) and finally youproblems with the trigger bandwidth when measuring at plus 18 GHz. The accurathe above procedure is about +/- 0.2 dB.

To have an idea of the linearity of the scope this procedure is repeated for sedifferent input powers.

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2.3.2 Results

The experimental results:

Fig. 11

On this graph we have a frequency axis in GHz and a relative amplitude axis inThe numbers 0,6 and 12 in the trace titles refer to 0 dBm, -6 dBm and -12 dBm oput power set. I hereby remark that 0 dBm corresponds to the full scale of the scFrom these measurements we can see clearly that the oscilloscope suffers fromnonlinearities above 20 GHz, but only at rather high powers (-6 dBm). When we zin on the data we see that the nonlinear compression stays within 0.2 dB for all polower than -6dBm and for all frequencies lower than 15 GHz. For a clear understing I remark that the nonlinearity we’re looking at is certainly not a static one andnever be compensated by any kind of lookup table (this kind of correction is alrepresent in the scope).

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2.4 “back-to-back” versus “power measurement”

2.4.1 Comparison of the results

Now that we have results from both procedures we can do a crosscheck on thplitude part of the scopes transfer function. Of course we have to use the resultspower measurement without compression, corresponding to the -12 dBm curve (wno change in this “transfer function” when we use even lower powers). Both resultsplotted on the same grid:

Fig. 12

We have a frequency axis in GHz and a transfer axis in dB. We see a close cspondence between the two results (about 0.2 dB deviation from DC until 30 GHz)ter 30 GHz the results begin to have a greater difference, but these results cneglected because of the fact that the connectors involved are 3.5mm-precision,means that repeatability is reduced for frequencies above 30 GHz.

Although the new calibration method seems very promising because of the phawell as the amplitude information, we may not forget that the new method reliessome important assumptions. We did suppose that the two oscilloscopes used in tperiment are identical. To tackle this problem I already designed an experiment wby this assumption is no longer needed. Unfortunately this experiment is very haperform in practice becausethree oscilloscopesare needed on the same place at tsame time. Another problem is the remaining uncertainty on the phase (cfr. Eq.11effort to tackle this is described in the following chapter. A last but certainly not leproblem is of course the validity of the rather simple model we use.Thinking abou

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verification of the model I checked a very important consequence. It is easy to sthat when we switch the functionalities of the sending and receiving oscilloscopehave to find exactly the same result for the amplitude transfer function, even wheassume both oscilloscopes have different characteristics. Indeed, we can writeeven when we assume the two oscilloscopes are different, the only thing we haveis to give a different index to the terms due to the receiver and to the sender. Whedo this Eq.10, which gives the spectrum of the measured pulse, becomes:

In this two equations we associate the indices 1 and 2 with the two different oscscopes. In Eq.12 scope 1 is the receiving one and in Eq.13 it is the sending one. Efor the factor K, independent of frequency and easy to determine, we see that thedifference in transfer characteristic due to the non symmetricity can be noticed inphase but can never be noticed in the amplitude! This is the case because the onference between the right sides of Eq.12 and Eq.13 is that the conjugate has to bewith the other S. To do this experiment we can even eliminate the connector reproibility by doing the two experiments (scope 1 receiving and sending) without openthe connection between the two scopes, the difference between the two experimening achieved by software means. The result of my experiment is shown on the foing graph:

Fig. 13

Here we have frequency in GHz on the x-axis and the difference between themeasured amplitude characteristics in mdB on the y-axis. We see that the maximumference between the two measurements is only about 200 mdB until a frequen26 GHz. This is in very good agreement with the predicted identity. The comparisothe phases of the two measurements could also be very interesting but to do this I wneed more measurements. The reason that the comparison of the two phases ismore difficult is the fact that we can never compare the time axes of the two oscscopes. Indeed, we can never say what the exact delay is of our pulse relative to tput connectors because the time axis we see on the scope is determined by thecircuitry. This means that the linear on frequency dependent part of the phase canbe determined. Although we do not care about this unknown delay because it willer influence our measurements, it makes it very hard to compare the phase behathe two experimental transfer functions.

V2osc Fs( ) H1 Fs( )H2 Fs( )S1 F( )S2conj Fs( )

K2

25Voffset---------------------=

V1osc Fs( ) H2 Fs( )H1 Fs( )S2 F( )S1conj Fs( )

K1

25Voffset---------------------= (Eq.12)

(Eq.13)

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3. Phase distortion uncertainty

3.1 Introduction

As mentioned in the comment after Eq.11 the phase of S(f), the part of the trafunction due to the finite sampling aperture, can never be determined by the propback-to-back calibration procedure. The way this problem is tackled is by say“computer modelling suggests that the phase of H(f) is much greater than the phaS(f) and that it dominates the oscilloscope’s response” (sic). Because I could nevhappy with the word “suggests” I searched for a way to put a limit on this phase of Sstarting from as few assumptions as possible. Although the limit I found can be tofor certain applications, I’m convinced that the concepts developed in the theoredescription can be very useful in future treatments of the problem.

