on that space.

TABLE I
Legend for Traces of Figures 8 and 9
Trace Number1234567
SymbolX₂X₁γβi dξ₂/dτ dξ₁/dτ
run No. 5
signal 7½ Vrms 7½ Vrms π ptop 35.6 m cps
noise 16 Vrms15 Vrms π/9 ptop[5] sine wave
DC 00
power s/n 1/41/4 81/1 01/2[6]
terminal value π/4 π/4
run No. 6
signal 7½ Vrms7½ Vrms π ptop 35.6 m cps
noise 000[7] sine wave
DC -30V0
power s/n ∞∞ ∞ 0 ∞
terminal value π/4π/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.

NETWORKS OF NPO’S

A network of NPO’s may constitute anything from a SOM to a preprogrammed detector, depending upon the relative amount of preprogramming included. Two methods of preprogramming are: (1) Feeding a signal out of a permanent storage into some of the inputs of the network of NPO’s. This a priori copy need not be perfect, because the SOM will measure the angles Θᵢ anyhow. (2) Feedback, which, after all, is just a way of taking advantage of the storage inherent in any delay line. (We implicitly assume that any reasonable physical realization of an NPO will include a delay T between the x input and the ξ output which is not less than perhaps 10⁻¹ times the time constant of the internal feedback loop in the γ computation.)

Simulation of channels that possess a discrete component requires feedback path(s) to generate the required free products of the finitely generated groups. Then, such a SOM converges to a maximal subgroup of the group describing the symmetry of the signal that is a free product available to this SOM.

Because a single NPO with 1 ≤ n₀ ≤ K₀ is isomorphic (provides the same input to output mapping) to a suitable network of NPO’s with n₀ = 1, it suffices to study only networks of NPO’s with n₀ = 1.

[Figure 10] is largely self-explanatory. Item a is our schematic symbol for a single NPO with n₀ = 1. Items b, d (including larger feedback loops), and f are typical of artificial intelligence networks. Item c is employed to effect the level changing required in order to apply the three channels in cascade algorithm to the solution of one-dimensional coding problems. Observe that items c and e are the only configurations requiring the γ output. Item d may be used as a limiter by making T⁻¹ high compared to the highest frequency present in the signal. Observe that item e is the only application of NPO’s that requires either the ξ₂ or β outputs. Item f serves the purpose of handling higher power levels into and out of what effectively is a single (larger) NPO.