Carrying and Identifying Algebraic Structures in Analog Waveforms

This paper advances our Structural Communication Paradigm by bringing it closer to practice.

Our previous paper argued that transmitting and receiving Structures, together with their Modulations and Distortions, is the foundation of intelligent communication: they carry semantics, meaning, and perceptual objects. We also explained that Structures are represented by Relations, and thus that the mechanism of transmitting a Structure rests on deriving Relations from a given Structure on the sender side, transmitting those Relations through a channel, and identifying the Structure from the possibly altered Relations on the receiver side.

The mathematical language we used to anchor the framework was that of Algebraic Structures and the Equality Relations in their signature. While such a formulation already lends itself to certain practical applications, the question we ask here is different: how can the framework, and its underlying mental model, be embedded in the realm of continuous analog signals?

The new paper addresses this question. We show that one can construct systems of differential equations whose solution trajectory encodes a desired algebraic structure. We consider a transmission scenario in which these solutions determine the waveform of the transmitted signal, and we show how a receiver, observing only the altered wave, can recover both the structure and the channel distortions. We see this step as an enabler for applications such as wireless communication, scanning, sensing, diagnostics, bio-signal recognition, and many others.

The research and the paper are currently under development, but the approximate content will include:

• The transition from algebraic groups to differential equations

• How differential equations can encode the relations of algebraic structures

• How a wave can carry a structure

• Pilot-free identification of algebraic structures and distortions carried in a wave

• Possible extensions and applications

If the first paper covered roughly 70% of TRL-1 and 30% of TRL-2, the new paper is expected to cover 70% of TRL-2 and 30% of TRL-3.

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