MECHANIZATION OF THE NPO_Self-Organizing Systems, 1963

Analog computer elements were used to simulate one NPO which was

tested in the experimental configuration shown in Figure 6. The NPO

operates upon a channel which is artificially generated from the two

noise generators i₁ and i₂ and the signal generator i₀ (i₀ may also be

a noise generator). The NPO accepts the inputs labelled X₁ and X₂ and

provides the three outputs Ξ₁, Ξ₂, and γ. X₁ is the linear combination

of the outputs of generators i₁ and i₀, similarly X₂ is obtained from

i₂ and i₀.

[Illustration: Figure 6—Experimental test configuration for the

simulation of an NPO]

Obviously, i₀ is an important parameter since it represents the memory

relating the spaces X₁ and X₂. Ξ₁ has the property that the magnitude

of its projection on i₀ is a maximum while Ξ₂ to the opposite has a

zero projection on i₀. γ is the detected version of the eigenvalue of

Ch(X₂,X₁).

In the companion paper it was shown how one can provide a Euclidean

geometrical representation of the NPO. This representation is shown in

Figure 7 which shows the vectors i₀, i₁, i₂, X₁, X₂, Ξ₁, Ξ₂, and the

angles Θ₁, Θ₂, and γ. The length of a vector is given by

|X| = κₓ(2πε)⁻¹ᐟ² ∈ H(X)

and the angle between two vectors by

|Θ(X₁,X₂)|-sin⁻¹ ∈ -R(X₁,X₂).

The three vectors i₀, i₁, i₂ provide an orthogonal coordinate system

because the corresponding signals are random, _i.e._,

R(i₀,i₁,i₂) ≡ 0.

As external observers we have a prior knowledge of this coordinate

system; however, the NPO is given only the vectors X₁ and X₂ in the i₀

⨉ i₁ and i₀ ⨉ i₂ planes. The NPO can reconstruct the entire geometry

but the actual output Ξ obviously is constrained to lie in the plane of

the input vector X. The following formulas are typical of the relations

present.

|Ξ₁|

tan β = ————

|Ξ₂|

cos Θ = cos 2β csc 2γ

cos 2β

cos 2Θ₁ = -1 + 2 ———————

1-cos 2γ

cos Θ = cos Θ₁ cos Θ₂.

[Illustration: Figure 7—Geometry of the NPO]

[Illustration: Figure 8—NPO run number 5]

[Illustration: Figure 9—NPO run number 6]

We have obtained a complete description of the NPO which involves 74

formulas. These treat the noise in the various outputs, invariances of

the NPO and other interesting features. A presentation of these would

be outside of the scope of this paper and would tend to obscure the

main features of the NPO. Thus, we show here only a typical sample of

the computer simulation, Figure 8 and Figure 9. Conditions for these

runs are shown in Table I. Run No. 6 duplicates run No. 5 except for

the fact that i₁ and i₂ were disabled in run No. 6.

Observe that all our descriptions of the NPO and the space it is to

decompose have been time invariant while the signals shown in the

simulation are presented as functions of time. The conversion may be

effected as follows: Given a measurable (single-valued) function

x = x(t)t ∊ T

where

μ(T) > 0

we define the space

X = {x = x(t) ∍ t ∊ T}

and a probability distribution

μ(x⁻¹(X′))

P(X′) = —————————— X′ open ⊂ X

μ(T)

on that space.

TABLE I

Legend for Traces of Figures 8 and 9

---------+-------+-------+--------+-----+----------+-------+--------

Number | 1 | 2 | 3 | 4 | 5 | 6 | 7

---------+-------+-------+--------+-----+----------+-------+--------

Symbol | X₂ | X₁ | γ | β | i | dξ₂/dτ | dξ₁/dτ

---------+-------+-------+--------+-----+----------+--------+-------

| | | | | | |

| | | | | | |

| | | | | | |

DC | 0 | 0 | | | | |

| | | | | | |

power s/n| 1/4 | 1/4 | 81/1 | | | 0 | 1/2[6]

| | | | | | |

value | | | π/4 | π/4 | | |

---------+-------+-------+--------+-----+----------+--------+-------

| | | | | | |

| | | | | | |

noise | 0 | 0 | 0[7] | |sine wave | |

| | | | | | |

DC | -30V | 0 | | | | |

| | | | | | |

power s/n| ∞ | ∞ | ∞ | | | 0 | ∞

| | | | | | |

value | | | π/4 | π/4 | | |

---------+-------+-------+--------+-----+----------+--------+-------

[5] Observed from Oscillogram

[6] Computed

[7] Observed from Oscillogram

Then (X,p(X)) is a stochastic space in our usual sense and x(T) is a

stochastic variable. Two immediate consequences are:

P(X) is stationary (P(X) is not a function of t ∊ T), and no question

of ergodicity arises.