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) =
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.