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A Statistical Framework and Analysis for Perfect Radar Pulse Compression

Neil K. Chada, Petteri Piiroinen, Lassi Roininen

TL;DR

The paper addresses fundamental limits of perfect radar pulse compression by casting radar coding as a statistical experiment comparison problem and modeling the target with Itô measures whose structure is governed by a variance function $|\sigma|^2$. It proposes a Bayesian hierarchical framework, derives finite-dimensional conjugate updates via inverse-Wishart priors for the covariance, and extends to infinite dimensions where the posterior for $|\sigma|^2$ is non-Gaussian and characterized by generalized inverse-Wishart limits. A key result shows posterior variances for two codes with equal Fourier-modulus responses are identical, linking code design to posterior uncertainty. This work bridges radar coding and rigorous Bayesian analysis, enabling uncertainty quantification and suggesting avenues for non-Gaussian priors and distance-based comparisons for design.

Abstract

Perfect radar pulse compression coding is a potential emerging field which aims at providing rigorous analysis and fundamental limit radar experiments. It is based on finding non-trivial pulse codes, which we can make statistically equivalent, to the radar experiments carried out with elementary pulses of some shape. A common engineering-based radar experiment design, regarding pulse-compression, often omits the rigorous theory and mathematical limitations. In this work our aim is to develop a mathematical theory which coincides with understanding the radar experiment in terms of the theory of comparison of statistical experiments. We review and generalize some properties of the Itô measure. We estimate the unknown i.e. the structure function in the context of Bayesian statistical inverse problems. We study the posterior for generalized $d$-dimensional inverse problems, where we consider both real-valued and complex-valued inputs for posteriori analysis. Finally this is then extended to the infinite dimensional setting, where our analysis suggests the underlying posterior is non-Gaussian.

A Statistical Framework and Analysis for Perfect Radar Pulse Compression

TL;DR

The paper addresses fundamental limits of perfect radar pulse compression by casting radar coding as a statistical experiment comparison problem and modeling the target with Itô measures whose structure is governed by a variance function . It proposes a Bayesian hierarchical framework, derives finite-dimensional conjugate updates via inverse-Wishart priors for the covariance, and extends to infinite dimensions where the posterior for is non-Gaussian and characterized by generalized inverse-Wishart limits. A key result shows posterior variances for two codes with equal Fourier-modulus responses are identical, linking code design to posterior uncertainty. This work bridges radar coding and rigorous Bayesian analysis, enabling uncertainty quantification and suggesting avenues for non-Gaussian priors and distance-based comparisons for design.

Abstract

Perfect radar pulse compression coding is a potential emerging field which aims at providing rigorous analysis and fundamental limit radar experiments. It is based on finding non-trivial pulse codes, which we can make statistically equivalent, to the radar experiments carried out with elementary pulses of some shape. A common engineering-based radar experiment design, regarding pulse-compression, often omits the rigorous theory and mathematical limitations. In this work our aim is to develop a mathematical theory which coincides with understanding the radar experiment in terms of the theory of comparison of statistical experiments. We review and generalize some properties of the Itô measure. We estimate the unknown i.e. the structure function in the context of Bayesian statistical inverse problems. We study the posterior for generalized -dimensional inverse problems, where we consider both real-valued and complex-valued inputs for posteriori analysis. Finally this is then extended to the infinite dimensional setting, where our analysis suggests the underlying posterior is non-Gaussian.
Paper Structure (11 sections, 14 theorems, 101 equations, 1 figure)

This paper contains 11 sections, 14 theorems, 101 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contributions
  3. Outline
  4. Radar coding
  5. Bayesian Posterior Analysis
  6. Conclusion & Discussion
  7. finite dimensional analysis
  8. Characteristic functions
  9. Densities for complex Gaussian vectors
  10. Infinite-dimensional analysis
  11. Connection to radar equation

Key Result

Theorem 3.2

Assume the priori distribution of $|\,\sigma\,|^2$ is interpretable as an affine transform of inverse Wishart distribution, then the posteriori distribution of $|\, \sigma \,|^2$ can be interpreted as a generalized limit of affine transforms of inverse Wishart distributions of the similar type, give

Figures (1)

  • Figure 1: Various density plots of the inverse Wishart distribution with varying degrees of freedom. Red plot is for 3 degrees of freedom. Blue plot is for 2 degrees of freedom. Grey plot is for 1 degree of freedom.

Theorems & Definitions (41)

  • Definition 2.1
  • Definition 2.2: Itô measure with constant variance
  • Remark 2.3
  • Definition 3.1: Inverse Wishart distribution
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Conjecture 4.1
  • ...and 31 more