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On the Gaussian Limit of the Output of IIR Filters

Yashaswini Murthy, Bassam Bamieh, R. Srikant

TL;DR

This work rigorously analyzes when the output of a stable LTI system driven by non-Gaussian input is approximately Gaussian. Using Stein's method and Wasserstein-1 distance, it derives explicit bounds that depend on the system’s impulse response and input dependence, showing that Gaussianity emerges with rate $O(1/\sqrt{t})$ as the dominant pole approaches the edge of stability under independence, positive correlation with a real positive pole, or correlation decay. The paper also provides counterexamples illustrating that Gaussian limits can fail without these conditions, thereby clarifying the limits of the stochastic describing-function intuition. These results provide a rigorous foundation for the commonly observed Gaussianity of outputs from low-pass LTI systems and suggest directions for extending the theory to broader noise models.

Abstract

We study the asymptotic distribution of the output of a stable Linear Time-Invariant (LTI) system driven by a non-Gaussian stochastic input. Motivated by longstanding heuristics in the stochastic describing function method, we rigorously characterize when the output process becomes approximately Gaussian, even when the input is not. Using the Wasserstein-1 distance as a quantitative measure of non-Gaussianity, we derive upper bounds on the distance between the appropriately scaled output and a standard normal distribution. These bounds are obtained via Stein's method and depend explicitly on the system's impulse response and the dependence structure of the input process. We show that when the dominant pole of the system approaches the edge of stability and the input satisfies one of the following conditions: (i) independence, (ii) positive correlation with a real and positive dominant pole, or (iii) sufficient correlation decay, the output converges to a standard normal distribution at rate $O(1/\sqrt{t})$. We also present counterexamples where convergence fails, thereby motivating the stated assumptions. Our results provide a rigorous foundation for the widespread observation that outputs of low-pass LTI systems tend to be approximately Gaussian.

On the Gaussian Limit of the Output of IIR Filters

TL;DR

This work rigorously analyzes when the output of a stable LTI system driven by non-Gaussian input is approximately Gaussian. Using Stein's method and Wasserstein-1 distance, it derives explicit bounds that depend on the system’s impulse response and input dependence, showing that Gaussianity emerges with rate as the dominant pole approaches the edge of stability under independence, positive correlation with a real positive pole, or correlation decay. The paper also provides counterexamples illustrating that Gaussian limits can fail without these conditions, thereby clarifying the limits of the stochastic describing-function intuition. These results provide a rigorous foundation for the commonly observed Gaussianity of outputs from low-pass LTI systems and suggest directions for extending the theory to broader noise models.

Abstract

We study the asymptotic distribution of the output of a stable Linear Time-Invariant (LTI) system driven by a non-Gaussian stochastic input. Motivated by longstanding heuristics in the stochastic describing function method, we rigorously characterize when the output process becomes approximately Gaussian, even when the input is not. Using the Wasserstein-1 distance as a quantitative measure of non-Gaussianity, we derive upper bounds on the distance between the appropriately scaled output and a standard normal distribution. These bounds are obtained via Stein's method and depend explicitly on the system's impulse response and the dependence structure of the input process. We show that when the dominant pole of the system approaches the edge of stability and the input satisfies one of the following conditions: (i) independence, (ii) positive correlation with a real and positive dominant pole, or (iii) sufficient correlation decay, the output converges to a standard normal distribution at rate . We also present counterexamples where convergence fails, thereby motivating the stated assumptions. Our results provide a rigorous foundation for the widespread observation that outputs of low-pass LTI systems tend to be approximately Gaussian.

Paper Structure

This paper contains 15 sections, 2 theorems, 83 equations, 1 figure.

Table of Contents

  1. INTRODUCTION
  2. MODEL AND PRELIMINARIES
  3. MAIN RESULTS
  4. PROOFS
  5. $\{u_i\}$ is an independent sequence
  6. $\{u_i\}$ is positively correlated
  7. Counterexamples
  8. When $u_i$ are postiviely correlated and the dominant pole is in the left half plane
  9. When the dominant pole is purely imaginary and noise is positively correlated
  10. CONCLUSIONS
  11. Acknowledgments
  12. Structural Assumptions on the Noise Sequence in \ref{['eq:finalsystem_mb']}
  13. The System Response $G$
  14. Positively correlated $\{u_i\}$ with a real positive dominant root $r_1$
  15. General Correlation Decay

Key Result

Proposition 1

Consider the system defined in eq:finalsystem_mb. Suppose the input sequence $\{u_i\}$ satisfies the following conditions: Then, for all $t \geq 1$, the Wasserstein-1 distance between the normalized output $y_t / \sigma_t$ and the standard normal random variable $Z \sim \mathcal{N}(0,1)$ satisfies: where $\sigma_t^2 = \operatorname{Var}(y_t)$ is defined in eq:variance.

Figures (1)

  • Figure 1: A large class of nonlinear time-invariant dynamic systems (from $d$ to $z$) can be modeled as a Linear Time-Invariant (LTI) system in feedback with a memoryless nonlinearity $N(.)$.

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Remark 2