Hi
Thank you for the clarification and rf-tools link.
Agree with your calculation. That’s why I raised this question regarding a fixing PN degradation by Pendulum CNT-91.
Could you please explain where is the error in my reasoning of the experiment :
Karen, ra3apw
Hi
As HP found out back around 1973 or so, translating ADEV to phase noise
is not possible. This is true, even if you have the ADEV numbers for a variety
of Tau values as opposed to some sort of “average” kind of number.
There are a number of things ( like spurs ) that can strongly influence a counter
based ADEV reading, and have very little impact on a phase noise ( or signal to
noise reading. There also are ways the shape of the phase noise curve can
impact ADEV and have very little signal to noise impact for a specific signal.
By far the best way to do this is to properly measure phase noise at various
offsets from carrier. You can then look at the dbc/Hz numbers at each offset.
This lets you see what your devices are doing to the signal. You can then track
down the offending bit or piece and fix the problem.
The easiest way I know of to do phase noise is to quadrature lock two identical
sources into a double balanced mixer. You then put in a simple amplifier stage
to drive the mix down output into a sound card or spectrum analyzer. Total cost
if you already have a sound card should be < $50 ( US dollars …) for a DIY version.
That assumes you have the usual junk box parts and do a point to point wire
version.
Some example ADEV plots:
http://leapsecond.com/museum/manyadev.gif http://leapsecond.com/museum/manyadev.gif
http://leapsecond.com/museum/manyadev.gif http://leapsecond.com/museum/manyadev.gif
Some plots of a number of measurements:
http://www.leapsecond.com/pages/fe405/ http://www.leapsecond.com/pages/fe405/
Quick primer on phase noise measurement
https://www.npl.co.uk/special-pages/guides/gpg68_noise https://www.npl.co.uk/special-pages/guides/gpg68_noise
( The easy approach starts on page 21 :) )
Bob
On Jun 19, 2022, at 11:40 AM, Karen Tadevosyan via time-nuts time-nuts@lists.febo.com wrote:
Hi
Thank you for the clarification and rf-tools link.
Agree with your calculation. That’s why I raised this question regarding a fixing PN degradation by Pendulum CNT-91.
Could you please explain where is the error in my reasoning of the experiment :
Karen, ra3apw
time-nuts mailing list -- time-nuts@lists.febo.com
To unsubscribe send an email to time-nuts-leave@lists.febo.com
Bob,
Many thanks for the guidance you provide and the phase noise measurement
document.
Can you provide feedback on this reasoning: A counter is like an ADC but
in the frequency domain. So if you measure with 0.01 s tau you basically
average over 0.01 s so you can only observe "phase noise" (e.g. energy
that is not at the exact requested frequency) up to maximum 50 Hz from
the carrier. But as you measure the true frequency changes the
sensitivity of this measurement is extremely high. Translating the
amount of time spend at a certain frequency away from the carrier
(ADEV?) into a phase noise number in dBc is something I do not yet
understand.
With a (very good) spectrum analyzer you may be able to come close to
the carrier but as there is so much energy in the carrier it will be
difficult to observe phase noise energy closer than say 1 or 10 kHz (at
least not with the equipment I can afford) so any phase noise plot
created using a spectrum analyzer can not be better than the combined
phase noise of all LO's in the spectrum analyzer and will start at say 1
or 10 kHz.
For the frequencies between 50 Hz and 20 kHz the simplest option is to
use a second LO and a mixer and a slow (loop BW below 10 Hz)PLL to keep
the mixer in quadrature and feed the output of the mixer, after low pass
filtering, into a PC soundcard for FFT processing.
Erik.
On 19-6-2022 22:45, Bob kb8tq via time-nuts wrote:
Hi
As HP found out back around 1973 or so, translating ADEV to phase noise
is not possible. This is true, even if you have the ADEV numbers for a variety
of Tau values as opposed to some sort of “average” kind of number.
There are a number of things ( like spurs ) that can strongly influence a counter
based ADEV reading, and have very little impact on a phase noise ( or signal to
noise reading. There also are ways the shape of the phase noise curve can
impact ADEV and have very little signal to noise impact for a specific signal.
By far the best way to do this is to properly measure phase noise at various
offsets from carrier. You can then look at the dbc/Hz numbers at each offset.
This lets you see what your devices are doing to the signal. You can then track
down the offending bit or piece and fix the problem.