The only input for calculating a limit on the phase distortion (we do not care abthe linear part of the phase which corresponds with a delay) is the limited apertureof the sampling diodes (Tap) and the assumption that the model discussed in 2.2.3valid. The analytical expression for the limit is the following:

In this equation stands for the phase distortion (phase deviation from line

ty) and with f equal to the frequency. A plot of this limit with on the x axis th

normalized variable and on the y axis the maximum deviation from phase lin

ity in degrees:

Fig. 14

ϕ ω( )ωTap ωTap( )sin–

ωTap( )cos--------------------------------------------

atan≤ ωTap π≤for (Eq.14)

ϕ ω( )

ω 2πf=

2ωTap

π----------------

Ref Nr. : JV/1/28/92/001


isout 1

plexns of

) wep(t) ism-

nev-ined

of a

ateery

hase:

ing.he de-

forith a

When we assume a Tap of about 15 ps for a 54120 with a bandwidth of 20 GHz thmaximum deviation would be about 13 degrees for a frequency of 10 GHz and abdegree for a frequency of 5 GHz.

The mathematical theory is developed in what follows. Although not using comtheorems I do recommend not to continue reading if one starts feeling primary siga headache.

3.2 Mathematical theory

3.2.1 Mathematical consequence of the model

Just looking at the model we used to describe the physical meaning of p(t) (Eq.1can deduce some very important mathematical features. The first is the fact thatdifferent from zero only in an interval on the time axis with a width equal to the sapling diodes aperture time. Another very important feature is the fact p(t) is eitherer negative or never positive. Indeed, Eq.1 shows us that the sign of p(t) is determby the constant Voffset as Z(t) can never be negative (Z(t) describes the impedancetime variant resistor).

3.2.2 Useful lemma’s and notations

Some notations:

p(t), q(t) = pulse shapeP(ω), Q(ω) = Fourier transform of p(t), q(t)Φ(ω), Ψ(w) = Phase of P, Q

Lemma 1:

Because we are looking on a limit on the phase distortion we will have to eliminthe linear part of the phase. To be able to do this the following theorem will be vuseful, as it provides us the link between the pulse p(t) and the linear part of the p

Before giving the prove of this lemma first a few words on the physical meanTheτ parameter defined in Eq.15 has the dimension of time and can be seen as tlay associated with low frequencies. From Eq.15 we see thatτ is actually the “centre ofgravity” of the pulse p(t) on the time axis. Another interpretation especially suitedpositive pulses is thatτ can be seen as the mean value of t when p(t) is associated wprobability density function.

ωddΦ

ω 0=

–

p t( )tdt

∞–

∞

∫

p t( )dt

∞–

∞

∫----------------------- τ= = (Eq.15)

Ref Nr. : JV/1/28/92/001


The prove of the lemma:

Eq.18 gives the relationship betweenΦ(ω) and p(t).

P ω( ) p t( )e j ωt– dt

∞–

∞

∫=

P ω( ) p t( ) ωt( )cos dt

∞–

∞

∫ j p t( ) ωt( )sin dt

∞–

∞

∫–=

Φ ω( )( )tan

p t( ) ωt( )sin dt

∞–

∞

∫

p t( ) ωt( )cos dt

∞–

∞

∫------------------------------------------–=

(Eq.17)

(by definition) (Eq.16)

(Eq.18)

ωdd Φ ω( )( )tan

1

Φcos( )2--------------------

ωddΦ

= (derivative chain rule) (Eq.19)

ωddΦ

ω 0=ωdd Φ ω( )( )tan

ω 0=

=(Eq.20)

ωdd Φtan

p t( )t ωt( )cos dt

∞–

∞

∫

p t( ) ωt( )cos dt

∞–

∞

∫

p t( )t ωt( )sin dt

∞–

∞

∫

p t( ) ωt( )sin dt

∞–

∞

∫

+

p t( ) ωt( )cos dt

∞–

∞

∫

2--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------–=

(Eq.21)

By taking the derivative of Eq.18 we obtain

Now we combine Eq.15, Eq.20 and Eq.21 to evaluateτ

τωd

dΦ–ω 0=

ωdd Φtan

ω 0=

–

p t( )tdt

∞–

∞

∫

p t( )dt

∞–

∞

∫-----------------------= = = (Q.E.D)

Ref Nr. : JV/1/28/92/001


rier

ay)this

nceway

Lemma 2:

Prove of the lemma: trivial (should try it once).

