IEEE Solid-State Circuits Magazine - Summer 2015 - 62

FOMS LF Asymptote (dB)

180
175
170
165
160
155
150

2000

2005
Year

2010

2015

Figure 4: fomS trend over time (low-frequency asymptote). The fit line has a slope of
about 1 dB per year.

fs at FOMS = 150 dB (Hz)

1011
1010
109
108
107
106

2000

2005
Year

2010

2015

every three years). In this context, it
is worth noting that FoM w tends to
improve significantly faster, since
it de-emphasizes noise limits and is
thus catered toward technology-limited designs that benefit more directly
from feature size scaling. The work of
[19] extracted a FoM w improvement
rate of 2x every 2.5 years for deltasigma converters and only 1.8 years
for Nyquist converters. These numbers align well with the 2x energy
improvement for every 1.9 years that
was observed in [16]. The key takehome here is that the composition of
the figure of merit plays an important
role in the observed trends.
It is sometimes suggested that
part of the above-quantified improvements come from ever more "creative"
reporting of results and leaving out
significant power contributors (e.g.,
off-chip calibration units, I/O, etc.)
[10]. While this may certainly play a
role, it cannot explain the steady improvements observed in Figure 4,
which sum to an aggregate progress
of over 20 dB in less than two decades.
In the context of ever-improving
efficiency, it is useful to revisit the
fundamental limit discussion of [9].
There, we noted that a useful bound
on conversion energy is given by the
minimum energy it takes to drive a
sampling capacitor using an ideal
(class-B) amplifier [40], [41]:

Figure 5: fomS trend (high frequency asymptote). The fit line indicates doubling of the
conversion rate (fs) for every 1.8 years.

c

between 10 and 100 MHz, the efficiency begins to deteriorate and rolls
off with a slope of approximately
-10 dB per decade. As discussed in
[35], this indicates that the power dissipation scales with the square of the
conversion speed in this regime.
Also notice from the plot that
recent pipelined SAR designs ([36]-
[38]) set the peak performance near
the corner. The time-interleaved
SAR ADC reported in [39] marks the
fastest design and interestingly lies
almost exactly on the -10 dB/decade
roll-off portion of the envelope. The
envelope is positioned by taking the
average of the top five data points

62

su m m E r 2 0 15

to define the horizontal asymptote, and the average of the top five
designs along a -10 dB slope.
As we can see from Figure 3, the
contributions of the past five years
have pushed the asymptotes up and
to the right. It is interesting to quantify the rate of this movement. This
is done for the location of the lowfrequency asymptote in Figure 4. We
observe that the improvements have
followed a steady pace with minor
variations (likely due to the finite
size of the data set) from year to year.
Interestingly, the overall progress rate
comes out almost exactly to 1 dB per
year (or doubling of energy efficiency

IEEE SOLID-STATE CIRCUITS MAGAZINE

P m = 8 kT # SNR .
fS min

(2)

Approximating SNDR , SNR, assuming room temperature, and inserting this expression into the definition of FoM S gives:
FOM S,max = SNR ^dB h
1
m
16 kT # SNR
= - 10 log ^16 kT h = 192 dB.
+ 10 log c

(3)
Since this bound considers only
the sampling energy, it is clear that
we will likely never reach this number. A more practical limit may be
around 186 dB, which would be



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