Constructing Dampened LTI Systems Generating Polynomial Bases

TL;DR

This paper introduces a method to construct dampened LTI systems that generate polynomial bases, including Legendre polynomials, by using a delay re-encoder to approximate windowed impulse responses, applicable to various polynomial bases.

Contribution

It provides an alternative derivation of the LDN system and a general technique for creating dampened LTI systems for polynomial bases without closed-form solutions.

Findings

Derived an LTI system for Legendre polynomials

Introduced a delay re-encoder for damping

Applicable to arbitrary polynomial bases

Abstract

We present an alternative derivation of the LTI system underlying the Legendre Delay Network (LDN). To this end, we first construct an LTI system that generates the Legendre polynomials. We then dampen the system by approximating a windowed impulse response, using what we call a "delay re-encoder". The resulting LTI system is equivalent to the LDN system. This technique can be applied to arbitrary polynomial bases, although there typically is no closed-form equation that describes the state-transition matrix.