3.2.3 Prove of the limit

Mathematical formulation: (notations cfr. 3.2.2)

If a pulse p(t) has following properties:

then there exists an upperlimit for the phase deviation from linearity of its Foutransform.

In this descriptionτ is an arbitrary real number (associated with an arbitrary deland Ta corresponds with the diodes aperture time. The analytical expression forlimit is the following:

Prove:

Let q(t) be defined: with

Except for the normalization by making the area of q(t)=1 (which has no influeon the phase), this mathematical operation corresponds with delaying p(t) in such a

if p t( ) 0≥ for t T≤

p t( ) 0= t T>for

p t( )dt

T–

T

∫ 1=

if

if

then po t( )dt

0

T

∫ 12---≤pe t( )dt

0

T

∫ 12---= and

with pe t( ) p t( ) p t–( )+2

------------------------------= and po t( ) p t( ) p t–( )–2

------------------------------=

and

and

p t( ) 0= t τ τ Ta+,[ ]∉

p t( ) 0≥ t τ τ Ta+,[ ]∈

p t( )dt

∞–

∞

∫ ∞≠

Φ ω( ) ω ωddΦ

ω 0=

–ωTap ωTap( )sin–

ωTap( )cos--------------------------------------------

atan≤ for ωTa π≤ (Eq.22)

q t( ) p t τ+( )

p t( )dt

∞–

∞

∫---------------------= τ

p t( )tdt

∞–

∞

∫

p t( )dt

∞–

∞

∫-----------------------= (Eq.23)

Ref Nr. : JV/1/28/92/001


he-

s ofng am:

that its so called “centre of gravity” is coincident whit the time axis zero point. Matmatically expressed this operation has the following effect on the phase:

By delaying we succeeded in cancelling the linear part of the Mac Laurin seriethe phase function! To prove the validity of Eq.22 we can now concentrate on findilimit for Ψ(ω). This leads us to the following formulation of the mathematical proble

By construction q(t) has the following properties:

Prove that:

Derivation of the upperlimit:

We know that (Eq.28) and then also that

Eq.30 and Eq.31 lead us to the following identity:

Ψ ω( ) φ ω( ) ω ωddΦ

ω 0=

–= (Eq.24)

ωddΨ

ω 0=ωd

dΦω 0=

ωddΦ

ω 0=

– 0= = (Eq.25)

q t( ) 0= t Ta– Ta,[ ]∉

q t( ) 0≥ t Ta– Ta,[ ]∈

q t( )dt

∞–

∞

∫ 1=

q t( )tdt

∞–

∞

∫ 0=

(Eq.26)

(Eq.27)

(Eq.28)

Ψ ω( )ωTa ωTa( )sin–

ωTa( )cos----------------------------------------

atan≤ for ωTa π≤ (Eq.29)

Ψ ω( )( )tan

q t( ) ωt( )sin dt

Ta–

Ta

∫

q t( ) ωt( )cos dt

Ta–

Ta

∫-------------------------------------------–= Eq.18 and Eq.26 (Eq.30)

q t( )tdt

∞–

∞

∫ 0= q t( )ωtdt

Ta–

Ta

∫ 0= (Eq.31)

Ψ ω( )( )tan

q t( ) ωt( )sin dt

Ta–

Ta

∫ q t( )ωtdt

Ta–

Ta

∫–

q t( ) ωt( )cos dt

Ta–

Ta

∫-----------------------------------------------------------------------------–= (Eq.31)

Ref Nr. : JV/1/28/92/001


-

itmit

and also

When we now introduce qe(t) and qo(t) as we did for p(t) in lemma 2, we can rewrite Eq.32for all ω such that 2ωTa<π:

We can do this because (ωt-sin(ωt)) is an even and cos(ωt) is an uneven function onthe interval we’re looking at. Written in this form it is now simple to find an upperlimfor the left part of Eq.33 by finding an upperlimit for the nominator and an underlifor the denominator. To do so, we use the fact that cos(ωt) and (ωt-sin(ωt)) are mono-tonic positive functions on [0,Ta]. For the denominator:

And because of lemma 2 :

For the nominator we see:

And because of lemma 2:

Combining Eq.35 and Eq.37 results in

Ψ ω( )( )tan

q t( ) ωt ωt( )sin–( )dt

Ta–

Ta

∫

q t( ) ωt( )cos dt

Ta–

Ta

∫-----------------------------------------------------------= (Eq.32)

Ψ ω( )( )tan

qo t( ) ωt ωt( )sin–( )dt

0

Ta

∫

qe t( ) ωt( )cos dt

0

Ta

∫-----------------------------------------------------------= (Eq.33)


0

Ta

∫ qe t( )dt

0

Ta

∫

ωTa( )cos≥ (Eq.34)


