MECHANIZATION OF THE NPO

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.

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.,

	κ
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.

Figure 7—Geometry of the NPO

Figure 8—NPO run number 5

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
Trace Number	1	2	3	4	5	6	7
Symbol	X₂	X₁	γ	β	i	dξ₂/dτ	dξ₁/dτ
run No. 5
signal	7½ Vrms	7½ Vrms	π ptop		35.6 m cps
noise	16 Vrms	15 Vrms	π/9 ptop[5]		sine wave
DC	0	0
power s/n	1/4	1/4	81/1			0	1/2[6]
terminal value			π/4	π/4
run No. 6
signal	7½ Vrms	7½ Vrms	π ptop		35.6 m cps
noise	0	0	0[7]		sine wave
DC	-30V	0
power s/n	∞	∞	∞			0	∞
terminal value			π/4	π/4

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.