The easiest way I know of to do phase noise is to quadrature lock two identical
sources into a double balanced mixer. You then put in a simple amplifier stage
to drive the mix down output into a sound card or spectrum analyzer. Total cost
if you already have a sound card should be < $50 ( US dollars …) for a DIY version.
That assumes you have the usual junk box parts and do a point to point wire
version.
Some example ADEV plots:
http://leapsecond.com/museum/manyadev.gif http://leapsecond.com/museum/manyadev.gif
http://leapsecond.com/museum/manyadev.gif http://leapsecond.com/museum/manyadev.gif
Some plots of a number of measurements:
http://www.leapsecond.com/pages/fe405/ http://www.leapsecond.com/pages/fe405/
Quick primer on phase noise measurement
https://www.npl.co.uk/special-pages/guides/gpg68_noise https://www.npl.co.uk/special-pages/guides/gpg68_noise
( The easy approach starts on page 21 :) )
Bob
On Jun 19, 2022, at 11:40 AM, Karen Tadevosyan via time-nuts time-nuts@lists.febo.com wrote:
Hi
Thank you for the clarification and rf-tools link.
Agree with your calculation. That’s why I raised this question regarding a fixing PN degradation by Pendulum CNT-91.
Could you please explain where is the error in my reasoning of the experiment :
Karen, ra3apw
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Hi
Bob, many thanks for your explanations/recommendations and links.
According your advices I will try to make PN measurement on new SA with cross-correlation to get more clear picture of the output signals.
The next my step will be a quadrature method measurement and comparison of the results.
Karen, ra3apw
From: Bob kb8tq kb8tq@n1k.org
Sent: Sunday, June 19, 2022 11:46 PM
To: Discussion of precise time and frequency measurement time-nuts@lists.febo.com
Cc: Karen Tadevosyan ra3apw@mail.ru
Subject: Re: [time-nuts] Fixing PN degradation via ADEV measurement
Hi
As HP found out back around 1973 or so, translating ADEV to phase noise
is not possible. This is true, even if you have the ADEV numbers for a variety
of Tau values as opposed to some sort of “average” kind of number.
There are a number of things ( like spurs ) that can strongly influence a counter
based ADEV reading, and have very little impact on a phase noise ( or signal to
noise reading. There also are ways the shape of the phase noise curve can
impact ADEV and have very little signal to noise impact for a specific signal.
By far the best way to do this is to properly measure phase noise at various
offsets from carrier. You can then look at the dbc/Hz numbers at each offset.
This lets you see what your devices are doing to the signal. You can then track
down the offending bit or piece and fix the problem.
The easiest way I know of to do phase noise is to quadrature lock two identical
sources into a double balanced mixer. You then put in a simple amplifier stage
to drive the mix down output into a sound card or spectrum analyzer. Total cost
if you already have a sound card should be < $50 ( US dollars …) for a DIY version.
That assumes you have the usual junk box parts and do a point to point wire
version.
Some example ADEV plots:
http://leapsecond.com/museum/manyadev.gif
http://leapsecond.com/museum/manyadev.gif
Some plots of a number of measurements:
http://www.leapsecond.com/pages/fe405/
Quick primer on phase noise measurement
https://www.npl.co.uk/special-pages/guides/gpg68_noise
( The easy approach starts on page 21 :) )
Bob
On Jun 19, 2022, at 11:40 AM, Karen Tadevosyan via time-nuts <time-nuts@lists.febo.com mailto:time-nuts@lists.febo.com > wrote:
Hi
Thank you for the clarification and rf-tools link.
Agree with your calculation. That’s why I raised this question regarding a fixing PN degradation by Pendulum CNT-91.
Could you please explain where is the error in my reasoning of the experiment :
There is one 10 MHz OCXO with ADEV = 5 mHz
There are two boards (DUT1 and DUT2) which multiply 10 MHz OCXO signal by 2.5 using the PLL method
DUT1 has 25 MHz output signal with high PN (checking by air and by measurement of S/N)
DUT2 has 25 MHz output signal with low PN (checking by air and by measurement of S/N)
Experiment’s steps:
Step 1: DUT1 ADEV measuring gives me a value of 60 - 70 mHz instead of the expected 12.5 mHz (5 mHz x 2.5)
Step 2: DUT2 ADEV measuring gives me a value of 10 - 12 mHz which matches the expected 12.5 mHz (5 mHz x 2.5)
Step 3: based on ADEV values which in the first case (DUT1) are much greater than expected and in the second case (DUT2) coincide with the expected I conclude that PN of the output signal from DUT2 will be lower than from DUT1.
