IEEE Solid-States Circuits Magazine - Summer 2019 - 23

100

10-10

Timing:

TFIX = 0

(1)

T CC C C

Probability

100

10-20
(2)
10-30
10-40

T CF CF C F C

TFIX
-128

-16

N = 25
-2
0
2
Code Error
(a)

T Track and Hold

10-10
10-20

F Fixed Delay

TFIX = 0

(1)

T CC C C

(2)

T C FC F C F C M

10-30
10-40

128

Timing:

TFIX

Ts = 58 τ
-128

0
2
-2
Code Error
(b)

-16

C Comparator Decision

128

M Timing Margin

FIGURE 5: The effect of changing TFIX /x illustrated in metastability code error PMFs. (a) The TFIX /x parameter is increased while keeping N
constant. From the perspective of the MSBs, adding fixed delay is the same as adding timing margin that can be borrowed from the LSBs.
Therefore, large code errors caused by running out of time during MSB regeneration become less likely. However, with N kept constant, the
total probability of all metastable events (Pmeta) must remain the same. (b) The TFIX /x parameter is increased while keeping Ts /x constant. This
reduces N and increases the probability of all metastability errors.

Metastability Code Error
Distribution With Random Input
Signal and Noisy Comparator
Much like that of the synchronous
architecture, the model of the asynchronous variant combining comparator noise and metastability can
be understood as a straightforward
extension of the model for noise in
isolation. This noise-only model is
applicable to both synchronous and
asynchronous architectures because
it considers only the correctness of
comparator decisions and ignores
the timing of the ADC. Let's begin
with a review of the model for thermal noise in isolation.
The first step is to approximate
the input signal's continuous probability density function (PDF) using a

Reset = 00000000

100
10-10
10-20
10-30
Probability

output bits are zeros. The actual ADC
output code is always lower than the
ideal ADC output code, so all errors
are negative. But suppose we prefer
metastability code errors not to have a
bias; in other words, we prefer a code
error PMF with mean close to zero.
This can be accomplished by balancing the reset code. Figure 6 shows the
code error PMF with reset code equal
to 00000000, 11111111, or 01010101.
The balanced reset code with alternating zeros and ones leads to a code
error PMF with a near-zero mean.

Reset = 10101010

100
10-10
10-20
10-30

Reset = 11111111

100
10-10
10-20
10-30

-128

-16

-2

128

Code Error
FIGURE 6: A plot of metastability code error PMFs for different reset codes. The balanced
reset code (10101010) leads to a distribution with near-zero mean.

discrete PMF. This works well as long as
the step size is much smaller than
the thermal noise standard deviation. Then, for each discrete level in
the approximate input signal PMF, we
enumerate all 2B possible DAC voltage waveforms. Because of thermal
noise, all DAC trajectories are possible, even though some are vanishingly improbable. The probability of
a DAC trajectory can be calculated by
integrating the distribution of the input

plus noise either above or below the
DAC voltage at each decision, shown
in Figure 7(a) and (b) for a 3-b example. The product of the shaded areas
is the probability of that DAC voltage
waveform and the associated ADC
output code. Finally, we increment
the probability of the corresponding
code error by this amount (scaled by
the probability of the input level),
and that's how we build up the code
error PMF.

IEEE SOLID-STATE CIRCUITS MAGAZINE

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IEEE Solid-States Circuits Magazine - Summer 2019

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