IEEE Circuits and Systems Magazine - Q4 2022 - 40

By confining the ADC range to the useful signal range,
quantization errors are reduced for the same number of
levels. The benefit of calibration is greater for differential
cells, which have a much smaller useful signal range
as explained in Section 6.2.
Fig. 16 further shows that after range calibration is
applied, differential cells can tolerate an ADC with several
fewer bits of resolution than offset subtraction systems:
evidently, differential cells are more resilient to
quantization errors. This resilience can be understood
by again considering the proportional weight mapping
property of differential cells. Combined with the analog
subtraction of currents, this results in dot product
proportionality: the voltages at the ADC input are proportional
to the numerical values of the dot products
(or the slice-wise dot products). Offset subtraction systems
lack this critical proportionality, since an offset
must be subtracted after the ADC to obtain the true dot
products.
Dot product proportionality implies that the data
compression function of the ADC is effectively applied
to the numerical dot products. Therefore, the required
ADC resolution is directly connected to the neural network's
inherent sensitivity to data precision, which is
hardware-independent and is fully decoupled from Bout.
The effect is most striking with unsliced weights, where
Bout = 26.2 bits but high accuracy is maintained down to
BADC = 7 bits. This is close to the inherent precision sensitivity
of ImageNet neural networks, which is typically
about 8 bits [31].
Fig. 17 compares systems with offset subtraction
(without dot product proportionality) and differential
cells (with dot product proportionality) at a fixed
ADC resolution of 8 bits. Offset-subtraction systems
can only tolerate an 8-bit ADC when the array is small
(≤144 rows) and the weights are finely sliced (≤2 bits/
cell), which together bring Bout close to 8 bits. Differential
cells suffer almost no accuracy loss with an
8-bit ADC regardless of the bits per cell and array
size; this again illustrates that the accuracy is decoupled
from Bout. Dot product proportionality makes a
practical ADC resolution of 8 bits compatible with
a much larger analog resolution Bout. Equivalently,
much more computation can be done in the analog
domain before the signal is ever converted to digital.
This has significant consequences for energy efficiency,
discussed in Section 9.
7. Suppressing Error Propagation
As shown in Section 5, the accuracy loss due to accumulated
cell errors can be minimized by using sufficiently
precise memory cells and exploiting state-proportional
errors. With less precise cells, some prior work has

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