Constructing Dampened LTI Systems Generating Polynomial Bases
This work provides an incremental derivation method for LDN systems, potentially aiding researchers in signal processing or neural networks.
The authors tackled the problem of deriving the linear time-invariant (LTI) system for the Legendre Delay Network (LDN) by constructing an LTI system that generates Legendre polynomials and dampening it with a 'delay re-encoder' to approximate a windowed impulse response, resulting in an equivalent system to LDN.
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.