IEEE Circuits and Systems Magazine - Q3 2020 - 26

DSNG. As a result, the output sequence encodes the
discretized stochastic integration solution for the analytical solution, (2 /r) sin (rt / 2 ). The output DSS is also
shown in Fig. 15(b).
The stochastic integrator has been used in a stochastic
low-density parity-check (LDPC) decoder to implement
a tracking forecast memory in [49]. It solves the latch-up
problem in the decoding process with a low hardware
complexity.
For the FSM-based sequential circuits, however, the
DSS cannot work well because the probability consistently changes and a steady hyper-state is hard to reach.
The DSS could also violate the condition of the model
being a Markov chain due to the autocorrelation of the
encoded signal.
IV. Application of DSC
By using the stochastic integrators, DSC can implement a series of algorithms that involve iterative accumulations, such as an IIR filter, the Euler method and
a GD algorithm.

	

H (Z ) =

1
, (3)
2n Z + 1 - 2n

for a stable first-order low-pass IIR filter [50]. An oversampled T - R modulated bit stream is then used as the
input of the circuit. Since the generation of a T - R modulated bit stream can also be considered as a Bernoulli
process [51], the DSS can be used for IIR filtering as well.
Fig. 17 shows the filtering of mixed sinusoidal signals
of 4 Hz and 196 Hz. The mixture is first sampled at a sampling rate of 65.6 kHz and the sampled digital signal is
then converted to a DSS. The DSS is filtered by an 8-bit
ADDIE. As can be seen, the high-frequency signals are almost filtered out while the low-frequency signals remain.
B. ODE Solvers
The Euler method provides numerical solutions for
ODEs by accumulating the derivative functions [52]. For
an ODE dy/dt = f (t, y (t )), the estimated solution, yt (t ),
is given by

y (t )
p

A. IIR Filtering
In [50], it is shown that an ADDIE with different parameters can be used as a low-pass IIR filter with different
frequency responses, as shown in Fig. 16. Given the bit
width of the stochastic integrator n, the transfer function of such a filter in the Z-domain is given by

+

	

-

Figure 16. ADDIE circuit produces an exponential function
when the input is a sequence encoding a static number.

yt (t i + 1) = y i + 1 = y n + hf (t i, y i) (4)

with h as the step size and t i + 1 = t i + h. After k steps of
accumulation,
	

k-1

y k = y 0 + h / f (t i, y i). (5)
i=0

To use a stochastic integrator, let the pair of input DSS's
- erivative
(e.g., A and B in (2)) encode the discrete d
function, {f (t i, y i)}, such that E[A i - B i] = f (t i, y i). By doing so, the stochastic integrator implements (2) and approximately computes (5) with a step size h = 1/ 2 n. As
a result, the bit width of the counter does not only decide the precision of the computation but also the time
resolution. The integration in the Euler method is implemented by accumulating the stochastic bits in the DSS's
instead of real values.
Using this method, an exponential function can be
generated by a stochastic integrator. The ADDIE circuit,
as shown in Fig. 16, solves

1
0
-1

0

0.1

0.2

0.3

0.4

t (s)
(a) Original and Filtered Signals
Original Signal
Filtered Signal

0.4
0.2
0

0

50

100

150
f (Hz)

200

250

(b) Spectra of Original and Filtered Signals
Figure 17. The low-pass filtering results in (a) the time domain and (b) the frequency domain.
26 	

	

d y (t )
= p - y (t ), (6)
dt

which is the differential equation for a leaky integrator
model [53]. Thus, the ADDIE can be used to implement
neuronal dynamics in a leaky integrate-and-fire model.
The analytical solution of (6) is given by

IEEE CIRCUITS AND SYSTEMS MAGAZINE 		

THIRD QUARTER 2020
IEEE Circuits and Systems Magazine - Q3 2020

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