IEEE Circuits and Systems Magazine - Q3 2020 - 25

in DSC, which subsequently requires the use of different
computational elements. Specifically, a DSS can encode
a signal within [0,  1] in the unipolar representation or
within [-1, 1] in the bipolar or sign-magnitude representation. For the sign-magnitude representation, two
sequences are used to encode a signal. One encodes the
magnitude of the signal. Since the signal can have both
positive and negative values, another sequence is used
to represent the signs of the numbers.

cal results. Since the output DSS is generated by comparing the changing signal (as the integration results)
with uniformly distributed random numbers, the RNG
and the comparator inside the integrator works as a

y (t )

2) Sequential Circuits
The stochastic integrator is an important component
used in DSC for iterative accumulation. It implements
integrations of the signal encoded by accumulating
the stochastic bits in a DSS. Fig. 15 shows that a bipolar stochastic integrator can be used to perform numerical integration. In Fig. 15(a), the input digital signal
is cos (rt/ 2 ), sampled at a sampling rate of 2 8 Hz and
converted to a DSS by the DSNG. The other input of the
stochastic integrator is set to a bipolar stochastic sequence encoding 0 (a stochastic bit stream with half of
the bits being '1'). By accumulating the input bits, the
stochastic integrator provides an unbiased estimate to
the Riemann sum [48]. The integration is obtained by
recording the changing values of the counter. Since the
integrated results are directly provided by the counter,
a reconstruction unit is not required.
As can be seen from Fig. 15(b), the results produced
by the stochastic integrator is very close to the analytiTHIRD QUARTER 2020 		

X

DSNG

Z

Y

Signal
Reconstruction

zi

(a)

C. Stochastic Circuits
The DSC circuits can be combinational or sequential.
x (t )

1
0.5
0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

y (t )

1
0.5
0

Original Signal

DSS

1
z (t )

1) Combinational Circuits
A combinational circuit using DSS's as its inputs implements the composite function of the original stochastic function. For example, an AND gate, or a unipolar stochastic multiplier, computes z i = x i y i for each
pair of bits in the input DSS's that independently encode x i and y i (i = 0, 1, 2, f), respectively [46], due to
E[Z i] = E[X i Yi] = E[X i] E[Yi] = x i y i , where Z i, X i and Yi
are the ith bits in the output and input DSS's, respectively. Note that for a continuous signal, oversampling is
required to produce the discrete signals, x i and y i, for
accurate multiplication [46]. Fig. 14(a) shows the DSC circuit for a signal multiplication, and (b) shows the results
of multiplying two sinusoidal signals with frequencies
of 1.5 Hz and 2 Hz, both sampled at a rate of 2 10 Hz. The
sampled signals are converted to DSS's by two DSNGs
with independent Sobol sequence generators [47] as the
RNGs. The output DSS is reconstructed to a digital signal by a moving average [28]. The DSS can also be reconstructed by an ADDIE [1].

DSNG

x (t )

0.5
0

0

0.5

1

1.5

t (s)
Double Precision
Dynamic Stochastic
(b)
Figure 14. A frequency mixer by using a stochastic multiplier
with DSS's as the inputs: (a) circuit and (b) results.

Digital
DSNG
Signal
Bipolar Stochastic
Sequence Encoding '0'

Integration
Results

+

Output DSS

-
(a)

1
0.5
0
-0.5
-1

0

0.5

1

1.5

2

Analytical Results
Results by DSC

2.5

3

3.5

4

Original Signal
Output DSS

(b)
Figure 15. A bipolar stochastic integrator for numerical integration: (a) circuit and (b) results. The output DSS is decimated for a clear view.

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IEEE Circuits and Systems Magazine - Q3 2020

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