I can see this PN degradation using Pendulum CNT-91 only as R&S FSQ8 does not fixate any PN degradation between DUT1 and DUT2
Karen, ra3apw
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Erik,
A counter actually measures a number of phase measurements. Then, as you
process that you get a frequency readout based on the difference between
them (event-count divided by time between phase measurements). Now, as
you want to do frequency read-out, you can do a handful of filtering
mechanisms, and the CNT-91 can do the linear regression. This filtering
takes a number of samples and provides a filter to estimate frequency.
The consequence of that is that variations you see now have a different
scale than if you did the original calculation of only two
phase-samples. This creates a bias function and variations needs to be
corrected for to get numbers you can relate to the normal scale. It's
great for giving better frequency readings, but if you aim to quantify
the variations you end up fooling yourself.
Also, your assumption on observation frequency in Nyquist is wrong.
Turns out that aliasing of higher frequencies is very problematic. It is
only very recent instruments that can have the ability to avoid aliasing
(by using digital decimation), but a counter is not one of them, it is
fully exposed to the aliasing problem.
There is translations charts to convert the noise-amplitude of each
noise type into phase-noise and ADEV readings. If you have truely random
noise obeying the rules, you can convert between them. Toss in a spur,
and it works differently, and well, you need to convert those too
according to other rules. Look for "Enricos chart".
Noise types reaching for high frequencies compared to measurement tau0
will affect the resulting ADEV for sure. The bandwidth of that even
affects white phase modulation directly, and flicker phase modulation to
some degree.
So, a counter is really like an ADC for phase, with wide bandwidth input
and a sub-sampling mechanism (trigger/time-base). Through processing
frequency estimates can be provided. Aliasing occurrs in the
sub-sampling. Modern counters can provided estimation filters than goes
from a higher sub-sampling rate to a lower, which to some degree removes
aliasing, but not fully. These frequency estimation methods form a form
of decimation filter.
Cheers,
Magnus
On 2022-06-20 08:45, Erik Kaashoek via time-nuts wrote:
Bob,
Many thanks for the guidance you provide and the phase noise
measurement document.
Can you provide feedback on this reasoning: A counter is like an ADC
but in the frequency domain. So if you measure with 0.01 s tau you
basically average over 0.01 s so you can only observe "phase noise"
(e.g. energy that is not at the exact requested frequency) up to
maximum 50 Hz from the carrier. But as you measure the true frequency
changes the sensitivity of this measurement is extremely high.
Translating the amount of time spend at a certain frequency away from
the carrier (ADEV?) into a phase noise number in dBc is something I do
not yet understand.
With a (very good) spectrum analyzer you may be able to come close to
the carrier but as there is so much energy in the carrier it will be
difficult to observe phase noise energy closer than say 1 or 10 kHz
(at least not with the equipment I can afford) so any phase noise plot
created using a spectrum analyzer can not be better than the combined
phase noise of all LO's in the spectrum analyzer and will start at say
1 or 10 kHz.
For the frequencies between 50 Hz and 20 kHz the simplest option is to
use a second LO and a mixer and a slow (loop BW below 10 Hz)PLL to
keep the mixer in quadrature and feed the output of the mixer, after
low pass filtering, into a PC soundcard for FFT processing.
Erik.
On 19-6-2022 22:45, Bob kb8tq via time-nuts wrote:
Hi
As HP found out back around 1973 or so, translating ADEV to phase noise
is not possible. This is true, even if you have the ADEV numbers for
a variety
of Tau values as opposed to some sort of “average” kind of number.There are a number of things ( like spurs ) that can strongly
influence a counter
based ADEV reading, and have very little impact on a phase noise ( or
signal to
noise reading. There also are ways the shape of the phase noise
curve can
impact ADEV and have very little signal to noise impact for a
specific signal.By far the best way to do this is to properly measure phase noise at
various
offsets from carrier. You can then look at the dbc/Hz numbers at each
offset.
This lets you see what your devices are doing to the signal. You can
then track
down the offending bit or piece and fix the problem.The easiest way I know of to do phase noise is to quadrature lock two
identical
sources into a double balanced mixer. You then put in a simple
amplifier stage
to drive the mix down output into a sound card or spectrum analyzer.
Total cost
if you already have a sound card should be < $50 ( US dollars …) for
a DIY version.