0

Ta

∫ 12--- ωTa( )cos≥ (Eq.35)


0

Ta

∫ qo t( )dt

0

Ta

∫ ωTa ωTa( )sin–( )≤ (Eq.36)


0

Ta

∫ 12--- ωTa ωTa( )sin–( )≤ (Eq.37)

q t( ) ωt( )sin dt

0

Ta

∫

q t( ) ωt( )cos dt

0

Ta

∫----------------------------------------

ωTa ωTa( )sin–

ωTa( )cos----------------------------------------≤ (Eq.38)

Ref Nr. : JV/1/28/92/001


When we now use Eq.33 we finally conclude:

Ψ ω( )ωTa ωTa( )sin–

ωTa( )cos----------------------------------------

atan≤ ωTa π≤for Q.E.D.

Ref Nr. : JV/1/28/92/001


rorsjittere axisure-ndente ef-gerthe

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-

4. Time jitter compensation

4.1 Introduction

In 2.1 “time jitter” was recognized as being one of the two main systematic erencountered in an “equivalent time sampling oscilloscope”. The presence of timemeans that every sample taken by the oscilloscope can only be situated on the timwith a certain probability. This timing jitter causes systematic errors in the measments which are very hard to compensate because the jitter is very much depefrom as well the instrument as the quality of the trigger signal. This means that thfect of the jitter will be different whenever we use another instrument or another trigsignal! As far as I know there are only a few articles in technical literature covering

subject. The ones I could find are mentioned in the footnote below1. The results andconclusions of the methods used in the second and third articles will be discussecompared with the results of a method I developed myself. The two methods coverthese articles are the so called “median method” and “the compensated averageod”.The third article, written in 1962, just recently got my attention and is for the mment under investigation. Maybe it can lead me to a perfection of my algorithm, wis based on the “compensated average method”.

4.2 The median method

This method is described in the second article mentioned in the footnote. The idvery simple. Assume we have a strictly monotonic waveform x(t) which is being spled at a time Ts relative to the trigger event. Due to the jitter the value of our samwill not equal x(Ts) but will equal x(Ts-τ), with τ being a stochastic variable. When wlook at a lot of samples taken at the equivalent time Ts we know that about half of thesamples where taken at a time instant earlier then Ts and the other half taken at an instant later then Ts. Because of the strictly monotonic character of x(t) this also methat the value of half of the samples will be smaller then x(Ts) and the value of the oth-er half bigger. To have now a good estimation of x(Ts) out of our measurements it willbe sufficient to calculate the median of all sample values. It is easy to prove that thtimator is asymptotically unbiased for monotonic waveforms. Unfortunately mwaveforms we’re measuring are not monotonic (pulses, multisines, even most uniresponses). For all the non-monotonic waveforms the maxima and minima will betorted (kind of clipping), and no matter how many measurements we do to estimatmedian, this distortion can never be compensated. Although this is a disadvantaneed to be said that the algorithm has the advantage of being very simple to implewhile giving very good results in many practical cases. Because of its bias howevewill never be able to use it to do really precise measurements.

1. “On the problem of time jitter in sampling.”. A.V.Balakrishnan, IRE Transactions on information the-ory, April 1962.2. “The measurement and deconvolution of time jitter in equivalent time waveform samplers”,William L. Gans, IEEE Transactions on instrumentation and measurement, vol.IM-32, March 1983.3. “The effects of timing jitter in sampling systems.”, T. Michael Souders, Donald R. Flach, Charles Hagwood and Grace L. Yang, IEEE Transactions on instrumentation and measurement, vol.39,Feb.1991

Ref Nr. : JV/1/28/92/001


g”.sy tonal it-e wills add-ill in-ditivemewill

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4.3 The compensated average method

4.3.1 Method described in literature (footnote 2)

The most popular digital signal processing algorithm is undoubtedly “averaginThe purpose of this averaging is the elimination of noise on the signal. It is very eashow that the average of the measured signal is an unbiased estimator for the sigself when we only have the presence of regular additive noise. This means that wbe able to recover the signal out of the measurements no matter how many noise ied. Of course the number of averages needed to acquire a certain accuracy wcrease when the noise energy is increased. Unfortunately jitter noise is not adnoise. In what follows it will be shown that averaging jittered data will have the saeffect as applying a low pass filter on the real signal. The characteristic of the filterbe related to the characteristics of the jitter.

