Sequential logic based SOs

\label{sequential-logic-based-sos}
Since combinational logic can not effectively be used to implement the non–linear functions, such as tanh, absolute and exponentiation, the authors in [4], [14], [17], [15], [16] have proposed the stochastic elements based on sequential logic, or Finite State Machine, which give a novel approach to perform those functions in SC.
X
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Figure 3.3: General diagram of linear state transition
Fig.3.3 illustrates a generic linear FSM transition diagram. In this configuration, there are N = 2K states, where K is a postitive integer; X, Y are the input and output of this state machine, respectively. It is assumed that X is a stochastic sequence which represents the probability in uniform format PX that each bit being logic 1. Similarly, PY is the probability of observing logic 1 in uniform representation of the output stochastic sequence.
As we discussed about combinational logic based stochastic operators in the sections above, there are many functions which are not represented using the simple operators (Addition, Multiplication, Subtraction, etc.). As a result, it can not be realized these fuctions without difficulty. In this part, we will introduce FSM-SOs that can effectively overcome this problem.

Stochastic Tanh Function

\label{stochastic-tanh-function}
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Figure 3.4: State transition diagram of Stochastic Tanh function
Fig. 3.4 shows the diagram of Stochastic Tanh function [4] in which the output Y only depends on the current states Si:
Y = . 0, 0 ≤ i ≤ N/2 − 1 , where 0 i N 1. (3.7) 1, N/2 ≤ i ≤ N − 1 ≤ ≤ −
This configuration approximates a Tanh function as follows
N N
e 2 x − e− 2 x