[time-nuts] On some pitfalls of the dual mixer time difference method of horology

[Ulrich Bangert]

Hi folks,

Let me say sorry in advance for this lengthy posting but I did not
manage to make it shorter. 

If one reads the common literature on oscillator characterization then
he quickly finds out that one of the standard methods for oscillator
comparisons is the "Dual mixer time difference" method, or short: DMTD.
While most of you readers may know the principle of DMTD very well
please allow me to describe the method in a few words for those who are
not so familiar with it:

In the DMTD the signal of the OUT (Oscillator Under Test) as well as the
signal of a reference oscillator are mixed down to the low frequency
domain (say 1 to some ten Hz) by means of two double balanced mixers and
a third oscillator which I will call the transfer oscillator. The down
mixed low frequency signals have an interesting property. 

Let us assume the OUT signal changes its phase on its working frequency
of 10 MHz by an amount of 1 ps (expressed in absolute time) then this
absolute change can be expressed as a 1 ps / 100 ns = 1E-5 change
relative to the period length of 100 ns. Please note that a 1 ps change
in phase is orders of magnitude smaller than we are able to measure with
direct methods. 

The down mixed signal of the OUT experiences a relative phase shift of
1E-5 in this case as well because it is a fundamental property of mixing
to preserve phase/frequency information of the input signals to be mixed
(That is the reason why phase and frequency modulation can be used with
superhet receivers). 

Now comes the clue: While the relative phase shift stays the same, the
phase shift expressed in absolute time is MUCH bigger now because the
relative phase shift value now applies to a much smaller frequency and
therefore to a much bigger period length! Say we have a down mixed
frequency of 1Hz then the 1E-5 relative phase shift is 1E-5 * 1 s = 10
microseconds absolute. Wow! We have amplified the effect from the domain
of un-measurability to something that can be measured with a garden
variety counter.  

Who has followed the explanation closely up to now might argue that the
down mixed signal preserves the phase/frequency information of the OUT
as well as that of the transfer oscillator so that any phase/frequency
change that we observe on the down mixed signal may relate to the OUT or
the transfer oscillator or both of them. This argument is correct!  And
that is why we have the second mixer which is used to down mix the
reference oscillator's signal with the transfer oscillator. 

Assume the transfer oscillator experiences a phase shift. Then this
phase shift is the SAME in both down mixed signals. If we make a time
difference measurement between the two down mixed signals any phase
shift of the transfer oscillator is believed to cancel out completely.
This is what the DMTD is all about and usually in schematic diagrams
explaining the DMTD we find two zero crossing detectors for the down
mixed signals followed by a TIC (time interval counter) measuring the
time between a zero crossing of one signal to the zero crossing of the
second signal. 

Now it seems we have really created the universal workhorse of horology:
We have amplified the effect to be measured by a factor of 1E7 by simply
mixing down the signals to 1 Hz and the transfer oscillator's influence
completely cancels out due to the difference measurement. That is how we
find it described in the textbooks!

I do not know how many of you readers really built a DMTD circuit or
used one. I know at least that some of you are planning to build
circuits like that and that TVB owns an instrument (a TSC 5110) that
works according this principle. 

I built a DMTD and made measurements with it on the few good oscillators
that I own. While my experiments have shown that the principle works
INDEED as described they also have shown that the DMDT has some pitfalls
which you will find ABSOLUTELY NOTHING about in the textbooks. I get the
impression that a lot of the authors explaining the method simply reecho
what the have read elsewhere and that only a very small number of
experts have an experience of their own with this method. I would like
to explain what I have found out and discuss this stuff with you.

First Pitfall of DMTD: Transfer oscillator effects do NOT cancel out
completely

The explanation above gave you the impression that any effect in phase
or frequency of the transfer oscillator cancels out due to the
difference measurement between the down mixed signals, didn't it? And
hey, this argument is not really wrong. But we need to be precise: Any
transfer oscillator related effect cancels out completely if we look at
the two down mixed signals at the SAME time! This is due to the fact
that at the same time the transfer oscillator is in the same state
concerning both channels.

But do we really do so? No, we do not! We look at one signal when its
own zero crossing takes place and we look to the second signal when its
own zero crossing takes place. With 1 Hz beat notes the zero crossings
may be up to 500 ms apart of each other. No means of "at the same time",
but anything between 0 and 500 ms. Of course: We may have a certain hope
that the transfer oscillator's properties do not change completely
within the maximal time of 500 ms. However, the principal idea that the
transfer oscillator is in absolutely the same state concerning both
channels is wrong because we do not look at both channels at the same
time and for that reason its effects will not cancel completely but only
up to a certain degree. 