The mathematical theory of these statements:Notations: x(t) = real signal

xm(t,i) = one measured waveform (measurement i)a(t) = expectation of the measured waveform

Because of the presence of time jitter we will be able to write the following:for all instants t0 xm(t0,i) = x(t0-τi) (Eq.39)

In this expressionτi is the ith realization of a stochastic variableτ, modelling the pro-cess of time jittering, which can be characterized by a probability density function pτ).The expectation of the measured waveform is, by definition:

Eq.39 tells us that a(t) (= the value we measure when we have applied an innumber of averages) is actually the product of the convolution between the real sx(t) and the probability density function of the jitter noise p(t). When we look at tequation in the frequency domain we have the following:

Written in this form we see that the measured signal is a filtered version of the onal signal, with the characteristic of the filter being equal to P(ω), the Fourier transformof the jitter noise probability function. P(ω) being the Fourier transform of a probabili

ty density function has the following properties1:

In this equationστ equals the standard deviation of the time jitter. From Eq.41can see that P(ω) indeed has the characteristics of a low pass filter. The idea develo

in the article by Gans is to measure p(τ) and to calculate P-1(ω) in the useful band. Themain problem of course is how to measure p(τ). To do this Gans needs a kind of calibration signal that has the form of a less or more perfect slope and which is fully

1. “The Fourier integral and its applications.”, Papoulis, Mc Graw-Hill Book Company, Inc. 1962.

a t( ) x t τ–( ) p τ( )dτ∞–

∞

∫= (Eq.40)

A ω( ) X ω( )P ω( )= (Eq.41)

P 0( ) 1=

ω

2

dd

P ω( ) στ2–=

(Eq.42)

Ref Nr. : JV/1/28/92/001


ltage

ring.izedoiselfill

ce thethe

nera-act

trac-wefol-

mea-

s twote ofthe

his isis av-n beoises:

per-well as

lentnt. The

chronized with the signal we want to measure. By having a perfect slope the vodistribution at a certain time point will be equal to p(aτ) divided by a normalizing con-stant, with ‘a’ being equal to the value of the slope at the time point we’re measuUnfortunately severe problems arise when we want to make such a fully synchronslope. By saying fully synchronized I mean that there may be no “relative” phasenbetween the calibrating slope and the signal itself. This condition is very hard to fuat microwave frequencies because we would need to trigger on the signal to produslope, which means that there already will be time jittering (“phase noise”) betweensignal and the slope before they enter the scope. A second problem will be the getion of a good slope itself. A third problem, less difficult to solve however, is the fthat we have to know exactly how much white noise is present.

Although practically not suited for my purposes, the basic idea seemed very attive to me. Main reason for this is the fact that there is no theoretical reason whycould not build an asymptotically unbiased estimator based on this principle. Thelowing paragraph covers the work I did on this topic.

4.3.2 The “non-stationary sigma” method (developed by myself)

The basic idea of this method is to use the stochastic properties of the differentsurements of the signal itself to make an estimation of p(τ). By doing this the need for acalibration slope signal has totally disappeared. The algorithm I developed needsignals as input (specified later) and it returns an asymptotically unbiased estimaX(ω) as well as P(ω). The two input signals are the average of all measurements andaverage of all measurements squared (first squaring, then averaging). In fact tequivalent by using the averaged time signal and the standard deviation around theraged signal, which will be time dependent (note that this is the main -and it caproved to be the only- difference in the appearance of white stationary additive nand jitter noise). The mathematical explanation and derivation of the final equation

Notations: x(t) : real undistorted signala(t) : expectation of the measured signals(t) : expectation of the square of the measured signalp(t) : time jitter probability density functionX(ω), A(ω), S(ω), P(ω) : Fourier transforms of x,a,s,p

In practice we will be able to estimate in an unbiased manner a(t) and s(t) byforming many measurements and taking the average of these measurements astaking the average of the square of the measurements.

Derivation of the final equations:In the way I tackle the problem there are two unknowns: x(t) and p(t), or equiva

to these X(ω), P(ω). The only way to find out what they are is to build two independeequations containing these two unknowns together with things we can measurederivation of two such equations is explained in the following:

A second equation is found by calculating the spectrum of s(t):

A ω( ) X ω( )P ω( )= (see Eq.41)A first equation of course:

s t( ) x2 t τ–( ) p τ( )dτ∞–

∞

∫=By definition: (Eq.43)

In the frequency domain: S ω( ) X X∧( ) ω( )P ω( )= (Eq.44)

Ref Nr. : JV/1/28/92/001


and

ns.very

Pheepidlyansen-

s fol-

In Eq.43 we see that the spectrum of S(ω) equals the product of P(ω) and the convo-lution of X(ω) with itself. The final set of equations in X and P we have to solve:

For clarity I repeat that in this set of equations P and X are the unknowns and AS can be measured.

Solving the equations:Until now I could not find any method in literature to solve such a set of equatio

Nevertheless it was possible for me to develop a numerical algorithm that workedwell both on simulated as well as on experimental data.