That assumes you have the usual junk box parts and do a point to
point wire
version.Some example ADEV plots:
http://leapsecond.com/museum/manyadev.gif
http://leapsecond.com/museum/manyadev.gifhttp://leapsecond.com/museum/manyadev.gif
http://leapsecond.com/museum/manyadev.gifSome plots of a number of measurements:
http://www.leapsecond.com/pages/fe405/
http://www.leapsecond.com/pages/fe405/Quick primer on phase noise measurement
https://www.npl.co.uk/special-pages/guides/gpg68_noise
https://www.npl.co.uk/special-pages/guides/gpg68_noise( The easy approach starts on page 21 :) )
Bob
On Jun 19, 2022, at 11:40 AM, Karen Tadevosyan via time-nuts
time-nuts@lists.febo.com wrote:Hi
Thank you for the clarification and rf-tools link.
Agree with your calculation. That’s why I raised this question
regarding a fixing PN degradation by Pendulum CNT-91.Could you please explain where is the error in my reasoning of the
experiment :* There is one 10 MHz OCXO with ADEV = 5 mHz
* There are two boards (DUT1 and DUT2) which multiply 10 MHz OCXO
signal by 2.5 using the PLL method
* DUT1 has 25 MHz output signal with high PN (checking by air
and by measurement of S/N)
* DUT2 has 25 MHz output signal with low PN (checking by air
and by measurement of S/N)
Experiment’s steps:
* Step 1: DUT1 ADEV measuring gives me a value of 60 - 70 mHz
instead of the expected 12.5 mHz (5 mHz x 2.5)
* Step 2: DUT2 ADEV measuring gives me a value of 10 - 12 mHz
which matches the expected 12.5 mHz (5 mHz x 2.5)
* Step 3: based on ADEV values which in the first case (DUT1) are
much greater than expected and in the second case (DUT2) coincide
with the expected I conclude that PN of the output signal from DUT2
will be lower than from DUT1.
I can see this PN degradation using Pendulum CNT-91 only as R&S FSQ8
does not fixate any PN degradation between DUT1 and DUT2Karen, ra3apw
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On 6/20/22 2:39 AM, Magnus Danielson via time-nuts wrote:
So, a counter is really like an ADC for phase, with wide bandwidth
input and a sub-sampling mechanism (trigger/time-base). Through
processing frequency estimates can be provided. Aliasing occurrs in
the sub-sampling. Modern counters can provided estimation filters than
goes from a higher sub-sampling rate to a lower, which to some degree
removes aliasing, but not fully. These frequency estimation methods
form a form of decimation filter.
Cheers,
Magnus
An intruiging thought as I drink my first cup of coffee (meaning it's
not well thought out)..
jumping off from "counter is similar to an ADC for phase" - is there a
time domain equivalent for Nyquist criterion? Certainly there's the
cycle ambiguity.. you know when the zerocrossing occurred, but not how
many are in between (although a counter usually does). For everything
else there is a frequency/time duality, so I suspect there is. The
criterion is usually explained in terms of information - so there should
be an equivalent "has all the information" statement for counters/gate
widths/precisions.
Hi
A “proper” phase noise analyzer will get you down to < -165 dbc/Hz. It also will preserve
the frequency spectrum ( no sampling / nyquist roll over ). What you get at 132 Hz offset
is the (DSB) noise at that offset and only that noise.
With a counter ( as Magnus mentions ) the sampling process looks at and sums up all the
noise in the input bandwidth. If it’s a 0.01 second sample, you get noise from 100Hz, 200Hz,
300Hz on out to whatever the counter’s input bandwidth is.
Since the counter likely has a bandwidth of a couple hundred MHz and you are measuring
something like 10 or 25 MHz, you sum up a whole lot of “noise floor” noise. Even if the counter
input amp has a zero db noise figure and the sampler does as well, you “fold in” 200 MHz worth
of that noise. This makes the sensitivity of the counter to low levels of phase noise pretty poor.
If you didn’t have this sort of limitation, your counter might well have 20 fs resolution rather than
20 ps …. (and no, this isn’t the only reason you have limited resolution)
Bob
On Jun 19, 2022, at 10:45 PM, Erik Kaashoek via time-nuts time-nuts@lists.febo.com wrote:
Bob,
Many thanks for the guidance you provide and the phase noise measurement document.