Some authors (seldom to be found) will show you schematics that include
phase shifting elements in the OUT's or the reference oscillator's
signal path BEFORE the mixers. By means of phase shifting one of the
original signals the zero crossings of BOTH down mixed signals can be
forced to happen at the same time or at least app. the same time. In
this case the transfer oscillator's effects do indeed cancel out and you
may assume that this author knows what he's talking about. 

But the measurement itself gets more complicated this way because for
the comparison of the oscillators we do not only the have to take the
TIC's measurement into account but also the phase shift that we now have
applied artificially which is not measured easily with the same
precision. If you see a TIC being part of a DMDT system WITHOUT phase
shifting elements then be prepared that the author has not the definite
in-depth knowledge about his topic. However, if DMTD is THAT standard
and common in horology one would expect this property of the DMTD being
discussed in hundreds of scientific articles available in the internet.
But it is not. I suggest you search yourself a bit for that! If you are
lucky then you may come across the very ONE SINGLE source of information
about this fact that I have been able to find at

http://tmo.jpl.nasa.gov/progress_report/42-143/143K.pdf

The fact that this topic is covered by only one publication is why I
think real experience with the DMTD method is the domain of a very few
experts. Greenhall is one of the really big guns in horology and has
published a lot of intelligent stuff about it. He clearly shows in this
publication that a time interval counter is not sufficient for the DMDT
method without artificial phase shifting. Instead two independent
time-tag counters are necessary and a bunch of mathematics that most of
these DMDT people do not even have an idea about. 

Second Pitfall of DMTD: Decreasing slope to noise ratio counteracts the
magnifying effect of down mixing

In a textbook I once read the remarkable sentence: "When it comes to
precise timing measurements the slope to noise ratio and not the signal
to noise ratio is the true figure of merit" Remarkable because I found
it in a textbook about instrumentation in nuclear physics. And
remarkable because it mentions the "slope to noise ratio" which I had
never heard about before. 

For a better understanding of what's coming next I suggest you download
 
http://www.ulrich-bangert.de/AMSAT-Journal.pdf

from my homepage. This article is in German but you are not expected to
read it. However, it contains some graphics which I would like to refer
to.

Consider the case that we need to make a timing measurement on a
sinusoidal signal, for example to determine its period length. While he
may not be able to explain completely why, every technician would decide
to use the zero crossings of the signal as the timing reference. That
is: He would build a zero crossing detector and measure the time between
two zero crossings from negative to positive values. 

Had we a noise-free signal available then there were no problem at all
because the noise-free signal crosses the zero line at a sharp defined
point in time. However, noise-free signals are an idealization not given
with real-world signals. There is always a certain amount of noise,
sometimes more, sometimes less, a fact that documents itself in the well
known signal to noise figure. 

In "Abbildung 6" on page 12 I have drawn a noisy sinusoidal signal
crossing the zero line. Please note that I have chosen a real bad signal
to noise figure but that has just been done to show you the principle of
this effect. The effect itself I am explaining now takes place at every
signal to noise figure even at very good ones.

Because the amplitude of the sinusoidal signal contains amplitude noise
it becomes immediately clear that the signal now has a certain chance to
cross the zero line also at times before the cross point of the noise
free signal as well as behind that. It may even cross the zero line
several times. 

>From geometrical considerations concerning the slope of the underlying
sinusoidal signal we can see that a certain amount of amplitude noise
directly translates into phase noise when it comes to measurements of
zero crossings and that the slope is the "translation factor". In the
first graphics I have made the slope to 1 giving a 1:1 translation of
amplitude noise into phase noise.

Now consider "Abbildung 7" on page 13. Here, almost everything is the
same. The signal has the same amplitude and the same signal to noise
ratio. However it has only half the frequency. Due to that it has only
half of the slope at the zero crossing than the first signal and you see
nicely how 1 part of amplitude noise now roughly translates into 2 parts
of phase noise. It becomes clear, that if everything else stays the same
the slope of the wave at the zero crossings and therefore nothing else
than the wave's frequency decides how precise we can measure the time of
its zero crossings.

Perhaps you do already see what that means for DMTD? Even if we consider
the mixing process as being noise free which is a VERY optimistic
assumption then every factor by which we down mix the signal and by
which we magnify the effect to be measured will be counteracted by a
decrease in slope to noise ratio in the same measure.

For those who feel that this objection is pure academic I suggest the
following experiment: If you own a counter that can do statistics, take
it and lock it to the best frequency reference that you have available.
Now take the best synthesizer generator that you have available, for
example a HP3325 and lock it to the same reference. Set the generator to
1 Hz sine and let the counter make say 100 measurements on period length
and then display the standard deviation of the measurements. I am sure
you will be surprised. 

