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Near-ideal predictors and causal filters for discrete time signals

Nikolai Dokuchaev

TL;DR

The paper addresses forecasting and high-/low-frequency filtering of deterministic discrete-time signals exhibiting spectrum degeneracy. It develops a framework that replaces noncausal ideal transfer functions with causal polynomials $\psi_d(z)=\sum_{k=0}^d \frac{a_k}{(1-z^{-1})^k}$, achieving near-ideal performance on spectra with gaps. A key theoretical contribution is a Stone–Weierstrass–driven density result guaranteeing arbitrary accuracy of the causal approximants on the relevant frequency sets, together with explicit time-domain representations as iterated causal sums $h_k$. The authors also propose practical implementation strategies, including finite-data fitting schemes for the auxiliary coefficients $\eta_k$ and constructive choices for $\psi_d$ in the $T=1$ case, with extensions to low-frequency and shifted spectra and guidance for numerical computation and robustness.

Abstract

The paper presents linear predictors and causal filters for discrete time signals featuring some different kinds of spectrum degeneracy. These predictors and filters are based on approximation of ideal non-causal transfer functions by causal transfer functions represented by polynomials of Z-transform of the unit step signal.

Near-ideal predictors and causal filters for discrete time signals

TL;DR

The paper addresses forecasting and high-/low-frequency filtering of deterministic discrete-time signals exhibiting spectrum degeneracy. It develops a framework that replaces noncausal ideal transfer functions with causal polynomials , achieving near-ideal performance on spectra with gaps. A key theoretical contribution is a Stone–Weierstrass–driven density result guaranteeing arbitrary accuracy of the causal approximants on the relevant frequency sets, together with explicit time-domain representations as iterated causal sums . The authors also propose practical implementation strategies, including finite-data fitting schemes for the auxiliary coefficients and constructive choices for in the case, with extensions to low-frequency and shifted spectra and guidance for numerical computation and robustness.

Abstract

The paper presents linear predictors and causal filters for discrete time signals featuring some different kinds of spectrum degeneracy. These predictors and filters are based on approximation of ideal non-causal transfer functions by causal transfer functions represented by polynomials of Z-transform of the unit step signal.
Paper Structure (9 sections, 5 theorems, 45 equations)

This paper contains 9 sections, 5 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Problem setting
  3. The main result
  4. Representation of operators $p_d(\cdot)$ in the time domain
  5. On numerical implementation of Theorems \ref{['ThP']}-\ref{['ThF']}
  6. A possible choice of $\psi_d$ for predictors in Theorem \ref{['ThP']} setting
  7. Low frequency and other signals
  8. Proofs
  9. Concluding remarks

Key Result

Lemma 1

For $\bar{\Omega}\in (0,\pi)$, let the function $\zeta:[-\pi,\pi]\to {\bf C}$ be defined either as $\zeta(\omega)=e^{i\omega T}$ or as $\zeta(\omega)={\mathbb{I}}_{|\omega|\ge \bar{\Omega}}$. Then, for any $\varepsilon>0$, there exists a integer $d=d(\varepsilon,T)>0$ and $\psi_d\in\Psi_d^{\bf R}$ s

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Lemma 2
  • Corollary 1