y ≈ e N
N , (3.8)
2 x + e− 2 x
where y = 2PY − 1, x = 2PX − 1 are the
bipolar representation of the input sequence X and the output
sequence Y , respectively.
Stochastic
Exponentiation
Function
\label{stochastic-exponentiation-function}
… …
… …
… …
Figure 3.5: State transition diagram of Stochastic Exponential function
This function that is illustrated in
Fig.3.5 is configured as
[4]:
. 0, 0 ≤ i ≤ N − tt − 1
Y = , where 0 i N 1 (3.9) 1, N − tt ≤ i ≤ N
− 1
By defining x = 2PX − 1 to be the bipolar representation
of the input sequence, and y = PY to be the unipolar
format of the output sequence, we have an approximation as follows
. e−2Gx, 0 < x ≤ 1
y , where 0 < tt << N.
(3.10) 1, −1 ≤ x ≤ 0
…
\label{section-13}
…
\label{section-14}
…
\label{section-15}
Figure 3.6: State transition diagram of Stochastic Absolute function
Stochastic
Absolute Value
Function
\label{stochastic-absolute-value-function}
Stochastic Absolute Function was propose in
[14] with the state transition
diagram is as Fig.3.6. The output Y is determined by current
state Si as
0, i is odd when 0 ≤ i ≤ N/2 − 1 OR
Y =
i is even when N/2 ≤ i ≤ N − 1
1, i is even when 0 ≤ i ≤ N/2 − 1 OR i is odd
when N/2 ≤ i ≤ N − 1
, where 0 ≤ i ≤ N − 1; 0 < tt
<< N. (3.11)
And this configuration approximate an absolute function in which the
input x = 2PX − 1 and the output y = 2PY − 1
are the bipolar formats, and
y = |x| (3.12)
Stochastic
Exponentiation Function on an Absolute
Value
\label{stochastic-exponentiation-function-on-an-absolute-value}
𝑋
𝑆0
𝑌0 = 0
X …
…
…
𝑋
X
𝑋
𝑆𝐺−1
𝑌𝐺−1 = 0
X
𝑋
𝑆𝐺
𝑌𝐺 = 1
X
X
… …
… …
… …
𝑋
Figure 3.7: State transition diagram of Stochastic Absolute Exponential
function
This mixed function whose state transition diagram is shown in
Fig.3.7. This kind of FSM based
stochastic function can be implemented by setting as follows,
. 1, tt ≤ i ≤ N − tt − 1
Y = , where 0 i N 1; 0 < tt
<< N (3.13) 0, i < tt OR i
> N − tt − 1
The approximate function in this configuration
[17],
[15] is
y = e−2G|x|, (3.14)
where x = 2PX −1 and y = PY are the bipolar
representation of the input sequence X and the unipolar format of
the output stream Y , respectively.
Simulation
and analysis
\label{simulation-and-analysis}
As all FSM–based Stochastic elements are build using the approximation
[15], there are certainly the
variation of these linear functions in comparison with the theoretical
ones. According to the configuration that we introduced, the number of
states will be considered as a principal that affects on the output.
Here the tanh and absolute functions will be considered as our case
study.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
x
bipolar
Figure 3.8: Simulation results of the Stochastic absolute function when
consider different number of states
It can be seen in Fig.3.8 that the
approximation results are closer to the desired one, i.e., approximation
error decreases, when the number of states increases from 8 to 64. And
this increasing equally affects on the malformation at x = 0: the
distortion can be minimized by using more states of FSM. However, one
thing must be remarked on is the incresing of states corresponds to the
increasing of complexity of implementation.
In order to evaluate the Stochastic tanh function, it can be illustrated
in Fig.3.9 when the numbers of state
are 4 and 16. Fig.3.9a represents a
poor result for the design using FSM with only 4 states, while
Fig.3.9b with 16 states provides a
better result where the approximation function is closer to the
theoretical one. This difference between two state selections are
considerable. In other word, we obtain a good result of this
approximation function for large values of states. Nervertheless, as the
first simulation, an issue is how to choose an amenable number of states
for our design. Fortunately, relying on the approximation error, the
authors in [15] also
recommended this such value corresponding to each FSM–based stochastic
function: 8–state FSM for the Stochastic tanh function, 16–state FSM
for the Stochastic Exponentiation function, 32–FSM for the Stochastic
Exponentiation function on an Absolute values, and 8–FSM for the
Stochastic absolute function.
1
1
0.8 0.8
0.6
0.4
0.6
0.4
0.2 0.2
0 0
−0.2 −0.2
−0.4 −0.4
−0.6 −0.6
−0.8 −0.8
−1
−1 −0.8 −0.6 −0.4 −0.2
x 0 0.2 0.4 0.6 0.8 1
bipolar
−1
−1 −0.8 −0.6 −0.4 −0.2
x 0 0.2 0.4 0.6 0.8 1
bipolar
-
(b)
Figure 3.9: Simulation results of the Stochastic tanh function: (a)
using 4–state FSM and (b) using 16–state FSM
Chapter
4
Digital Filter and Image Processing Using Stochastic Computing
This chapter is to address the architecture of digital filter and image
processing circuits using SC. The basic operators relied on
combinational logic circuitry will be employed for the stochastic FIR
filter. Meanwhile, the FSM–based stochastic elements will be
effectively investigated in two important applications: edge detection
and median filter based noise reduction. The soft–error is equally
injected into the internal circuit with different levels to evaluate the
fault–tolerace of stochasic approach. Our work will demonstrate that
stochastic implementations can be considerably more noise–tolerant and
consume less hardware than conventional ones.