Signal Processing with Pulse Trains: An Algebraic Approach- Part II

TL;DR

This paper introduces an algebraic framework for signal processing using pulse trains generated by the integrate and fire converter, establishing a field structure that enables linear operations directly in the pulse domain.

Contribution

It defines multiplication of IFC pulse trains based on time domain operations and proves it forms a field, facilitating linear signal processing in the pulse domain.

Findings

Pulse domain multiplication forms an Abelian group.

Pulse multiplication corresponds to pointwise multiplication of signals.

The algebraic structure supports linear signal processing in pulse trains.

Abstract

The integrate and fire converter (IFC) enables an alternative to digital signal processing. IFC converts analog signal voltages into time between pulses and it is possible to reconstruct the analog signal from the IFC pulses with an error as small as required. In this paper, we present the definition of multiplication in pulse trains created by the IFC based on time domain operations and prove that it constitutes an Abelian group in the space of IFC pulse trains. We also show that pulse domain multiplication corresponds to pointwise multiplication of analog signals. It is further proved that pulse domain multiplication is distributive over pulse domain addition and hence it forms a field in the space of IFC pulse trains, which is an important property for linear signal processing.