On Mon, 30 Jul 2018 16:32:40 -0700 (PDT), gyansorova@gmail.com wrote:

>On Tuesday, July 31, 2018 at 7:33:57 AM UTC+12, Steve Pope wrote:
>> <gyansorova@gmail.com> wrote:
>> 
>> >Good point and closely related is the reason why zero crossing detectors
>> >or hard limiters do not work. You just transfer uncertainty in amplitude
>> >to uncertainty in phase. This is why FM radio at poor SNRs (carrier to
>> >noise ratios) is better with no limiter at all.
>> 
>> Either FM or FSK can be demodulated after hard limiting but there
>> is a performance loss, since you no longer have optimal (e.g.
>> maximum-likelyhood) detection as a possibility.  You have
>> lost the pulse shape.
>> 
>> But in many situations the performance loss is not that bad.
>> 
>> Steve
>
>Depends on the CNR. Fine at high CNR ratios. There is no filtering action however in limiting. The noise is still there, just converted to another type, from additive to phase

A distinction in this case is that the signal of interest is already
NRZ, i.e., "limited" to rectangular pulses.   The only impairment is
the jitter that is already present, none is added by the processing.

On Tuesday, July 31, 2018 at 7:33:57 AM UTC+12, Steve Pope wrote:
> <gyansorova@gmail.com> wrote:
> 
> >Good point and closely related is the reason why zero crossing detectors
> >or hard limiters do not work. You just transfer uncertainty in amplitude
> >to uncertainty in phase. This is why FM radio at poor SNRs (carrier to
> >noise ratios) is better with no limiter at all.
> 
> Either FM or FSK can be demodulated after hard limiting but there
> is a performance loss, since you no longer have optimal (e.g.
> maximum-likelyhood) detection as a possibility.  You have
> lost the pulse shape.
> 
> But in many situations the performance loss is not that bad.
> 
> Steve

Depends on the CNR. Fine at high CNR ratios. There is no filtering action however in limiting. The noise is still there, just converted to another type, from additive to phase

<gyansorova@gmail.com> wrote:

>Good point and closely related is the reason why zero crossing detectors
>or hard limiters do not work. You just transfer uncertainty in amplitude
>to uncertainty in phase. This is why FM radio at poor SNRs (carrier to
>noise ratios) is better with no limiter at all.

Either FM or FSK can be demodulated after hard limiting but there
is a performance loss, since you no longer have optimal (e.g.
maximum-likelyhood) detection as a possibility.  You have
lost the pulse shape.

But in many situations the performance loss is not that bad.

Steve

On Thursday, July 26, 2018 at 4:28:07 AM UTC+12, Randy Yates wrote:
> We have a requirement to measure a 32.768 KHz TTL output quickly 
> and with a certain accuracy. 
> 
> If one used an N-bit ADC sampling at F samples per second, what is the
> relationship between T_m and F_delta, where T_m is the minimum
> measurement time (say, in seconds) for a maximum frequency error of
> F_delta (say, in PPM)?
> 
> For this discussion assume the frequency is stable over the measurement
> time. Also assume the only noise is the quantization noise of the ADC.
> 
> What is making my head hurt is some (seemingy) contradictory pieces of
> information I've come across over the years:
> 
>   1. If the input signal was noiseless and known to ba a sinusoid, it
>   only requires 3 samples to determine the frequency.
> 
>   2. The signal isn't noiseless, so I think we're getting into some
>   estimation theory here? 
> 
>   3. How can you square up the time-frequency uncertainty principle,
>   which I take to mean that in order to reduce the uncertainty in
>   frequency of a measurement, we have to increase the measurement
>   time (with some magic proportion involved), with 1? It sames that
>   if the assumptions of 1 were made, we can make aribtrarily faster
>   measurements by increasing the sample rate.
> 
> Can you guys set me straight?
> -- 
> Randy Yates
> Embedded Linux Developer
> http://www.garnerundergroundinc.com

Good point and closely related is the reason why zero crossing detectors or hard limiters do not work. You just transfer uncertainty in amplitude to uncertainty in phase. This is why FM radio at poor SNRs (carrier to noise ratios) is better with no limiter at all.

On Thursday, July 26, 2018 at 6:05:42 AM UTC-7, Randy Yates wrote:

(snip)

> OK, thanks for that Steve, but the fundamental question I have is:
> how small can we make the "observation window" and still attain
> a estimation certain accuracy (2PPM at 32.768 kHz)?

More obvious to me is a high-speed counter, hopefully as
low-noise and frequency stable as possible.

Count pulses on the high-speed counter between edges of the 32768kHz
oscillator.

You can then do usual statistical analysis on the count values.
Mean will give you the frequency. Standard deviation will have a 
term related to how far you are from an integer frequency, and a 
term from noise (jitter).  With enough counts, you can get accurate
enough results.  Mean improves with the square root of the number
of counts. I think SD does also, but I haven't thought about it
for a while.