Can you provide feedback on this reasoning: A counter is like an ADC but in the frequency domain. So if you measure with 0.01 s tau you basically average over 0.01 s so you can only observe "phase noise" (e.g. energy that is not at the exact requested frequency) up to maximum 50 Hz from the carrier. But as you measure the true frequency changes the sensitivity of this measurement is extremely high. Translating the amount of time spend at a certain frequency away from the carrier (ADEV?) into a phase noise number in dBc is something I do not yet understand.
With a (very good) spectrum analyzer you may be able to come close to the carrier but as there is so much energy in the carrier it will be difficult to observe phase noise energy closer than say 1 or 10 kHz (at least not with the equipment I can afford) so any phase noise plot created using a spectrum analyzer can not be better than the combined phase noise of all LO's in the spectrum analyzer and will start at say 1 or 10 kHz.
For the frequencies between 50 Hz and 20 kHz the simplest option is to use a second LO and a mixer and a slow (loop BW below 10 Hz)PLL to keep the mixer in quadrature and feed the output of the mixer, after low pass filtering, into a PC soundcard for FFT processing.
Erik.
On 19-6-2022 22:45, Bob kb8tq via time-nuts wrote:
Hi
As HP found out back around 1973 or so, translating ADEV to phase noise
is not possible. This is true, even if you have the ADEV numbers for a variety
of Tau values as opposed to some sort of “average” kind of number.
There are a number of things ( like spurs ) that can strongly influence a counter
based ADEV reading, and have very little impact on a phase noise ( or signal to
noise reading. There also are ways the shape of the phase noise curve can
impact ADEV and have very little signal to noise impact for a specific signal.
By far the best way to do this is to properly measure phase noise at various
offsets from carrier. You can then look at the dbc/Hz numbers at each offset.
This lets you see what your devices are doing to the signal. You can then track
down the offending bit or piece and fix the problem.
The easiest way I know of to do phase noise is to quadrature lock two identical
sources into a double balanced mixer. You then put in a simple amplifier stage
to drive the mix down output into a sound card or spectrum analyzer. Total cost
if you already have a sound card should be < $50 ( US dollars …) for a DIY version.
That assumes you have the usual junk box parts and do a point to point wire
version.
Some example ADEV plots:
http://leapsecond.com/museum/manyadev.gif http://leapsecond.com/museum/manyadev.gif
http://leapsecond.com/museum/manyadev.gif http://leapsecond.com/museum/manyadev.gif
Some plots of a number of measurements:
http://www.leapsecond.com/pages/fe405/ http://www.leapsecond.com/pages/fe405/
Quick primer on phase noise measurement
https://www.npl.co.uk/special-pages/guides/gpg68_noise https://www.npl.co.uk/special-pages/guides/gpg68_noise
( The easy approach starts on page 21 :) )
Bob
On Jun 19, 2022, at 11:40 AM, Karen Tadevosyan via time-nuts time-nuts@lists.febo.com wrote:
Hi
Thank you for the clarification and rf-tools link.
Agree with your calculation. That’s why I raised this question regarding a fixing PN degradation by Pendulum CNT-91.
Could you please explain where is the error in my reasoning of the experiment :
Karen, ra3apw
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Bob, Magnus,
Thanks, clear. A counter is for ADEV, not for phase noise.
I made a test setup to learn how to use the mixer/PLL approach.
First using 10MHz from both outputs of a DSS (Rigol DG990) to observe
the DC shift with shifting the phase between the two signal.
Then by modulating one output with FM or PM.
There is a low pass filter after the mixer to get rid of the 10 MHz and
its harmonics but the LPF is measured flat till about 10kHz.
The output signal from the mixer was kept within 10% of the full voltage
swing to stay in the (hopefully) linear range.
Using PM creates a low frequency output from the mixer that is
proportional to the phase shift (region 0-1 degree) and constant in
amplitude with change of frequency. Also when using external modulation
from an audio signal generator created the expected behavior with drive
level and no frequency impact
Using FM with 0.1 Hz frequency deviation the mixer output amplitude
decreases very fast with increasing frequency (range 0.1 to 10 Hz)
Also when using 1 Hz or more frequency deviation. The higher frequency
deviation leads to higher output levels as expected.
Can someone help me understand how this FM signal (0.1 to 1000 Hz
modulation and 0.1 to 1 Hz frequency deviation) translates to the
calibration example mentioned in the document on phase noise measurement
as linked by Bob. (0.1 Hz deviation at 1 kHz rate leading to a sideband
(at 1kHz?) level of -86 dBc)
At a 1kHz rate I see (yet) no output from the mixer where at 1Hz there
is a lot of output. Why is this output frequency dependency?