The algorithm is based on the equation in P we get after elimination of X:

Solving Eq.44 would give us P(ω) and using Eq.40 together with the knowledge ofwe are able to derive X(ω). Unfortunately there are some big problems in solving tequation. First of all P(ω) is only implicitly present. Another problem is the fact that wsee the appearance of the division of A by P. Because P is lowpass, 1/P will be rarising and even reaching for infinity for frequencies above a certain limit. This methat all noise appearing at frequencies higher then this limit will be amplified tremdously. The recursive algorithm I developed takes all this into account and works alows: (indices refer to the iteration)

A ω( ) X ω( )P ω( )=

S ω( ) X X∧( ) ω( )P ω( )=

X ω( ) A ω( )P ω( )-------------=

S ω( ) AP--- A

P---∧

ω( )P ω( )=

P ω( ) S ω( )AP--- A

P---∧

ω( )-----------------------------– 0=

(cfr. Eq.40)

(substitution in Eq.43)

(Eq.45)

P0 ω( ) 1=

PN 1+ ω( ) PN ω( ) α ω( )RN ω( )+=

PN 2+ ω( )PN 1+ ω( )PN 1+ 0( )----------------------=

initialization

actualization

normalization

iteration

With: RN ω( ) S ω( )A

PN------- A

PN-------∧

ω( )----------------------------------- PN ω( )–=

α ω( ) α

1 λA ω( ) 2

------------------+----------------------------=

residua (cfr. Eq.44)

relaxation + noise stabilization

Ref Nr. : JV/1/28/92/001


the

oump-

en-his-ad

rove-eter

insind

implend a

e ef-dent

filterthetedthe

liza-42).

hite

y re-ed sig-

Parametersα andλ are there to stabilize the algorithm, their values depend onspectrum of A and of the S/N ratio. More about this follows.

Some explanation on the algorithm:First we initialize by saying P0(ω)=1. This is equivalent with assuming there is n

jitter present. Then we calculate the residue (cfr. Eq.44) associated with this asstion and we use the residue to make a correction on PN(ω). To see that you can actuallyuse this residue as a correction term assume thatα(ω)=1, by doing this the algorithmwould become:

In this recursive algorithm we recognize an ordinary contraction on P. Problemscountered with this simple algorithm (which was tried out first) led to the more sopticated algorithm. A first problem with this contraction was the fact that we hconvergence problems when the phase dispersion of A was to large. Much impment could be achieved by using relaxation. This relaxation is controlled by paramα. For pulse like signals the phase dispersion is usually small enough to putα=1 (no re-laxation), for other signals we use a smallerα. Although we can achieve convergencemuch more cases by use of parameterα, it need to be said that the algorithm still failwhen applying it on signals with a very large phase distortion, like for example all kof multisine signals.

A second problem concerning convergence was due to noise, caused by the sfact that we can only perform a finite number of averages to measure A and S. To fisolution for this I searched in some literature handling on compensation of noisfects in deconvolution and I tried applying one of the simplest frequency depen

weighting functions1 on the residua. This weighting function contains theλ parameter.Combining the relaxation with the weighting function results in the definition ofα(ω).For most signals, and always for pulses, the weigthing function will be a lowpasswith a cut off frequency that equals the frequency where the signal power equalsλparameter. By makingλ just a little greater then the noisefloor (which can be estimaquite well) we will be able to automatically cut away all frequency components ofresidua where the S/N ratio of A is smaller then 0dB.

In the algorithm we also see a normalization on P before iterating. This normation brings in very important a priori knowledge: the fact that P(0) equals 1 (Eq.Omitting this normalization also causes convergence problems.

Effect of additive white noise:In experimental data we will always have the presence of white noise. This w

noise will cause minor problems for our algorithm. This can be shown as follows.In practice we will be able to estimate a(t) and s(t), by taking the average of man

alizations of the measured signal and the squared measured signal. The measurnal in this case is no longer defined by Eq.39, but will equal xm(t,i) = x(t-τi) + n(t,i). In

this definition n is the ith realization of white noise. With this new definition we now

1. “Study and performance evaluation of two iterative frequency-domain deconvolution techniques.”,Bidyut Parruck and Sedki M. Riad, IEEE-TIM, Vol. IM-33, No.4, Dec. 1984.

