Stochastic number (SN) representation

\label{stochastic-number-sn-representation}
To transform analog variables in the conventional domain to probabilities in the stochastic domain (Fig.1.1), there are generally [8] three methods:
Unipolar
This is the simplest form of linear mapping, in which, considering a quantity E in range 0 ≤ E ≤ V , a binary variable A is used to represent E with probability p as follows:
p = p(A = 1)= E/V (1.2)
Therefore, the maximum value of the quantity E (E = V ) is represented by a logic level which is always ON , similarly, the zero value is represented by a logic level which is always OFF . And a randomly fluctuating logic level represents the intermediate values.
Assume that the value A is noted at every clock interval with the value at the ith clock pulse being
Ai(Ai = 0, 1). Thus the estimation of the generating probability p, pˆ, in N clock intervals is:
n
pˆ = (1/N ) . Ai (1.3)
i=1
The expected value of this estimation, Exp(pˆ), is given by
Exp(pˆ) = p. (1.4)
The variance which is determine the accuracy of the estimation, var(pˆ), is given by [8]
var(pˆ) = Exp .(pˆ − p)2. = p(1 − p)/N. (1.5)
And the standard deviation of the estimated value of p is
σ(pˆ) = [p(1 − p)/N ]1/2. (1.6)
Thus the accuracy of the estimation is dependent on the number of the clock intervals. The error is zero when the generating probability is equal to zero or one, corresponding to deterministic sequences. And when p is equal to 0.5, the error takes the maximum value. From (1.6), the error is inversely proportional to the square root of the number of clock interval N . Hence the compromise between computational speed and accuracy is clearly illustrated: when the clock intervals N increase, or the sample is longer, the error will be less, however, it also means that it takes longer time in calculation.
Two-line bipolar
In this representation, the quantity E is within −V ≤ E ≤ V , two sequences of logic levels on separate lines, one representing positive quantities and the other negative, will be used. On the supposition that the line which has the probability is weighted positively is called the UP line and the negatively weighted line is called DOWN [19], we have
E
V = p(UP line ON ) − p(DOWN line ON ) (1.7)
Hence the UP line always ON and the DOWN line always OFF represent the maximum positive value of E. Similarly, the UP line always OFF and the DOWN line always ON represent the maximum negative value. The intermediate values will occur on one or both lines. And zero is represented by both lines being OFF.
Single-line bipolar
This is the mapping of most interest to the present work. In this representation, a quantity E in the range −V ≤ E ≤ V is represented by the mapping as follows:
p(ON ) = p(C = 1) =
  1. 1 E
+
  1. 2 V
(1.8)
where C is a binary variable on a single line. This transform shows that p(ON ) = 1 for maximum positive value E = V , and p(ON ) = 0 for maximum negative value E = −V . And zeros are represented
with an equal probability of being ON or OFF. It can be reduced from (1.8), the inverse transformation
E/V = 2p − 1. (1.9)
The variance of the estimation of E/V , var .Eˆ/V . is given by [9]
var .Eˆ/V . = 4p(1 − p)/N = [1 − (E/V )2]/N (1.10)
and its standard deviation:
σ .Eˆ/V . = [4p(1 − p)/N ]1/2 = .[1 − (E/V )2]/N . (1.11)
Therefore the conclusion about the accuracy of this mapping is similar to one at the first unipolar representation.