Table 2.3: Examples for Stochastic scale adder based on 10–WBtt
Table 2.3 shows that Stochastic scale adder give us the good results which are close to the expected ones, and promises a noticeable operator based on LFSSR in Stochastic computing.

2.3.4 Markov Chain based SNG

\label{markov-chain-based-sng}
Besides using Randomizer unit [18] as an ideal random bit stream generator, for the purpose of reducing the system cost in practice, the synthesizing techniques has been recently considered. The first approach is to synthesize combinational logic that transforms source probabilities into different target probabilities. Weikang et al. [20] investigated this technique in considering three scenarios with respect to the input set S. Nevertheless, the efficacy of transforming probabilities with combinational logic is bounded by the cost of the input stochastic sources. For instance, it is shown in Fig.5 in [20] that seven independent input stochastic sequences are required to generate the disired probability 0.119, i.e, the domination of random sources in combinational circuit.
As a result, the authors in [21] has lately proposed a novel synthesizing approach based on sequential logic or Finite State Machine (FMS) that emulate Reversible Markov chains. A Markov chain is defined as a discrete-time stochastic process that satisfies the following Markov property: given the present state Xn, the transition probability to the next state Xn+1 depends only on the present state. This feature is illustrated as
P {Xn+1 = j |Xn = i, Xn−1 = in−1, …, X0 = i0} = Pij , n ≥ 0. (2.16)
Reversible property indicates that Markov chain follows the Detailed Balance condition that illustrates relation between stationay probabilities and transition probabilities [21]:
πiPij = πj Pji, ∀i, j ∈ S, i ƒ= j (2.17)
where S is state space of the system. Saraf et al. [21] also concluded that the amenable number of independent input random sources is three. Therefore, this approach gives the capacity to reduce hardware cost of random sources (or controllable) used in stochastic system. It is trully considerable in comparison with the approach relied on combinational logic.
As an example, we will implement this technique with expected probability Poutput = 0.7 = 7/10 = a/b, and allowed number of random sources K = 3. The transition probability ratios are consequently powers of two upto 2K−1 = 23−1 = 4, and we have
a = 7 = 1 + 2 + 4
b = 10 = 1 + 1 + 2 + 2 + 4.
Hence,
a0 = 1, a1 = 1, a2 = 2, a3 = 2, a4 = 4
Son = {0, 2, 4} = {0, 3, 4} = … = {1, 2, 4}.
(2.18)
1/2 0
1/4 0
3/4