PN 1+ ω( ) S ω( )A

PN------- A

PN-------∧

ω( )-----------------------------------= (Eq.46)

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cannearity

thatrise

nderentq.49

nstantin

the

abol-ts ofegli-on fornto

a par-

have to calculate the relation between x(t), the unknown, and a(t), s(t), which wemeasure. a(t) equals the expectation of the measured signal and because of the liof the expectation operator a(t) will equal the expectation of x(t-τi) plus the expectationof n(t,i). Because n(t,i) is white noise its expectation will equal zero which meansa(t) is still given by Eq.40. In frequency domain this will result in Eq.41. Problems ahowever with s(t). Indeed, let <...> denote the expectation operator:

The only assumption made when going from Eq.47 to Eq.48 is the fact that n aτare independent stochastic variables, which certainly is the case due to their diffnature.σn equals the standard deviation of the added white noise. As we see in Ethis σn is squared and added as a constant to Eq.43. In frequency domain this cowill show up as aδ-distribution. The final set of equations in the frequency domawhich describes the relation between what we measure (A(ω), S(ω)) and the unknowns(X(ω), P(ω), σn) now becomes:

When we put this S(ω) in the algorithm we immediately see that the presence ofwhite noise causes the residua RN(ω) to be different from the “ideal residua” only forωequals zero. This means that the actualized estimate PN+1 cannot be trusted at DC. Tosolve this, I need to make one assumption, namely that P(ω) can be approximated verywell by a parabolic function near DC. Eq.42 proves us that P always contains a paric part in its Mac Laurin series. In practical cases the two first frequency componenP always lie in an interval around DC where the third and higher order terms are ngible, which means that we can use these two components to make an interpolatifinding the PN+1(0) value before normalization takes place. The algorithm taking iaccount the effect of added white noise:

In this algorithmωS is the sampling frequency,α(ω) and RN(ω) are defined as previ-ous. For the interpolation we use the two estimated values nearest DC and we use

s t( ) xm2 t i,( )⟨ ⟩ x2 t τi–( ) 2x t τi–( )n t i,( ) n2 t i,( )+ +⟨ ⟩= =

s t( ) x2 t τi–( )⟨ ⟩ 2 x t τi–( )⟨ ⟩ n t i,( )⟨ ⟩ n2 t i,( )⟨ ⟩+ +=

s t( ) x2 t τ–( ) p τ( )dτ∞–

∞

∫ σn2+=

(Eq.47)

(Eq.48)

(Eq.49)

A ω( ) X ω( )P ω( )=

S ω( ) X X∧( ) ω( )P ω( ) σn2δ ω( )+=

P0 ω( ) 1=

PN 1+ ω( ) PN ω( ) α ω( )RN ω( )+=

PN 2+ ω( )PN 1+ ω( )PN 1+ 0( )----------------------=

initialization

actualization

normalization

iteration

PN 1+ 0( ) 34---P ωS( ) 1

4---P 2ωS( )+= DC interpolation

Ref Nr. : JV/1/28/92/001


the

r in-iven angerecha-and

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andnebe

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ion).theirintoltingatesfority of

abolic function with a derivative equal to zero in the origin. This leads us torespective weighting coefficients.

A final comment:Undoubtedly many questions concerning this algorithm are as yet unsolved. Fo

stance we can ask for the accuracy of the reconstructed spectrum we can obtain, gcertain number of averages. Another problem is the fact that the algorithm no loconverges when phase dispersion is too large. Where is the limit and what is the mnism causing this problem? What is the influence of the width of the time windowof the sampling time?

The only thing I can really say for sure about this algorithm is that it really doegood job, in many practical cases even much better then the previously defined mestimator. To prove this statement I applied both algorithms on simulated jitter datacompared the calculated estimates of both algorithms with each other. At last I maexperimental set up in which I measured a step recovery diode pulse and I appliereconstructor on data acquired with a minimum and a lot of jitter noise. By compathe reconstructed signals in both cases I could see that the reconstructed shape wsensitive to the amount of jitter, hereby proving in practice that the whole theorythe algorithm works very well. Unfortunately I only had the time to perform just oexperiment (I had to join the Belgian Army). More relevant experiments need todone in the future. More details on the experimental set up will be given later on.

4.4 Comparison of “non-stationary sigma” versus “median”

4.4.1 About the simulation software

In order to compare the performance of both methods I had to write software slating the real measurements. The idea: we define an analytically described pulsconvenience I used the impulse response of a third order Butterwort lowpass fifrom which we can derive a sampled version. This pulse will correspond with the thretical x(t). Then we will construct several time jittered versions out of it by evaluatx(t) not at instants nTs but at instants nTs+τ, with τ being a random number, differenfor each different version as well as for each sample. By applying the right numerecipes we can construct many different probability density functions for thisτ out ofthe RANDOM numbers created by the computer (these have a uniform distributThen I calculate the average of all time jittered versions as well as the average ofsquare. These two functions will be the estimates for a(t) and s(t) and will be putthe algorithm, after being transformed into the frequency domain by an FFT, resuin A(ω) and S(ω). When the algorithm has converged I compare the resulting estimfor X(ω) and P(ω) with their real analytically calculated value. Then I do the samethe estimates resulting from the median method, so I can finally compare the qualboth methods.

