[SI-LIST] Re: Bounded vs Unbounded jitter (was : Jitter transfer vs. accumulation)

• From: "Alfred P. Neves" <al.neves@xxxxxxxxxxx>
• To: <Chris.Cheng@xxxxxxxxxxxx>, <si-list@xxxxxxxxxxxxx>
• Date: Mon, 26 Mar 2007 14:07:38 -0700
```When I was doing a lot of applications work in the jitter area this
concept was troubling to many engineers.  I did some research on this
issue and have some results to share.  BTW, there is an entire section
on Central Limit Theorem in the book "Probability and Random Processes
for Electrical Engineering" Alberto Leon-Garcia that I referenced
before.

First, the Gaussian probability density function (pdf) is used to model
many processes which is justified by the Central Limit Theorem.  It is
also important to differentiate that although the Guassian pdf used to
model the process is unbounded by definition, the actual process itself
may not be actually unbounded, here is an example why:

Consider the simple case of measuring the Johnson noise across a simple
resistor.   The noise across the resistor is modeled using a Gaussian
process.   The Central Limit theorem states that given a sequence of
random variables, X1, X2, etc., to Xn with finite mean u and finite
variance sigma-squared and let Sn be

Sn=X1+X2+...Xn...

Sn would be the total Thermal or Johnson noise modeled across the
resistor.   X1, X2...   is the contribution of each charged particle
random motion in the resistive material. The theorem specifically states
that as n becomes large, it "approximates" a Gaussian random variable.
n, in this case and most practical cases is not infinite since there are
countable number of charged particles, but as n---> infinity, or the
resistor size approaches infinitely large in the limit, the
"approximation" to Gaussian distribution fit to thermal voltage noise
becomes better.  There are, however, a finite number of charged
particles, so at some point the model breaks down for a finite size
resistor with a countable number of charged particles.

The practical aspect of these concepts is selecting a suitable noise
source for generation of jitter for RX tolerance testing.  I like to use
peak/RMS ratio to describe how deep the tails of a process modeled with
Gaussian pdf is.   To my knowledge at this date, the only 2 groups of
people who have generated either a jitter source, or a noise source with
tested "random-ness" or very high peak/RMS ratio is Agilent and Noisecom
.   Noisecom actually specifies a Peak/RMS ratio (or peak/1sigma where
mean=0) to their noise source.  Agilent specifically tested their jitter
generation tool, measuring the quality of the random-ness or peak/RMS
factor.   An interesting application of this is that you need greater
than 7 peak/RMS ratio for making BER measurements down to 10E-12.  You
have to use a very good quality noise source to create very high
peak/RMS jitter accordingly.

Alfred P. Neves      <*)))))><{

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-----Original Message-----
From: si-list-bounce@xxxxxxxxxxxxx [mailto:si-list-bounce@xxxxxxxxxxxxx]
On Behalf Of Chris Cheng
Sent: Monday, March 26, 2007 12:24 PM
To: si-list@xxxxxxxxxxxxx
Subject: [SI-LIST] Bounded vs Unbounded jitter (was : Jitter transfer
vs. accumulation)

Art,
In an effort to focus the original message to bandwidth trade-offs in =
PLL, I propose starting this bounded vs. unbounded jitter discussion in

-----Original Message-----
From: si-list-bounce@xxxxxxxxxxxxx
[mailto:si-list-bounce@xxxxxxxxxxxxx]On Behalf Of art_porter@xxxxxxxxxxx
Sent: Monday, March 26, 2007 11:36 AM
To: steven@xxxxxxxxxxxxxxxxxxxx; si-list@xxxxxxxxxxxxx
Subject: [SI-LIST] Re: Jitter transfer vs. accumulation

Here is a thought experiment that helps me think about "random" jitter,
= =3D which I first encountered when thinking about noise in the ancient
days = =3D when most digital hardware was so slow we didn't have to

Could the TIE of a transition ever reach 1000 years? If you argue that =
=3D it could, postulate a condition under which it could do so. (If the
PDF = =3D is truly normal and unbounded, then of course it could.)=3D20

Or think about some other randomly distributed parameter, such as =3D
heights of people. As you look at a larger and larger sample of =3D
individuals, the PDF gets more and more normal looking. But has there =
=3D ever been a 0.0005-inch tall adult human being, or a 700-foot tall
=3D adult? The normal distribution works fine as a mathematical insight
into = =3D physical processes such as jitter or noise, as long as you
don't force = =3D your thinking out of some normal range of bounds, such
as the mass that = =3D can be supported by a skeleton and musculature,
or the height to which = =3D the heart can pump blood.=3D20

The central limit theorem reminds us that many phenomena appear Gaussian
= =3D only because they are an accumulation of the effects of a large
number = =3D of non-random processes. I suspect the same is true of most
jitter in = =3D the real world.=3D20

Art Porter=3D20
Agilent Technologies     =3D20

-----Original Message-----
From: si-list-bounce@xxxxxxxxxxxxx [mailto:si-list-bounce@xxxxxxxxxxxxx]
= =3D On Behalf Of Steven Kan
Sent: Monday, March 26, 2007 11:56 AM
To: si-list@xxxxxxxxxxxxx
Subject: [SI-LIST] Re: Jitter transfer vs. accumulation

> From: "Alfred P. Neves" <al.neves@xxxxxxxxxxx>
> Subject: [SI-LIST] Re: Jitter transfer vs. accumulation
> Date: Sat, 24 Mar 2007 09:48:38 -0700
>
> This estimator, peak-peak jitter, is not a good estimator of the
> process since it continues to increase since the process is =3D
Gaussian
> and collecting more samples digs deeper into the tails of the
> distribution.    The process is by definition unbounded.
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

finite=3D20 samples may be a "best fit" with the Gaussian, can we really
say that = =3D any=3D20 process is truly unbounded when applied to
real-world phenomena that =3D occur=3D20 in real-world products? If I
apply the constraint that I need to examine = =3D a=3D20 given process
over the life of the product (or the life of the user or = =3D the=3D20
life of the Earth), does that then put bounds the process(es) and =
the=3D20 resulting statistics?

I can see the argument from the math side, e.g. "unbounded 'by =3D
definition'",=3D20 but do the 'definitions' include practical
constraints?

My empirical gold-bar ratio is still zero.=3D20

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