For those of you who have not this equipment at hand: I just made the
experiment with my HP5370A and my HP3325A. The HP3325A was set to 1 Hz
sine and 1 Vpp output. The HP5370A was set to period measurement and
sample size = 100.  

The first 100 measurements gave a standard variation from sample to
sample of 434 microseconds! 434 MICROSECONDS? Where has the famous 20 ps
resolution of the 5370 gone? The second hundred samples deliver a
standard variation of 289 from sample to sample, the third 363
microseconds, so the first measurement can not have been this wrong. I
do not need to remember you that the standard deviation is kind of
typical error. Assumed a Gaussian distribution of errors the real error
of a single measurement may even be as big as +/- 3 times this value.
Just to check that nothing is defect now set the 3325 to 10 MHz and
watch the counter display a standard variation of 30-40 ps as we are
used from it. This all is taking place just by means of different slope
to noise ratio and nothing else. 

So, while down mixing the OUT signal from 10 MHz to 1 Hz may have
increased the effect to be measured to 10 microseconds which we had
expected to be VERY EASY to measure, the true problem now turns out that
we must measure a resolution of about 10 microseconds on a 1 Hz signal
with a very low slope to noise figure! 

I once computed that on 10 MHz a signal to noise ratio of 20 dB is
sufficient to measure the zero crossings of a sinusoidal signal with an
uncertainity of about 1 ns. With some simple considerations one can
easily compute that for a 1 Hz signal a stunning signal to noise ratio
of 160 dB is necessary for the same precision. With a frequency relation
of 7 orders of magnitude the slope of the 1 Hz signal is 7 orders of
magnitude smaller than that of the 10 MHz signal. In order to get the
same slope to noise ratio the root mean square level of the noise has to
decrease by 7 orders of magnitude which give rise to a necessary
decrease of 140 dB in noise power.

Have you been told that by one of the friendly authors who yarn about
DMTD? There aren't no such things in reality as nanosecond resolution
timing measurements on 1 Hz sinusoidal signals! And yes, the sinusoidal
form of the signal IS part of the problem! But in case you are going to
think about making a "digital" signal out of the sine by some clever
trick, you are on the wrong track because it takes the translation of
amplitude noise to phase noise just from one point to another point in
the apparatus but does not solve it. And before you give it a try on
yourself: The trigger circuits in modern counters are already pretty
tough!

As it turns out there IS a clever way to handle this situation at least
up to the principal limits. Again there ARE a very few specialists who
know about this problem very well but they are rare to find animals. If
you ever want to build a DMTD system by yourself then be sure that you
have read 

Dick / Kuhnle / Sydnor: "Zero-crossing detector with sub microsecond
jitter and crosstalk" 

before! Some people will tell you that you need "low noise zero crossing
detectors" but only these guys will explain you in full detail how to
build them! This text is a bit difficult to get from the net. If you do
not manage to download it please ask me for help. Basically the authors
use a cascaded chain of low pass filters and combinations of
non-limiting and limiting amplifiers to increase the slope of the signal
in several steps while at the same time they try to filter out as much
noise as possible. As you can see from the title anything better than 1
microsecond jitter (!) is considered state of the art.

Third Pitfall of DMTD: Phase corruption due to mutual crosstalk

Given 2 signals in the same circuit there will be a mutual crosstalk
between them. Crosstalk means that a damped version of signal 1 rides on
top of signal 2 and that a damped version of signal 2 rides on top of
signal 1. Similar to the case of noise there may be more or less
crosstalk which documents in the crosstalk damping figure.

Let us consider how crosstalk can influence oscillator stability
measurements. Let us first assume that we have 2 signals of the SAME
frequency and what happens if there is crosstalk between them. 

If both signals have the same phase then "signal 2 riding on the top of
signal 1" means that the amplitudes of both signals add up at every
point. Depending on the crosstalk damping figure the result will be a
wave which's amplitude at every point in time is a little bit greater
than that of signal 1 alone.  

If both signals are 180 degree out of phase then "signal 2 riding on the
top of signal 1" means that the amplitudes of both signals subtract at
every point. Depending on the crosstalk damping figure the result will
be a wave which's amplitude at every point in time is a little bit
smaller than that of signal 1 alone.  

The cases of 90 degree and 270 degree phase shift are a bit more
complex. Signal 2 now has its maximal and minimal amplitude at the
points where signal 1 has its zero crossings. "Riding on top" in this
case means that the zero crossings of the combined signal are shifted a
little bit to left or right just depending on the sign of signal 2. 

