Einstein points out that everything must
be measured relative to something else. In the context of jitter,
my question is, "Relative to what?" To investigate that question I
shall first discuss a general problem of statistical
representation.
A general problem of statistical representation
Suppose I ask you to measure the standard
deviation of some electronic signal. You probably remember that
standard deviation is the root-mean-square (rms) measure of
deviations from the mean (average value). The first thing you must
do, then, is capture some samples from the signal and find its
mean. Then you can begin looking at differences from that mean and
commence calculating the rms deviation. What if, at the outset of your calculations, you begin with the
wrong mean value? Obviously your calculations, and everything based
on them, would be incorrect. Let me illustrate the difficulty of
finding the correct mean with an example. Figure 1 depicts a
100-MHz sine wave. The horizontal scale is 2 ns/div. Figure 1 - One full cycle of a
repetitive waveform contains enough information to reconstruct the
entire signal histogram.
In this simple example the signal repeats every 10 ns.
Figure 1 exploits that fact, sampling one exact period of the
waveform as shown in the yellow-shaded region. Every cycle is the
same, so there is no point in taking additional data. The histogram
at right (rotated 90 degrees) represents the various vertical
values sampled. From the histogram you may extract the mean and
other parameters you seek. Repetition makes statistical measurement
easy. Every individual cycle of the signal contains all the
information you need. The informational content of a non-repetitive signal is more widely
dispersed. Accurate statistical measurement of a continuous,
non-repetitive signal requires data spanning multiple cycles of the
lowest-frequency components present in the signal. Going back to the signal in Figure 1,
imagine what might happen if you don't capture enough data? What if
you come from an advanced galaxy where 10 ns seems like a
R-E-A-L-L-Y L-O-N-G T-I-M-E? If you do not know that the signal is
going to repeat, then, after capturing data samples for only one or
two ns, you might become bored, stop recording, and call it a day. Depending on where in the signal
cycle your little burst of captured samples occurs; the signal
might appear deceptively quiescent. For example, Figure 2 samples a
region only 1.5 ns wide, taken at a location near the trough. The
sampled values are taken from the yellow-shaded region at the left
side of the screen. Figure 2 - A short burst of samples taken near the trough produces
a non-representative histogram. The mean value averaged over that
short burst of samples (yellow region) represents only the lower
extreme of signal excursion. That's not right. Even worse, for the
samples inside the yellow region, the deviation around that mean is
too compact. Given only this limited sampling of data you may
(erroneously) conclude that this signal exhibits few if any
deviations from the mean. In a second example, a short burst of samples captured near the
zero crossing presents an entirely different histogram (Figure 3).
This histogram correctly predicts the mean, but exhibits a
misleading trend: all the data points within the central yellow
region ascend in a continuous straight line. A financial analyst
faced with such data might predict that the signal goes up forever.
As you know, probably all too well, that never happens. Figure 3 - Near the zero crossing, this signal trends up linearly. The mean value averaged over that
short burst of samples (yellow region) represents only the lower
extreme of signal excursion. That's not right. Even worse, for the
samples inside the yellow region, the deviation around that mean is
too compact. Given only this limited sampling of data you may
(erroneously) conclude that this signal exhibits few if any
deviations from the mean. In a second example, a short burst of samples captured near the
zero crossing presents an entirely different histogram (Figure 3).
This histogram correctly predicts the mean, but exhibits a
misleading trend: all the data points within the central yellow
region ascend in a continuous straight line. A financial analyst
faced with such data might predict that the signal goes up forever.
As you know, probably all too well, that never happens.
So, how much data must you capture? Mathematically, if the most slowly-moving features of the signal
undulate at frequency f0, then the required data capture time
Tcapture is given by: Tcapture = (several times)x(1/f0) [1] Taken to an extreme, as f0 approaches DC, the required capture time
soars to infinity. Therefore, if your signal includes significant
components that go all the way down to 0 Hz, you must capture data
for all time. Yep. For all time. Talk about inconvenient! It would be better for you if jitter had a
limited bandwidth that did not extend to zero frequency. Is there such a thing as DC jitter? Set up a wideband wireless transmitter and receiver at either end
of a large warehouse. To make the numbers easy, let's suppose they
operate at a baud rate of 1 GHz. Lock the receiver onto the
transmitted bit stream. If the warehouse is 100 feet long
there are at any one time about 100 bits of information stored in
the space between the transmitter and receiver (radio waves in air
travel at a speed of approximately 1 foot/ns). Now pick up the
transmitter and move it closer to the receiver, cutting the
distance in half (Figure 4). From the perspective of the receiver
information now arrives 50 bit times earlier than previously.
