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On Spectral Approach to the Synthesis of Shaping Filters

Konstantin A. Rybakov

TL;DR

The paper develops a spectral-form framework for synthesizing shaping filters that produce a Gaussian random process with a prescribed rational PSD, establishing a direct transfer-function relation and enabling exact mean-square error control in continuous time. By representing the impulse response as a two-dimensional spectral expansion with basis functions, it introduces a two-dimensional non-stationary transfer function $W$ that can be truncated to a finite size for practical use, while preserving accuracy via Parseval-based error analysis. It unifies descriptions across differential equations, impulse responses, and spectral forms, and provides concrete constructions for aperiodic, oscillatory, and Dryden-type turbulence blocks, including methods to whiten and handle non-stationary systems. The framework yields implementable algorithms with quantified convergence, offering a powerful tool for statistical modeling and turbulence synthesis in engineering contexts.

Abstract

This paper describes various approaches to modeling a random process with a given rational power spectral density. The main attention is paid to the spectral form of mathematical description, which allows one to obtain a relation for the shaping filter using a transfer function without any additional calculations. The paper provides all necessary relations for the implementation of the shaping filter based on the spectral form of mathematical description.

On Spectral Approach to the Synthesis of Shaping Filters

TL;DR

The paper develops a spectral-form framework for synthesizing shaping filters that produce a Gaussian random process with a prescribed rational PSD, establishing a direct transfer-function relation and enabling exact mean-square error control in continuous time. By representing the impulse response as a two-dimensional spectral expansion with basis functions, it introduces a two-dimensional non-stationary transfer function that can be truncated to a finite size for practical use, while preserving accuracy via Parseval-based error analysis. It unifies descriptions across differential equations, impulse responses, and spectral forms, and provides concrete constructions for aperiodic, oscillatory, and Dryden-type turbulence blocks, including methods to whiten and handle non-stationary systems. The framework yields implementable algorithms with quantified convergence, offering a powerful tool for statistical modeling and turbulence synthesis in engineering contexts.

Abstract

This paper describes various approaches to modeling a random process with a given rational power spectral density. The main attention is paid to the spectral form of mathematical description, which allows one to obtain a relation for the shaping filter using a transfer function without any additional calculations. The paper provides all necessary relations for the implementation of the shaping filter based on the spectral form of mathematical description.
Paper Structure (6 sections, 77 equations, 4 figures, 2 tables)

This paper contains 6 sections, 77 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Description of the Shaping Filter by Differential Equations
  3. Description of the Shaping Filter Based on the Impulse Response Function
  4. Description of the Shaping Filter Based on the Spectral Form
  5. Approximate Modeling of Random Processes and Error Analysis
  6. Conclusions

Figures (4)

  • Figure 1: Various forms of mathematical description
  • Figure 2: The shaping filter for the Dryden turbulence model
  • Figure 3: Trajectory of the random process $x_1(t)$ with power spectral density $S_1(\omega) = |H_1(\mathrm{i} \omega)|^2$
  • Figure 6: Trajectory of the random process $x_4(t)$ with power spectral density $S_4(\omega) = |H_4(\mathrm{i} \omega)|^2$