And of course the phase shift between signal 1 and signal 2 may be
anything in between creating a mixture of amplitude and phase
deformation effects. All of that deformation effects are not serious in
the sense that they are static in time. Every period of the combined
signal bears the same deformation.

But now let us consider the cases that the two signals are not of the
SAME frequency but have different frequencies which are very close to
each other. This is exactly the situation when we are going to compare
two very good oscillators! Now the situation is different in that signal
1 does not ride "static" on signal 2. Instead of having a constant phase
relation as with same frequency signals, the phase of signal 1 now moves
along the phase of signal 2 with an velocity that is given by the beat
frequency of the two signals and the peaks of signal 1 are sometimes at
the peaks of signal 2 but also sometimes at the zero crossings of signal
2. Due to this crosstalk the combined signals are both amplitude AND
phase modulated by their counterpart. That is what is meant by phase
corruption due to crosstalk. 

Now that we understand how there is phase corruption, let us compute how
big this phase corruption really is. In order to say how big the phase
corruption is, we need to say for which crosstalk damping figure we are
going to compute it. In situations like this I say: 100 dB. 100 dB is a
handy number in that we have real world examples available of what 100
db means: 

If you buy a good coaxial cable, this may have a shielding effectiveness
of 80 dB at radio frequencies. If you spend some bucks more you can get
a shielding effectiveness of 90 dB. 100 dB shielding is top and only
possible with double shielding and I do not remember to have seen a
shielding effectiveness been advertised better than 110 dB. So, 100 dB
are a VERY good isolation of two electrical circuits from each other and
are surely very hard to realize on the same printed circuit board or
within the same device. So, if 100 dB may be considered state of the art
in isolation let us see what -100 dB in mutual crosstalk means.

Assume a signal having the amplitude 1 V and the frequency 1 Hz. A
second signal that is damped 100 dB to this signal has an amplitude 1E-5
V. The 1 Hz signal has a slope of 2*Pi at the zero crossing, so the
damped signal riding on top will shift the zero crossing by
approximately +/-1E-5/(2*Pi) which is app. +/-0.6 milli degree and a +/-
1.6E-13 s effect expressed in absolute time on a 10 MHz carrier. This
should be very easy noticed on high resolution instruments as the TSC
5110!

AllanChart.Pdf shows a simulated oscillator comparison of two sources
that are absolutely stable but have a limited (sn+n)/n of about 80 dB
and a mutual crosstalk of -100 dB. Because the sources itself are stable
the only source of un-stability in the measurement is the jitter due to
amplitude to phase noise conversion and we would expect the tau-sigma
diagram to decline at a -1 slope from some starting point with white
amplitude noise. Instead we receive this! The two frequencies have been
chosen to be 1/16 Hz apart, therefore the big peak in Allan deviation at
approximately 8 s. You may be curious to ask how an effect that is 100
dB down the carrier can have THAT big influence with a (sn+n)/n of 80
dB. One would expect everything below -80 dBc to be "buried" in noise.
But here you must remember that for the (sn+n)/n figure the TOTAL noise
energy is used. If you look at the noise energy as a function of
frequency, which you do if you look at the signal with a spectrum
analyzer then the display for an 80 dB (sn+n)/n signal may look pretty
similar to SpecChart.Pdf showing that a -100 dBc carrier will make a
prominent peak in he spectrum. And with every 20 dB less isolation the
effect will be one order of magnitude more prominent in the
tau-sigma-diagram.

>From this example we see that mutual crosstalk should be one of our
biggest enemies in oscillator characterization. Pitfalls 1 and 2 may be
addressed by clever circuitry and mathematics but concerning crosstalk
we are at a fundamental border were we do not easily decrease it.  

The effects of mutual crosstalk on oscillator stability measurements
seem to be not well known in horology or at least I missed to find any
hints on it. That crosstalk in general has a serious impact on precise
phase measurements can however be read in: 

http://iram.fr/IRAMFR/TA/backend/phasemeter/index.html

I would like to hear from you if you ever experienced one of the
pitfalls yourself. Especially the owners of equipment like the TSC 5110
are asked to explain if they ever noticed crosstalk effects. Or do these
instruments perhaps involve some tricks that I am not aware of? 

One of the ideas I have is: If it is already clear that the zero
crossings of OUT and reference channel ARE apart from each other why do
I need the second channel WHILE I measure the zero crossing of the first
channel? Perhaps equipment like the TSC 5110 uses a very high isolation
switch to keep the second signal completely out of the box while it
measures the zero crossing of the first channel? And then re-computes
them to the same epoch as described in the Greenhall paper?

Best regards

Ulrich Bangert
df6jb at ulrich-bangert.de
Ortholzer Weg 1
27243 Gross Ippener
Germany 
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