User guide

Mathematical Background
Bit errors are caused by noise, and the Q-factor describes the signal-to-noise ratio
at the decision circuit.
It is possible to calculate the Q-factor from a limited number of measured BER vs.
threshold data points. It is also possible to calculate expected bit error rates from
the Q-factor. This is a method for predicting very low bit error rates (typically below
10
-11
) that would take a long time to measure.
The Q-factor is calculated as:
where μ
1,0
is the mean level of the 1 and 0 rails, respectively, and σ
1,0
is the standard
deviation of the noise distribution on the 1 and 0 rails.
The μ
1,0
and σ
1,0
values are calculated from a selected range of data points. This
calculation is correct if the noise distribution has Gaussian characteristics. Then,
the bit error rate can be expressed as:
where D is the decision threshold, μ
1,0
and σ
1,0
are the mean and standard deviation
of the 1 and 0 rails, and erfc (x) is the complementary error function.
This formula is the sum of two terms. It considers the probabilities of deciding that
a "0" has been received when a "1" was sent, and that a "1" has been received
when a "0" was sent.
For the following calculations, the assumption is made that the BER is dominated
by only one of the terms noted above, depending on whether the threshold is closer
to the 1 or 0 rail.
For the complementary error function
1
2p
erfc(x) =
x
8
- b
2
/2
e
d b
1
x 2 p
- x
2
/2
e
an inverse logarithmic approximation exists:
erfc(x)
1
2
Log
-1
1.192 - 0.6681 x - 0.0162 x
2
where x = Log(BER).
Advanced Analysis 6
Agilent J-BERT N4903B High-Performance Serial BERT 283