In your case a transition occurs as time advances by every period (no missi=
ng transitions). The closed form solution then gives extremely simple expre=
ssions for the variance in reference time "P11" and in the period "P22" at =
the end of 'M' consecutive observed transitions.  Kalman's notation for mea=
surement error
variance is 'R' -- that includes combined effects of all contributors; cloc=
k jitter, interpolation error, etc.).  Then
                P11 =3D R (2/M) (2M-1)/(M+1)    P22 =3D R (2/M) (6)/(M^2-1)
It is of interest to express RMS errors (square roots of the variances) for=
  large 'M' -- denoting square root of 'M' and 'R' as 'm' and 'r' respectiv=
ely the RMS error in reference time and in the period are essentially  2r/m=
  and  2(1.732)r/(Mm)  respectively.  This latter expression, rigorously de=
rived and quite easy to compute, is the idealized result you want. All you =
need is a value set for 'r' (make it conservative to be safe) and determine=
 the lowest value of 'm' that will give you your allowable error in the per=
iod (period is obviously the reciprocal of frequency).

There are a few details, probably of minor importance to you.  The solution=
 came from forming the least squares expression
                         [inverse {(H'H)}] H' z
(where ' here represents matrix transposition), followed by known expressio=
ns for sums of consecutive integers and their squares.  A slightly differen=
t covariance matrix is first formed by defining 'X(1)' as the initial refer=
ence time.  That matrix is then propagated to final reference time using a =
simple 2x2 transition matrix, producing the results just given.  Another ch=
ange that occurs with that propagation is a sign reversal of accompanying o=
ff-diagonal terms P12=3DP21 (also of less importance).  The navigation and =
tracking example is very similar but, again, involving other parameters (e.=
g., distance) that attach greater significance to initial-vs-final time (e.=
g., error in initial position carries implications differing from error in =
final position).

On Thu, 26 Jul 2018 13:51:42 -0400, Randy Yates
<randyy@garnerundergroundinc.com> wrote:

>theman@ericjacobsen.org (Eric Jacobsen) writes:
>
>> On Thu, 26 Jul 2018 07:14:16 -0700 (PDT), makolber@yahoo.com wrote:
>>
>>>
>>>> 
>>>> OK, thanks for that Steve, but the fundamental question I have is:
>>>> how small can we make the "observation window" and still attain
>>>> a estimation certain accuracy (2PPM at 32.768 kHz)?
>>>>
>>>
>>>is the answer here?
>>>
>>>
>>>http://www.dtic.mil/dtic/tr/fulltext/u2/a167992.pdf 
>>>
>>>mark
>>
>> That's for sinusoids.   This isn't a sinusoid, so assumptions may not
>> apply.
>
>I could just filter it... all that matters is the fundamental frequency.
>
>> I think the thing to do would be to pick a potential methodology or
>> two and then do some analysis on those to see whether you can get to
>> your desired spec or not with those methodologies.  If so, go for it,
>> if not, look for another method.
>>
>> e.g., take the jitter spec on the clock and see how much averaging it
>> might take to get down to 2ppm by timing edges, then how many periods
>> that would require, and see if it's a practical approach.   Or lock a
>> PLL to known reference signal and measure the difference.
>
>Ewe - sample clock jitter. Hadn't thought about that...

More importantly, jitter in the clock you're measuring.   The circuit
generating the clock should be analyzable for jitter expectations, or,
if it's coming out of an oscillator or PLL-conditioned oscillator, it
should already have jitter specs.

For that matter, it should have frequency stability specs, too, I'd
think.

Steve mentioned another approach to model jitter.   It's pretty easy
to see on an oscilloscope, too, to get a basic characterization for a
particular unit.   Digital scopes make it easy to evaluate jitter
excursions.

>> A few practical methods have been suggested.  I think it just takes a
>> little work from there.   There's unlikely to be a cookbook formula to
>> plug in and give a general answer to a specific problem.
>
>I appreciate all these practical suggestions, but I wanted to get a 
>theoretical, somewhat idealized result before I went further. E.g.,
>if the theory revealed I could never get better than 15 seconds, it
>ain't worth pursuing.

And what I'm suggesting is that you can be led down the garden path
that way, or spend a lot of time doing analyses that aren't that
useful toward the actual problem.   The CRB analyses for frequency
estimation are pretty well known and understood and generally are for
sinusoids in AWGN.   That's not what you have.   You're right, you
could turn it into one, but then you also need to include the effects
and characteristics of your filter, e.g., it's Noise Figure, etc.,
etc.   The CRB isn't really a universal concrete tool, anyway, but is
a good reference point.

