IEEE Solid-States Circuits Magazine - Summer 2019 - 28

equalizer (FFE) and decision feedback
equalizer (DFE). We assume 4-level
pulse amplitude modulation (PAM4)
signaling with symbols [−1, −1/3, 1/3,
1] and a representative channel with
14-dB loss at Nyquist, which gives the
blue pulse response in Figure  13(b).
The FFE has five precursor and 15
postcursor taps, which are optimally
adapted to yield the values in Figure  13(c). The pulse response at the
FFE output is shown in red in Figure 13(b). The first FFE postcursor is
assumed to be handled by the one-tap
DFE. For simplicity, we neglect quantization effects in the equalizers.
Figure 14(a) looks at the inner workings of the FFE, specifically showing
how it processes incoming probability densities. The delay-add operations lead to a weighted convolution
of the PDFs/PMFs associated with
each sample. As a cartoon example,
we show here the PDFs of the ADC's
quantization error, but this process
works similarly for independent distributions containing all nonidealities,

Starting slow and then speeding up the ADC
(i.e., decreasing N), we eventually hit a corner
where metastability begins to cause errors
more significant than the thermal noise.
noise. Figure  12(a) shows probabilities computed with the combined
model for metastability and noise at
TFIX /x = 8, and Figure  12(b) shows
probabilities approximated by convolving the metastability-only and
noise-only PMFs at several values of
TFIX /x. Figure  12(a) superimposes
a plot of Pmeta versus N to illustrate
that designing for Pmeta may be overly
conservative. For example, if we want
the probability of code errors with
magnitude greater than 8 LSB to be
10 -15, we should design for N = 4.3.
If we instead design for Pmeta = 10 -15,
then we would use N = 8.6. By considering the range of code errors that
matter, as well as the code error PMF,
we learned that we can run the ADC

about 40% faster (after including fixed
delays in the total conversion time).

Application Example:
ADC-Based Serial Link
Ultimately, the error PMF computed
in the previous section must be
interpreted at the system level and
translated into a metric that is useful for the application. To illustrate
this process using an example, we
consider an ADC-based serial link,
which commonly employs a timeinterleaved asynchronous SAR ADC.
Our analysis is similar to [2], except
that we work with our PMF model to
find the bit error rate (BER).
Figure 13(a) shows the considered
receiver with a digital feedforward

Thermal Noise

Metastability

FFE

Channel

DFE

(a)
30

20

0.8
0.6
Coefficient

25
Cursor Value (LSB)

1

Sampled Channel
Pulse Response
Post-FFE
Pulse Response
Handled by DFE

15
10

0.2

5

0

0

-0.2

-5
40

50

60

70
80
Sample Index
(b)

90

100

First FFE Post-Cursor Is
Nulled Due to DFE

0.4

-0.4

-5

0

5
Tap Number
(c)

10

15

FIGURE 13: (a) A system model for ADC-based serial link analysis. (b) A graph of pre- and post-FFE pulse responses. (c) A graph of adapted
FFE coefficients.

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