Signal Prediction by Derivative Samples from the Past via Perfect Reconstruction

TL;DR

This paper develops a method for signal prediction using derivative samples and perfect reconstruction in shift-invariant spaces, reducing sampling rates and enabling accurate signal recovery from limited past data.

Contribution

It introduces a new sampling formula based on nonuniform derivative samples, establishes conditions for complete interpolation, and proposes an efficient prediction algorithm validated with practical examples.

Findings

Derived a sampling formula for derivative samples in shift-invariant spaces.

Established necessary and sufficient conditions for complete interpolating sequences.

Validated the prediction method with cubic splines and Daubechies scaling functions.

Abstract

This paper investigates signal prediction through the perfect reconstruction of signals from shift-invariant spaces using nonuniform samples of both the signal and its derivatives. The key advantage of derivative sampling is its ability to reduce the sampling rate. We derive a sampling formula based on periodic nonuniform sampling (PNS) sets with derivatives in a shift-invariant space. We establish the necessary and sufficient conditions for such a set to form a complete interpolating sequence (CIS) of order $r-1$. This framework is then used to develop an efficient approximation scheme in a shift-invariant space generated by a compactly supported function. Building on this, we propose a prediction algorithm that reconstructs a signal from a finite number of past derivative samples using the derived perfect reconstruction formula. Finally, we validate our theoretical results through practical examples involving cubic splines and the Daubechies scaling function of order 3.