IEEE Solid-States Circuits Magazine - Summer 2019 - 26

treg,2

Ts - TFIX

P (t reg,1 + t reg,2 2 Ts - TFIX)

SU M M E R 2 0 19

t /t

0.5
0

t /t
(a)

100

t /t

100

treg,2

10-10
10-20

treg,1 + treg,2

treg,1

10-10
10-20

0.5

t /t

100

treg,1 + treg,2

10-10
10-20

t /t
(b)

100

FIGURE 10: The PDFs of MSB regeneration time, second MSB regeneration time, and their
sum on (a) linear and (b) log scales for the 3-b example. The PDF of the sum is the convolution of the individual PDFs. The probability mass inside the purple and hashed green areas
of Figure 9 is calculated by integrating PDFs in the shaded purple and green areas (visible
only on the log scale).

100

Noise Only
Metastability Only
Combined
Convolved

10-5
10-10
10-15

- P (t reg,1 2 Ts - TFIX) .
We already know the distributions of t reg,1 and t reg,2, so now we
just need the distribution of their
sum. Because these random variables
are independent, the distribution of
their sum is the convolution of their
distributions. Figure 10 shows the
PDFs of all three (t reg,1, t reg,2, and
their sum) and shows shaded areas
in purple and green corresponding to
the purple and hashed green areas in
Figure 9. Once we take the difference
of these shaded areas, we have calculated the probability that we run out
of time during second MSB regeneration, given some input level and DAC
trajectory. Based on Figure 8, running
out of time during the second MSB

treg,2

Probability

and we can obtain its probability by
finding the probability mass in the
hashed green area and subtracting
the probability mass in the purple
area. The purple area represents the
event t reg,1 + TFIX 2 Ts . Therefore, the
probability of running out of time
during the second MSB regeneration
(case 3) is

fT/t(t/t)

FIGURE 9: A depiction of what it means
to run out of time during second MSB
regeneration in terms of the sample space
of independent random variables t reg,1 and
t reg,2 . The hashed green area represents the
event in which t reg,1 + TFIX + t reg,2 exceeds the
conversion time. However, we're interested
only in the case when this happens and
t reg,1 + TFIX fits into the conversion time. The
purple area represents the event in which
t reg,1 + TFIX does not fit. By subtracting the
probability mass in the purple area from
the probability mass in the hashed green
area, we obtain the probability of the event
of interest.

treg,1

fT/t(t /t)

Ts - TFIX

fT/t(t/t)

0.5

fT/t(t /t)

Ts - TFIX

treg,1

100

fT/t(t /t)

fT/t(t/t)

10-20
-10

-8

-6

-4

-2
0
2
Code Error

FIGURE 11: A code error PMF overlay of metastability-only (black dots), noise-only (red),
combined model (blue), and the convolution of noise-only and metastsability-only PMFs
(green). From a 7-b ADC with ENOB = 6, TFIX /x = 8, and N = 5.

regeneration corresponds to a code
error of −1, so we increment the probability of this code error accordingly.
We repeat this procedure for all five
ways we can run out of time (as well
as for the on-time case), and then we
repeat for every DAC trajectory at every input level. Needless to say, this
can take a while.
Figure 11 shows code error PMFs for
a 7-b ADC with an effective number of

IEEE SOLID-STATE CIRCUITS MAGAZINE

bits (ENOB) of 6, TFIX /x = 8, and N = 5.
Here, ENOB is just a proxy for the ratio of the comparator input-referred
noise to the ADC full-scale range. It
is calculated from the signal-to-noise
ratio, taking signal to be a full-scale
sinusoid and noise to be the sum of
quantization noise and comparator
input-referred noise (other nonidealities are not considered). What's interesting is that the model combining

IEEE Solid-States Circuits Magazine - Summer 2019

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