A little time with an oscilloscope to characterize the jitter, if it
isn't on a datasheet already, would probably go further toward giving
you an idea of measurement limitations using simple, robust methods.

>mark, I am picking through that paper. Thanks for that.
>
>Also, thanks for your post, navaide - I'm going to try to dig up your
>paper and have a closer look.
>-- 
>Randy Yates
>Embedded Linux Developer
>http://www.garnerundergroundinc.com

Randy Yates  <randyy@garnerundergroundinc.com> wrote:

>spope384@gmail.com (Steve Pope) writes:

>> I think in this case the length of the observation window is
>> important, but one does not need to sample continuously
>> throughout the observation window -- all you need is the
>> time-position of one edge near the beginning of the window,
>> another near the end of the window, and a count of the number
>> of in-between edges.  This gives you the frequency.

>> I've used this for frequency-tracking, it can work perfectly well.

>OK, thanks for that Steve, but the fundamental question I have is:
>how small can we make the "observation window" and still attain
>a estimation certain accuracy (2PPM at 32.768 kHz)?

With the TTL signal (as opposed to a waveform that is actually
designed for this sort of purpose), I would model it based on the dV/dT
of the signal during its transitions (probably, the high-to-low
transition is faster, so always use that one).  Then I'd include
some additive gaussian noise with an amplitude of maybe 100 to 200 
millivolts.

(Although it may be fruitful to measure the noise, or just eyeball it
on a scope.)

Put together these numbers to get a jitter component along the
time axis on each negative edge.  Then make sure the sample rate of
your comparator is sufficiently high such that the accuracy
will be noise-limited rather than sample rate limited.
If that is possible.

This should give you the answer you're looking for.  Or at least
the average accuracy as a function of observation length; you'd have to 
look at it a bit more to estimate the worse case accuracy.

The above ignores the accuracy/jitter of your sampling clock,
which might also be limiting.

Steve

theman@ericjacobsen.org (Eric Jacobsen) writes:

> On Thu, 26 Jul 2018 07:14:16 -0700 (PDT), makolber@yahoo.com wrote:
>
>>
>>> 
>>> OK, thanks for that Steve, but the fundamental question I have is:
>>> how small can we make the "observation window" and still attain
>>> a estimation certain accuracy (2PPM at 32.768 kHz)?
>>>
>>
>>is the answer here?
>>
>>
>>http://www.dtic.mil/dtic/tr/fulltext/u2/a167992.pdf 
>>
>>mark
>
> That's for sinusoids.   This isn't a sinusoid, so assumptions may not
> apply.

I could just filter it... all that matters is the fundamental frequency.

> I think the thing to do would be to pick a potential methodology or
> two and then do some analysis on those to see whether you can get to
> your desired spec or not with those methodologies.  If so, go for it,
> if not, look for another method.
>
> e.g., take the jitter spec on the clock and see how much averaging it
> might take to get down to 2ppm by timing edges, then how many periods
> that would require, and see if it's a practical approach.   Or lock a
> PLL to known reference signal and measure the difference.

Ewe - sample clock jitter. Hadn't thought about that...

> A few practical methods have been suggested.  I think it just takes a
> little work from there.   There's unlikely to be a cookbook formula to
> plug in and give a general answer to a specific problem.

I appreciate all these practical suggestions, but I wanted to get a 
theoretical, somewhat idealized result before I went further. E.g.,
if the theory revealed I could never get better than 15 seconds, it
ain't worth pursuing.

mark, I am picking through that paper. Thanks for that.

Also, thanks for your post, navaide - I'm going to try to dig up your
paper and have a closer look.
-- 
Randy Yates
Embedded Linux Developer
http://www.garnerundergroundinc.com

On Thu, 26 Jul 2018 07:14:16 -0700 (PDT), makolber@yahoo.com wrote:

>
>> 
>> OK, thanks for that Steve, but the fundamental question I have is:
>> how small can we make the "observation window" and still attain
>> a estimation certain accuracy (2PPM at 32.768 kHz)?
>>
>
>is the answer here?
>
>
>http://www.dtic.mil/dtic/tr/fulltext/u2/a167992.pdf 
>
>mark

That's for sinusoids.   This isn't a sinusoid, so assumptions may not
apply.

I think the thing to do would be to pick a potential methodology or
two and then do some analysis on those to see whether you can get to
your desired spec or not with those methodologies.  If so, go for it,
if not, look for another method.

e.g., take the jitter spec on the clock and see how much averaging it
might take to get down to 2ppm by timing edges, then how many periods
that would require, and see if it's a practical approach.   Or lock a
PLL to known reference signal and measure the difference.


A few practical methods have been suggested.  I think it just takes a
little work from there.   There's unlikely to be a cookbook formula to
plug in and give a general answer to a specific problem.