IEEE Circuits and Systems Magazine - Q3 2020 - 24

D. Limitations
Although the SC circuits are much simpler than conventional arithmetic circuits, a large sequence length is
required to achieve a high accuracy. This requirement
leads to poor performance and low energy efficiency.
Fig. 11 shows the root mean square error (RMSE) encoding a number in the unipolar representation using different types of "random" numbers. The pseudorandom
numbers are generated by a 16-bit LFSR with randomly
generated seeds. The RMSEs are obtained from 1,000 trials using different sequence lengths. As shown in Fig. 11,
to achieve an RMSE of 1 # 10 -2, a sequence length of 2 12
bits is required for the LFSR generated sequences. For
the stochastic Sobol sequences, the length is significantly reduced to 2 6 or 64 bits to achieve a similar accuracy,
although it is still not as compact as an equivalent 5-bit
fixed-point number that results in a similar RMSE. Meanwhile, low precision (e.g. at an RMSE of 1 # 10 -2 or larger)
can be tolerated in many applications such as inference
in a machine learning model [12]-[17], [29]-[33], digital
signal processing [34]-[40] and image processing [41]-
[44]. Hence, progressive precision (PP) can be employed
and the Sobol sequence length can be as low as 8 bits in
an application in a CNN inference engine [45].

100

Despite the fact that some of these implementations
achieve a lower energy consumption with PP compared
with conventional circuits, the application of conventional SC (CSC) is still quite limited.
III. Dynamic Stochastic Computing
In a DSC system, as shown in Fig. 12, the stochastic sequences encode varying signals instead of static numbers. Consider a discrete-time digital signal {x i} within
[0, 1] (i = 0, 1, 2, f). A DSS encoding {x i} satisfies that
E[X i] = x i , where X i is the ith bit in the DSS. Thus, every bit in the sequence can have a different probability
to be 1.
A. DSS Generation
A DSS can be generated by a dynamic SNG (DSNG), as
shown in Fig. 13. It is similar to a conventional SNG except that the input is a varying signal instead of a static
number. The RNG generates uniformly distributed random numbers within [0,1]. Given a discrete signal, {x i},
as shown in Fig. 13, it is compared with a random number one sample at a time. If x i is larger than the random
number, a '1' is generated; otherwise, the DSNG generates a '0'. Generating every bit in the DSS is a Bernoulli
process (in the ideally random case), so the expectation
getting a '1' is equal to the corresponding sample value
from the discrete digital signal. The discrete signal can
either be sampled from an analog signal or read from a
memory device.

RMSE

10-1

B. Data Representations in DSC
Similar to CSC, all the aforementioned mapping schemes
can be used to encode numbers within certain ranges

10-2
10-3
10-4

Sobol Sequence
Pseudorandom Sequence
0

2

4
6
8
10
Sequence Length (2N )

RNG
12

{xi}

Figure 11. RMSE of the encoded numbers with different
types of stochastic sequences.

Digital Signals
(From ADCs or
Storage)

DSNGs
Dynamic
Stochastic
Sequences

A
B

A


IEEE Circuits and Systems Magazine - Q3 2020

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