IEEE Circuits and Systems Magazine - Q3 2020 - 27
y (t ) = p - ( p - y 0) e - t, (7)
where y 0 is provided in the initial condition, i.e.,
y (0) = y 0 . This model is commonly seen in natural
-p rocesses such as neuronal dynamics [53]. Fig. 18
shows the results produced by the ADDIE when y 0 = 1
and p = 0.
For a set of ODEs,
Z dy 1 (t)
]]
= y 2 (t ) - 0.5,
dt
[ dy 2 (t)
(8)
]
= 0.5 - y 1 (t ),
t
d
\
with the initial values y 1 (0 ) = 0 and y 2 (0 ) = 0.5, a numerical solution can be found by the circuit in Fig. 19.
The upper integrator computes the numerical estimate of y 1 (t ) by taking the output DSS from the lower
integrator and a conventional stochastic sequence
encoding 0.5 as inputs. So the difference of the two input sequences approximately encodes y 2 (t ) - 0.5, the
derivative of y 1 (t ). Simultaneously, the output DSS of
the upper stochastic integrator encodes the numerical
solution of y 1 (t ), which is used to solve the second differential equation in (8). Thus, the stochastic integrator
can be viewed as an unbiased Euler solution estimator
[54]. The results produced by the dynamic stochastic circuit is shown in Fig. 20. 8-bit up/down counters
are used in both stochastic integrators and the RNG is
implemented by a Sobol sequence generator [47]. Since
the results are provided by the counter in the stochastic integrator, an explicit reconstruction unit, as shown
in Fig. 12, is not required.
A partial differential equation, such as a Laplace's
equation, can also be solved by an array of stochastic
integrators with a finite-difference method.
C. Weight Update for an Adaptive Filter
DSC can be applied to update the weights in an adaptive digital filter. An adaptive filter system consists of a
linear filter and a weight update component, as shown
in Fig. 21. The output of the linear filter at time i is
given by
1
1
0.8
y1 (t )
DSC
Analytical
0.5
y (t )
0.6
0
0.4
0
2
4
6
0
2
4
6
8
10
12
8
10
12
1
0
y2 (t )
0.2
0
2
4
t
6
0.5
8
0
t
Figure 18. ADDIE used as an ODE solver. The solution is an
exponential function.
y1(t )
DSC
Analytical Solution
Figure 20. Simulation results of the ODE solver in Fig. 19.
They are compared with the analytical solutions.
+
p = 0.5
p = 0.5
-
xi
+
Linear Filter With
Weights wi
y2(t )
Figure 19. A pair of stochastic integrators solving (8) [54].
THIRD QUARTER 2020
-
+
+ t
i
ei
∆wi
-
yi
LMS Weight Update Unit
Figure 21. An adaptive filter.
IEEE CIRCUITS AND SYSTEMS MAGAZINE
27
IEEE Circuits and Systems Magazine - Q3 2020
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