MODELING AND BUDGETING OF TIMING JITTER AND NOISE - Advanced Signal Integrity for High-Speed Digital Design - page 552

where ρ T is the transition density, which is the ratio of the number of logic
transitions to the total number of bits transmitted (typically, ρ T is equal to 0.5),
and J ( t ) is the jitter distribution. The equation is the cumulative distribution
function of the timing jitter, which takes into account the probability of actually
making a transition on any given bit (since the jitter will be zero if the signal
remains at a given level).
To calculate the BER we need a model for the jitter distribution. We can
use equation (13-4) for the random jitter sources, but it will not be accurate
for systems that also include deterministic (bounded) jitter sources such as ISI,
as Figure 13-7 demonstrates. In particular, the jitter histograms extracted from
the zero crossing points of the eye show bimodal distributions, which suggests
that the jitter is comprised of a combination of deterministic and Gaussian jitter
sources. So we need a model for deterministic jitter that we can combine with
the Gaussian random jitter model.
Deterministic jitter sources can fit a variety of distributions, examples of
which are shown in Figure 13-11 and described in the next section. In creating
system-level jitter budgets, we employ the dual Dirac model in equation (13-6) to
express the probability density function (PDF) of the deterministic jitter, DJ( t ):

δ ( t − DJ δδ / 2 )

2

δ ( t + DJ δδ / 2 )

2

DJ (t) = + (13-6)

where DJ δδ is the dual Dirac deterministic jitter (ps) and δ(t) is the Dirac delta

function,

0

t = 0

δ(t) =

1 t = 0

As the equation shows, the dual Dirac model, which is widely used in industry,
treats the deterministic jitter as though it is equally distributed at extreme values.
The deterministic jitter in real systems does not fit a dual delta distribution
function. The usefulness of the model is that it allows us easily to combine
the deterministic and random jitter distributions. The reason that the dual Dirac
model works is that we really only need to be accurate in estimating the jitter at
low BER, which is governed by the random jitter. As a result, we can use the
dual Dirac distribution model to shift the “tails” of the jitter distributions (the RJ)
to their proper locations. Conceptually, the dual Dirac model gives us a Gaussian
approximation to the outer edges of the jitter distribution when displaced by DJ δδ .
For a more in-depth treatment of jitter distributions and the dual Dirac model,
we refer the reader to a report by Stephens [2004] and a book by Li [2008]. We
provide an example that demonstrates the application of the dual Dirac model in
Section 13.3.
The total jitter PDF, JT( t ), is created by convolving the DJ and RJ models:

√ 2 πσ RJ e − t 2 / 2 σ RJ t −

dt

∞

1 DJ δδ

2

t + DJ δδ

2

JT (t) = RJ (t) ∗ DJ (t) = +

−∞