Constructing Dampened LTI Systems Generating Polynomial Bases

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. The same technique can be applied to arbitrary polynomial bases, although there typically is no set of closed form equations that can be used to construct the corresponding LTI systems.