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Frequency truncated discrete-time system norm

Hanumant Singh Shekhawat

TL;DR

The paper tackles the problem of computing the frequency-truncated norm $||\mathcal{G}||^2_{[\theta_1,\theta_2]}$ for discrete-time systems, a quantity arising in multirate DSP and model reduction. It develops stable-case results using the discrete-time Lyapunov solution and an anti-derivative involving $\log(I - e^{-j\theta}A)$, then extends to the general descriptor form $K(z)=(zE - A)^{-1}$ with regular pencils, handling poles on, inside, or outside the unit circle via tangent-half-angle substitutions and carefully defined anti-derivatives $f_d$ and $f_i$. The paper provides numerically viable computation schemes, including limit-free forms using $\psi_1$, matrix exponentials, and reduced-state expressions, plus a brief continuous-time extension. Collectively, these results enable robust, in-band norm evaluation in practical DSP pipelines and model-reduction workflows.

Abstract

Multirate digital signal processing and model reduction applications require computation of the frequency truncated norm of a discrete-time system. This paper explains how to compute the frequency truncated norm of a discrete-time system. To this end, a much-generalized problem of integrating a transfer function of a discrete-time system given in the descriptor form over an interval of limited frequencies is also discussed along with its computation.

Frequency truncated discrete-time system norm

TL;DR

The paper tackles the problem of computing the frequency-truncated norm for discrete-time systems, a quantity arising in multirate DSP and model reduction. It develops stable-case results using the discrete-time Lyapunov solution and an anti-derivative involving , then extends to the general descriptor form with regular pencils, handling poles on, inside, or outside the unit circle via tangent-half-angle substitutions and carefully defined anti-derivatives and . The paper provides numerically viable computation schemes, including limit-free forms using , matrix exponentials, and reduced-state expressions, plus a brief continuous-time extension. Collectively, these results enable robust, in-band norm evaluation in practical DSP pipelines and model-reduction workflows.

Abstract

Multirate digital signal processing and model reduction applications require computation of the frequency truncated norm of a discrete-time system. This paper explains how to compute the frequency truncated norm of a discrete-time system. To this end, a much-generalized problem of integrating a transfer function of a discrete-time system given in the descriptor form over an interval of limited frequencies is also discussed along with its computation.

Paper Structure

This paper contains 7 sections, 9 theorems, 20 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Stable Case
  3. General case
  4. Computation
  5. A brief note on the continuous time system
  6. Conclusions
  7. Appendix

Key Result

Theorem 2.1

Let discrete-time system $\mathcal{G}$ be stable and strictly proper and $G(\mathrm{e}^{\mathrm{j}\theta})=C(\mathrm{e}^{\mathrm{j}\theta} I-A)^{-1}B$ with A, B, C real matrices. Then, an anti-derivative $\int G^\sim(\mathrm{e}^{\mathrm{j}\theta}) G(\mathrm{e}^{\mathrm{j}\theta}) \;d\theta$ equals where $P$ is the unique solution of dlyap and $\log$ denotes the principal logarithm and $\theta\in

Figures (1)

  • Figure 1: A setup for multi-rate discrete signal processing

Theorems & Definitions (9)

  • Theorem 2.1
  • Theorem 3.1
  • Corollary 3.2
  • Theorem 3.3
  • Theorem 4.1
  • Corollary 4.2
  • Theorem 5.1
  • Theorem A.1
  • Theorem A.2