Is this a problem with the signal generator? Or the mixer?
Then I tried to use the modulated signal from the SG PLL locked to a
10MHz VCO. Results where the same. FM output signal is frequency
dependent, PM not.
Erik.
I found the info on how to calibrate and lots of other practical stuff in
here: https://martein.home.xs4all.nl/pa3ake/hmode/dds_pmnoise_pll.html
Erik
On Mon, Jun 20, 2022, 19:43 Erik Kaashoek erik@kaashoek.com wrote:
Bob, Magnus,
Thanks, clear. A counter is for ADEV, not for phase noise.
I made a test setup to learn how to use the mixer/PLL approach.
First using 10MHz from both outputs of a DSS (Rigol DG990) to observe the
DC shift with shifting the phase between the two signal.
Then by modulating one output with FM or PM.
There is a low pass filter after the mixer to get rid of the 10 MHz and
its harmonics but the LPF is measured flat till about 10kHz.
The output signal from the mixer was kept within 10% of the full voltage
swing to stay in the (hopefully) linear range.
Using PM creates a low frequency output from the mixer that is
proportional to the phase shift (region 0-1 degree) and constant in
amplitude with change of frequency. Also when using external modulation
from an audio signal generator created the expected behavior with drive
level and no frequency impact
Using FM with 0.1 Hz frequency deviation the mixer output amplitude
decreases very fast with increasing frequency (range 0.1 to 10 Hz)
Also when using 1 Hz or more frequency deviation. The higher frequency
deviation leads to higher output levels as expected.
Can someone help me understand how this FM signal (0.1 to 1000 Hz
modulation and 0.1 to 1 Hz frequency deviation) translates to the
calibration example mentioned in the document on phase noise measurement as
linked by Bob. (0.1 Hz deviation at 1 kHz rate leading to a sideband (at
1kHz?) level of -86 dBc)
At a 1kHz rate I see (yet) no output from the mixer where at 1Hz there is
a lot of output. Why is this output frequency dependency?
Is this a problem with the signal generator? Or the mixer?
Then I tried to use the modulated signal from the SG PLL locked to a 10MHz
VCO. Results where the same. FM output signal is frequency dependent, PM
not.
Erik.
Hi Jim,
On 2022-06-20 17:57, Lux, Jim via time-nuts wrote:
On 6/20/22 2:39 AM, Magnus Danielson via time-nuts wrote:
So, a counter is really like an ADC for phase, with wide bandwidth
input and a sub-sampling mechanism (trigger/time-base). Through
processing frequency estimates can be provided. Aliasing occurrs in
the sub-sampling. Modern counters can provided estimation filters
than goes from a higher sub-sampling rate to a lower, which to some
degree removes aliasing, but not fully. These frequency estimation
methods form a form of decimation filter.
Cheers,
Magnus
An intruiging thought as I drink my first cup of coffee (meaning it's
not well thought out)..
Enjoy!
jumping off from "counter is similar to an ADC for phase" - is there a
time domain equivalent for Nyquist criterion? Certainly there's the
cycle ambiguity.. you know when the zerocrossing occurred, but not how
many are in between (although a counter usually does). For everything
else there is a frequency/time duality, so I suspect there is. The
criterion is usually explained in terms of information - so there
should be an equivalent "has all the information" statement for
counters/gate widths/precisions.
Well, considering that optimum phase/time sensitivity is at the
through-zero of a sine, with the optimum slew-rate of the signal, you
have two observation points per cycle. You can view that as having
essentially two sample-points of phase per cycle. Similarly you will
have two optimal sample-points for amplitude in quadrature on the peaks
of the sine.
Now, using this fact, you have a Nyquistian type of relationship and
also upper phase-information frequency being that of the cosine itself,
since you can fit a modulaiton that pushes the rising edge one way and
the falling edge the other way. As you attempt a higher modulation
frequency you cannot distinguish that from the mirror frequency lower
than that frequency. Thus, they Nyquist frequency of modulation is the
carrier frequency.
But then again, the same can be said for any overtones, so you can
support higher modulation frequencies there, with the same basic rule.
However, sorting that out can be a bit tricky, considering non-linear
functions and intermodulations.
PS. IEEE Std. 1139-2022 made it through a formal approval after
balloting, so now it is off for last editorial touch-ups before
publishing. Good news. Look forward to put it into use.
Cheers,
Magnus
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