That's 50 bit times worth of peak-to-peak jitter. If I move back
and forth once a day, the same shift in timing recurs at a daily
rate (11.57 micro-Hz). Figure 4 - The space between transmitter and receiver
stores one hundred bits of information. (Ed. Note: that’s my wife,
Liz, holding a cardboard box with wooden dowels stuck in the top
for antennas. Cool use of light, no?).
To measure that kind of long-term
variation in timing you would have to sample a coherent data record
that spans the entire daily movement. That could require hundreds
of billions of samples. Astrophysical problems involve even more extreme amounts of jitter.
An Earthbound receiver listening to transmissions from Mars
experiences deviations in timing proportional to the distance
between the two planets. As the distance varies from 36 to 250
million miles the total variation in delay (total jitter) varies by
1129 sec, or about 20 minutes. That's a lot of jitter, but it
develops slowly. The distance between the planets undulates back
and forth at a rate commensurate with the differences in their
orbital frequencies. One complete undulation occurs every 778 days
(14.9 nano-Hz).
To measure that kind of long-term
variation in timing you would have to sample a coherent data record
that spans the entire daily movement. That could require hundreds
of billions of samples. Astrophysical problems involve even more extreme amounts of jitter.
An Earthbound receiver listening to transmissions from Mars
experiences deviations in timing proportional to the distance
between the two planets. As the distance varies from 36 to 250
million miles the total variation in delay (total jitter) varies by
1129 sec, or about 20 minutes. That's a lot of jitter, but it
develops slowly. The distance between the planets undulates back
and forth at a rate commensurate with the differences in their
orbital frequencies. One complete undulation occurs every 778 days
(14.9 nano-Hz).
Long-period undulations, on a scale of
time long compared to the tracking response time of your PLL, are
called "wander". Short-period undulations, on a scale of time short
compared to the tracking response time of your PLL, are called
"jitter". The boundary between wander and jitter depends totally on
the characteristics of your PLL. There is no other factor
distinguishing the two. One man's wander could easily be another
man's jitter, and vice-versa. To speak intelligently about jitter
testing, the discussion must include consideration of the tracking
bandwidth of the device receiving the jittery signal. As long as your PLL remains locked,
wander has little practical effect on system performance. Because
it has little practical effect, even if your data signal
incorporates massive amounts of wander you need not measure it, and
the captured data record need not represent it. That is important.
The wander in your signal is the only part that includes components
extending down to zero frequency. Once you throw out the
requirement to measure wander, the bandwidth of the remaining
portion of the jitter signal no longer extends down to zero
frequency. Any technique that measures only jitter, and excludes
wander, eliminates the need for infinite data records, leading to
this simplification: Your captured data record need be no
longer than several PLL response times. You may choose to aggregate many independent data records to build
a deeply detailed histogram of performance, but each individual
record need be no longer than several PLL response times. Practical measurement of jitter In the context of a serial data
communication system, assuming the receiver's PLL maintains lock,
all that matters in the receiver circuit, in terms of jitter, are
deviations between the instantaneous incoming data phase and the
receiver's PLL-generated data recovery clock.
Given an identical input waveform, theTIE@LEVELfunction measures almost the same deviations. The only discrepancy
is that the TIE function measures deviations between the
instantaneous incoming data phase and the TIE internal reference
clock, not the receiver's PLL-generated data recovery clock. To the extent that theTIE@LEVELfunction and the receiver's PLL generate different internal
reference clocks, because they use different tracking algorithms,
their perceptions of jitter will differ. If you want to measure
jitter the same way the receiver sees it, then program the TIE
function to mimic the PLL algorithm for its internal reference
clock generation: Use for TIE jitter measurement the same PLL
tracking algorithm as your receiver. That is not merely possible; it is the required method for jitter
measurement. It excludes wander in precisely the same way as your
PLL circuit. Provided that you take sufficiently long data records, this method
sidesteps all issues about the existence of low-frequency wander.
Whatever your receiver sees, your measurement sees also. I'll talk
more about that in my next article. Tying it all together There exists no generalized,
self-consistent way to measure the complete range of all jitter,
because there is no way to capture jitter components all the way
down to zero frequency. If you attempt to measure jitter at all
frequencies you will discover your readings just getting higher and
higher as you make your data records longer and longer. Even if
your input signal is perfect, your scope isn't. In the limit, as
you attempt to measure jitter over very long periods of time, you
just end up measuring noise in the scope's reference oscillator
that soars to infinity as the data record approaches infinite
length.
Instead of trying to measure "all jitter",
focus your attention on measuring the jitter of interest to your
receiver. Capture coherently-sampled data records several times
longer than the tracking response time of your receiver's PLL and
measure jitter against a reference clock generated using the same
PLL algorithm as your receiver. A PLL-based reference clock is the
relative signal against which Einstein would want you to measure
jitter. Best Regards, Dr. Howard Johnson
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