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The envelope of a complex Gaussian random variable

Sattwik Ghosal, Ranjan Maitra

TL;DR

This work provides explicit, tractable representations for the envelope distribution of a general 2D complex Gaussian vector, encompassing the PDF, CDF, MGF, and raw moments for the second-order generalized Beckmann family. It introduces and analyzes the Identical Quadrature Components (IQC) model and shows how Beckmann, Rayleigh, Rice, Hoyt, and related envelopes arise as special cases, with clear limiting distributions including Gaussian and noncentral chi-square forms. The authors validate exact formulae against Monte Carlo simulations, demonstrate computational advantages, and apply the framework to MRI magnitude data and wind-speed datasets, finding IQC often offers a better fit than Rice. Taken together, the results advance precise envelope modeling in signal processing and related applications, while highlighting identifiability issues and practical estimation strategies.

Abstract

The envelope of an elliptical Gaussian complex vector, or equivalently, the amplitude or norm of a bivariate normal random vector has application in many weather and signal processing contexts. We explicitly characterize its distribution in the general case through its probability density, cumulative distribution and moment generating function. Moments and limiting distributions are also derived. These derivations are exploited to also characterize the special cases where the bivariate Gaussian mean vector and covariance matrix have a simpler structure, providing new additional insights in many cases. Simulations illustrate the benefits of using our formulae over Monte Carlo methods. We also use our derivations to get a better initial characterization of the distribution of the observed values in structural Magnetic Resonance Imaging datasets, and of wind speed.

The envelope of a complex Gaussian random variable

TL;DR

This work provides explicit, tractable representations for the envelope distribution of a general 2D complex Gaussian vector, encompassing the PDF, CDF, MGF, and raw moments for the second-order generalized Beckmann family. It introduces and analyzes the Identical Quadrature Components (IQC) model and shows how Beckmann, Rayleigh, Rice, Hoyt, and related envelopes arise as special cases, with clear limiting distributions including Gaussian and noncentral chi-square forms. The authors validate exact formulae against Monte Carlo simulations, demonstrate computational advantages, and apply the framework to MRI magnitude data and wind-speed datasets, finding IQC often offers a better fit than Rice. Taken together, the results advance precise envelope modeling in signal processing and related applications, while highlighting identifiability issues and practical estimation strategies.

Abstract

The envelope of an elliptical Gaussian complex vector, or equivalently, the amplitude or norm of a bivariate normal random vector has application in many weather and signal processing contexts. We explicitly characterize its distribution in the general case through its probability density, cumulative distribution and moment generating function. Moments and limiting distributions are also derived. These derivations are exploited to also characterize the special cases where the bivariate Gaussian mean vector and covariance matrix have a simpler structure, providing new additional insights in many cases. Simulations illustrate the benefits of using our formulae over Monte Carlo methods. We also use our derivations to get a better initial characterization of the distribution of the observed values in structural Magnetic Resonance Imaging datasets, and of wind speed.
Paper Structure (38 sections, 8 theorems, 54 equations, 4 figures, 1 table)

This paper contains 38 sections, 8 theorems, 54 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main Results
  3. Background and Preliminaries
  4. Additional characterization of the second order Beckman distribution
  5. The cumulative distribution function
  6. The Moment Generating Function
  7. Moments
  8. Identifiability of parameters
  9. Limiting distributions
  10. Numerical aspects
  11. Application to Special Distributions
  12. The Rayleigh distribution
  13. The Rice distribution
  14. Rice distribution asymptotics
  15. The Hoyt distribution
  16. ...and 23 more sections

Key Result

Proposition 1

Under the framework of Theorem theo:2d.CDF, the CDF of $R$ is equivalently, for $u > 0$ and zero everywhere else. Here, ${}_1\mathbb{F}_1(\cdot,\cdot,\cdot)$ is the confluent hypergeometric function, or Kummer's function, of the first kind kummer1837abramowitzandstegun64.

Figures (4)

  • Figure 1: Relative frequency histograms of simulated values of $R$ and calculated densities of the IQC distribution for $\mu_1=15,\mu_2=10,\rho=0.25$ and four different values of $\sigma$. Densities calculated using \ref{['eq:2d.PDF']} are in solid lines, while densities calculated using Theorem \ref{['conjecture']} are in dashed lines.
  • Figure 2: The $q$-values, after controlling for false discoveries benjaminiandhochberg95, obtained upon fitting a Anderson-Darling test for normality to realizations from the second order IQC distribution with $\mu_1,\mu_2\in [0,250]$, $\sigma=1$ and $\rho=0.25,0.5,0.75$.
  • Figure 3: Relative frequency histograms of simulated values of $R$ and calculated densities of the IQC distribution for $\mu_1=1,\mu_2=0.75,\sigma=1$ and four different values of $\rho$. The limiting density calculated using \ref{['eq:rho.infty']} is displayed by the solid line.
  • Figure 4: (a) Magnitude 2D MR images of the physical phantom scanned using a spin-echo imaging sequence scanned with TE = 30 milliseconds and TR = 1. The highlighted square region is from where pixels were sampled to obtain MOM parameter estimates. (b) Distribution of the M0M-obtained $\hat{\rho}$ for each of the 18 design parameter settings.

Theorems & Definitions (22)

  • Remark 1
  • Claim 1
  • Proposition 1
  • Remark 2
  • Claim 2
  • Remark 3
  • Corollary 1
  • Remark 4
  • Claim 3
  • Claim 4
  • ...and 12 more