IEEE Solid-States Circuits Magazine - Spring 2019 - 93

code error for this outcome is +1,
and, due to the improbable second
MSB error, this outcome is less likely
than the error-free case.
Because the distribution of comparator input-referred noise is well
modeled by a Gaussian, every decision has some nonzero probability of
being a logic 1 or a logic 0. Even for
a deterministic input voltage, all 2 B
output codes are possible (although
some are diminishingly improbable).
To find the code error distribution,
we can consider each of the 2 B possible outcomes, find the corresponding
code error, and evaluate the probability. Then, we can increment the probability of the corresponding code
error by this amount as we iteratively
build the code error distribution.
Given a particular ADC output
code, the DAC voltage waveform that
leads there is determined. Each bit
of the output code selects between
two directions at each step in the
SAR decision tree. We can easily
build the DAC voltage waveform in

VFS
2

111
110
101
100
011
010
001
000

vID
0
V
- FS
2

this manner and then integrate the
distribution of v ID + v n, i at each step
to find the individual probabilities
of the decisions that give rise to
this DAC voltage waveform. Then,
the probability of the ADC output
code is the product of these individual probabilities. By extending this
method for deterministic inputs to
random inputs using total probability as described previously, we can
calculate the code error distribution
for our Gaussian signal in the presence of just thermal noise. Figure 8
shows the code error distributions
for several values of comparator
noise, parameterized by effective
number of bits (ENOB). 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 SNR, taking signal to be a fullscale sinusoid and taking noise to be
the sum of quantization noise and
comparator input-referred noise. It
is worth noting that these PMFs do
not differ significantly from those
predicted by conventional wisdom,
i.e. a model that just refers the comparator noise directly to the input
of the SAR ADC. However, it is still
worthwhile to review this detailed
thermal noise model because it is
readily e x tended to capt ure the
interaction between metastability and
thermal noise in the SAR loop.

Metastability and Thermal
Noise Combined
This framework for calculating the
code error distribution in the presence of noise turns out to be extremely
versatile. In our model for metastability, each comparator decision has
four possible outcomes. These are a
logic 1/0 that regenerates on time and
a logic 1/0 that regenerates one cycle
late. Hence, the SAR loop behavior
has 4 B possible outcomes. For each
outcome, there is a corresponding
DAC voltage waveform, which we can
build up step by step by following the
sequence of fast or slow, high or low
decisions, similar to Figure 7. Then,
to calculate the probability of some
outcome, we integrate the distribution of v ID + v n, i in the corresponding
voltage ranges that give rise to this
DAC waveform and find the product
of these individual probabilities (see
"Computing the Combined Probability Mass Function"). Figure 9 provides
an example.
Figure 9(a) shows the error-free
case, which results in the same DAC
voltage waveform as Fig ur e 7(a).
However, a correct decision now
entails not only resolving the correct bit, but also resolving it on time.
Hence, the red stripes around the
transition levels that lead to metastability are excluded from the voltage ranges leading to the error-free

(a)
100
111
110
101
100
011
010
001
000

vID
0
V
- FS
2
(b)

FIGURE 7: A 3-b example showing successive approximation of a deterministic
input in the presence of comparator noise.
(a) Each bit is resolved correctly, leading
to a code error of 0. (b) Thermal noise flips
the second MSB decision. The LSB is then
resolved correctly with high probability,
leading to a code error of +1.

ENOB = 5.5
ENOB = 6
ENOB = 6.5
ENOB = 7
ENOB = 7.5

10-5
Probability

VFS
2

10-10

10-15

10-20

-15

-10

-5

0
Code Error

FIGURE 8: Code error PMFs due to thermal noise across a range of ENOB. An 8-b ADC digitizing a Gaussian input signal with clipping probability 10−4.

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IEEE Solid-States Circuits Magazine - Spring 2019

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