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4.4.2 Results:

For the simulation the following parameters where chosen:

x(t) : impulse response of a third order lowpass Butterwort withωcut=1 rad/sec

analytically:

τ : Gaussian jitter noise withστ = .75 sec

Ts : 0.3 sec (sampling time)

Number of samples : 256

Start time: -10 sec (stop time = 66.5 sec)

Number of simulated waveforms used for averaging : 1000

On the following pages we see the results of such a simulation :

solid line : theoretical pulse (x(t))

dashed line : pulse reconstructed with my method

dot dashed line : pulse reconstructed with median method

x t( ) e t– 1

3------- 3

2-------t

sin3

2-------t

cos– e

t2---–

+=

x t( ) 0=

for t >= 0

for t < 0

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Reconstructed waveforms in time domain (Voltage vs. sec)

median estimated pulse clipped at max

my method introduces ringing

Deviation of reconstruction from ideal (Voltage vs. sec)

max deviation significantly lower withmy method

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Reconstructed spectra (dB vs. Hz)

my method fails totally above acertain frequency limit

Deviation of reconstruction from ideal (dB vs. Hz)

bias of median estimator

my method unbiased until a certain frequency limit

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Estimation of P(ω) vs. theoretical Gaussian (dB vs.Hz)

Deviation from theoretical curve (dB vs Hz)

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od asical

l

s

theitself,

need-ave-345

de-uctionation

4.4.3 Conclusions

Looking at the graphs shows both advantages and disadvantages of my methwell as the median method. All effects we notice are fully compatible with theoretpredictions. A summary of the characteristics:

Median method : advantages: - easy to implement- covers wide frequency spectrum

disadvantages: - biased estimates at maxima and minima (clipping effect)- biased spectrum at all frequencies (even at DC)

My method: advantages: - theoretically asymptotically unbiased (both time and frequency domain)- practically unbiased in frequency domain unti a certain frequency limit

disadvantages: - time consuming algorithm- ringing appears in time domain- information lost at certain frequency limit (depending on amount of jitter)- algorithm converges only for pulse like signal

About the time consumption of my algorithm, this is mostly due to the fact thataverage of the square of the measured signal can not be calculated in the scopebut has to ba calculated in a controller, which means that many data transfers areed. The calculation of the reconstructed spectrum itself out of the two calculated wforms takes less then a minute for the simulated data I used, performed on aworkstation with coprocessor.

Obviously the choice between using my method or the median method totallypends upon the actual application. Because my goal is to make a precise reconstrof the spectrum of the pulse received when applying the new back-to-back calibrmethod (cfr. Chapter 2), my method is ideally suited.

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Appendix

A.1. Hardware

A.1.1 HP8510B Network analyzer

Using a few samples (microstrip coupled line filter, transmission lines,...) I learnedhow to perform accurate measurements with the network analyzer. This concerned theuse of several calibration kits and techniques (sliding load,...). I also familiarized my-self with several kinds of precision connectors and their maintenance. When I thoughtto have enough experience in handling the 8510B I studied the use of the verificationkits (APC-7 and 3.5mm) and I succeeded in performing a successful verification for thethree calibration kits we possess.

A.1.2 Microwave lab at the university

As I am also responsible for the microwave environment at the ELEC department atthe Brussels University I helped starting up the new ‘microwave lab’ for the students.This concerned several things. I purchased a lot of documentation about microwave ma-terial manufacturers and suppliers (connectors, solid state devices, laminates,...). I alsoinvestigated the overall precision of the printed circuit board realization at the universi-ty to see how well suited this is to fabricate microwave designs (precision of plots, etch-ing,...). I studied and performed the so called ‘full sheet resonance method formeasuring the epsilon of several laminates.

A.1.3 HP54120 Broadband sampling oscilloscope

For this moment the only instrument we have and had at our disposal too do somenonlinear microwave measurements is the HP54120 in our lab. It is also the key instru-ment of the previous chapters, concerning the nonlinear calibration. Before being ableto say something about the instrument I first had to learn how to use it.

A.2 Software

A.2.1 Rocky Mountain Basic

Because we needed to perform many sophisticated measurements the use of RockyMountain Basic to set up automized experimental set ups was very useful to us. To beable to use it all instrument interfaces needed to be studied.

A.2.2 Microwave Design System

Many effort was put in the use of MDS. This software tool is very interesting for us.In the first place it gives us the possibility to simulate nonlinear circuits, which is veryimportant for our work, but we can also use it to help students as well as ourselvesbuilding and understanding microwave designs.

A.2.3 ANSI-C

Because the other three members of our group are programming in the C-language,it was very useful to me to follow a beginners course on this topic. This way I couldstay ‘compatible’ with them.

phd. activity report: oct.90 - july 91 -...

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